The transmutation method and boundary-value problems for singular elliptic equations

V. V. Katrakhov ; S. M. Sitnik

Contemporary Mathematics. Fundamental Directions, Singular differential equations, Tome 64 (2018) no. 2, pp. 211-426.

Résumé

The main content of this book is composed from two doctoral theses: by V. V. Katrakhov (1989) and by S. M. Sitnik (2016). In our work, for the first time in the format of a monograph, we systematically expound the theory of transmutation operators and their applications to differential equations with singularities in coefficients, in particular, with Bessel operators. Along with detailed survey and bibliography on this theory, the book contains original results of the authors. Significant part of these results is published with detailed proofs for the first time. In the first chapter, we give historical background, necessary notation, definitions, and auxiliary facts. In the second chapter, we give the detailed theory of Sonin and Poisson transmutations. In the third chapter, we describe an important special class of the Buschman–Erdéelyi transmutations and their applications. In the fourth chapter, we consider new weighted boundary-value problems with Sonin and Poisson transmutations. In the fifth chapter, we consider applications of the Buschman–Erdélyi transmutations of special form to new boundary-value problems for elliptic equations with significant singularities of solutions. In the sixth chapter, we describe a universal compositional method for construction of transmutations and its applications. In the concluding seventh chapter, we consider applications of the theory of transmutations to differential equations with variable coefficients: namely, to the problem of construction of a new class of transmutations with sharp estimates of kernels for perturbed differential equations with the Bessel operator, and to special cases of the well-known Landis problem on exponential estimates of the rate of growth for solutions of the stationary Schrödinger equation. The book is concluded with a brief biographic essay about Valeriy V. Katrakhov, as well as detailed bibliography containing 648 references.

Bibliographie

[1] Z. L. Abzhandadze, V. F. Osipov, Fourier–Fresnel Transform and some Applications, S.-Peterb. Univ., SPb., 2000 (in Russian)

[2] M. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, Russian translation, Mir, Moscow, 1979 | MR

[3] M. Abramowitz, I. Stegun, Handbook Of Mathematical Functions, Russian translation, Nauka, Moscow, 1979

[4] S. Agmon, A. Douglis, L. Nirenberg, Estimates Near the Boundary for Solutions of Elliptic Partial Differential Equations, Russian translation, Inostr. lit., Moscow, 1962

[5] Z. S. Agranovich, V. A. Marchenko, Inverse Scattering Problem, KHGU, Khar'kov, 1960 (in Russian)

[6] M. S. Agranovich, Elliptic Pseudodifferential Operators, Part 1, 2, Moscow, 2003, 2004 (in Russian)

[7] N. V. Azbelev, V. P. Maksimov, L. F. Rakhmatullina, Elements of the Modern Functional-Differential Equations Theory. Methods and Applications, Inst. Komp. Issl., Moscow, 2002 (in Russian) | MR

[8] Sh. A. Alimov, “Fractional powers of elliptic operators and isomorphism of classes of differentiable functions”, Diff. uravn., 8:9 (1972), 1609–1626 (in Russian) | MR | Zbl

[9] E. A. Arshava, “Inversion of integral operators by the method of integral identities”, Nauch. vedom. Belgorod. gos. un-ta. Ser. Mat. Fiz., 17/2:13(68) (2009), 18–29 (in Russian)

[10] N. I. Akhiezer, “To the theory of paired integral equations”, Uch. zap. Khar'kov. gos. un-ta, 80, 1957, 5–21 (in Russian)

[11] N. I. Akhiezer, Lectures on Integral Transforms, Vishcha shkola. Khar'kov State Univ., Khar'kov, 1984 (in Russian) | MR

[12] V. V. Babikov, A Method of Phase Functions in Quantum Mechanics, Nauka, Moscow, 1976 (in Russian)

[13] I. I. Bavrin, V. L. Matrosov, O. E. Yaremko, Transmutation Operators for Boundary-Value Problems, Integral Representations and Dependence Recovering, Prometey, Moscow, 2015 (in Russian)

[14] V. G. Bagrov, B. A. Samsonov, “Darboux transform for the Shrödinger equation”, Fiz. elem. chastits i atom. yadra, 28:4 (1997), 951–1012 (in Russian) | MR

[15] A. N. Baydakov, “A priori estimates of Hölder norms of solutions to quasilinear $B$-elliptic equations”, Diff. uravn., 23:11 (1987), 1923–1930 (in Russian) | MR

[16] A. G. Baskakov, Harmonic Analysis of Linear Operators, VGU, Voronezh, 1987 (in Russian)

[17] H. Bateman, A. Erdélyi, Higher Transcendental Functions, Russian translation, v. 1, Nauka, Moscow, 1966 | MR

[18] H. Bateman, A. Erdélyi, Higher Transcendental Functions, Russian translation, v. 2, Nauka, Moscow, 1966 | MR

[19] H. Bateman, A. Erdélyi, Higher Transcendental Functions, Russian translation, v. 3, Nauka, Moscow, 1967 | MR

[20] E. Beckenbach, R. Bellman, Inequalities, Russian translation, Mir, Moscow, 1965 | MR

[21] S. Bergman, Integral Operators in the Theory of Linear Partial Differential Equations, Russian translation, Mir, Moscow, 1964

[22] Yu. M. Berezanskiy, Expansions in Eigenfunctions of Self-Adjoint Operators, Naukova Dumka, Kiev, 1965 (in Russian)

[23] L. M. Berkovich, Factorization and Transformations of Differential Equations. Methods and Applications, RKHD, Moscow, 2002 (in Russian)

[24] L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Russian translation, Inostr. lit., Moscow, 1961

[25] O. V. Besov, V. P. Il'in, S. M. Nikol'skiy, Integral Representations of Functions and Embedding Theorems, Nauka, Moscow, 1975 (in Russian)

[26] A. V. Bitsadze, Equations of Mixed Type, Izd. AN SSSR, Moscow, 1959 (in Russian)

[27] A. V. Bitsadze, Some Classes of Partial Differential Equations, Nauka, Moscow, 1981 (in Russian)

[28] A. V. Bitsadze, V. I. Pashkovskiy, “To the theory of Maxwell–Einstein equations”, Dokl. AN SSSR, 216:2 (1974), 249–250 (in Russian) | MR | Zbl

[29] A. V. Bitsadze, V. I. Pashkovskiy, “On some classes of Maxwell–Einstein equation”, Tr. MIAN, 134, 1975, 26–30 (in Russian) | MR | Zbl

[30] A. Sh. Blokh, “On defining of a differential operator by its spectral matrix-function”, Dokl. AN SSSR, 92:2 (1953), 209–212 (in Russian)

[31] A. V. Borovskikh, “A formula of wave extending for one-dimensional nonhomogeneous medium”, Diff. uravn., 38:6 (2002), 758–767 (in Russian) | MR | Zbl

[32] A. V. Borovskikh, “A method of extending waves”, Tr. sem. im. I. G. Petrovskogo, 24, 2004, 3–43 (in Russian)

[33] B. Boyarskiy, “Generalized solutions to a first order system of differential equations of elliptic type with discontinuous coefficients”, Mat. sb., 43(85):4 (1957), 451–503 (in Russian) | MR | Zbl

[34] Yu. A. Brychkov, A. P. Prudnikov, Integral Transforms of Generalized Functions, Nauka, Moscow, 1977 (in Russian) | MR

[35] N. Bourbaki, Functions of a Real Variable, Russian translation, Nauka, Moscow, 1965

[36] V. I. Burenkov, Functional Spaces, RUDN, Moscow, 1989 (in Russian)

[37] V. I. Burenkov, M. L. Gol'dman, Methodical Materials to the Course of Functional Spaces, RUDN, Moscow, 1989 (in Russian)

[38] S. A. Buterin, “On a recovery of a convolution perturbation of Sturm–Lioville operator by spectrum”, Diff. uravn., 46:1 (2010), 146–149 (in Russian) | MR | Zbl

[39] Yu. N. Valitskiy, “On a transmutation operator for integral-differential operators of Volterra type”, Mathematical Physics, Naukova Dumka, Kiev, 1965, 23–36 (in Russian) | MR

[40] E. M. Varfolomeev, L. E. Rossovskiy, Functional-Differential Equations and Their Applications to the Study of Neuron Nets and Information Transmission by Nonlinear Laser Systems with Feedback, Textbook, RUDN, Moscow, 2008 (in Russian)

[41] G. N. Watson, A Treatise on the Theory of Bessel Functions, Russian translation, v. 1, Inostr. lit., Moscow, 1949

[42] A. A. Vasharin, P. I. Lizorkin, “Some boundary-value problems for elliptic equations with strong degeneration on the boundary”, Dokl. AN SSSR, 137:5 (1961), 1015–1018 (in Russian) | MR | Zbl

[43] I. N. Vekua, “On solutions to the equation $\Delta u+\lambda^2u$”, Soobshch. AN Gruz. SSR, 3:4 (1942), 307–314 (in Russian) | MR | Zbl

[44] I. N. Vekua, “Inversion of one integral transform and some of its applications”, Soobshch. AN Gruz. SSR, 6:3 (1945), 177–183 (in Russian) | MR | Zbl

[45] I. N. Vekua, “On a generalization of the Poisson integral on the plane”, Dokl. AN SSSR, 56:2 (1947), 229–231 (in Russian) | Zbl

[46] I. N. Vekua, New Methods for Solving Elliptic Equations, GITTL, Moscow–Leningrad, 1948 (in Russian) | MR

[47] I. N. Vekua, Generalized Analytic Functions, Nauka, Moscow, 1988 (in Russian)

[48] N. A. Virchenko, V. Gaydey, Classical and Generalized Multiparameter Functions, Kiev, 2008 (in Ukrainian)

[49] N. A. Virchenko, V. Ya. Rybak, Foundations of Fractional Integrodifferentiation, Kiev, 2007 (in Ukrainian)

[50] M. I. Vishik, V. V. Grushin, “Boundary-value problems for elliptic equations degenerated at the boundary of domain”, Mat. sb., 80:4 (1969), 455–491 (in Russian) | MR | Zbl

[51] V. S. Vladimirov, Generalized Functions in Mathematical Physics, Nauka, Moscow, 1979 (in Russian)

[52] V. Ya. Volk, “On inversion formulas for a differential equation with a singularity at $x=0$”, Usp. mat. nauk, 8:4 (1953), 141–151 (in Russian) | MR | Zbl

[53] I. K. Volkov, A. N. Kanatnikov, Integral Transforms and Operational Calculus, MGTU im. N. E. Baumana, Moscow, 2002 (in Russian)

[54] V. F. Volkodavov, V. N. Zakharov, Tables of Riemann and Riemann–Hadamard Functions for Some Differential Equations in $n$-Dimensional Euclidean Spaces, Samara, 1994 (in Russian)

[55] V. F. Volkodavov, M. E. Lerner, N. Ya. Nikolaev, V. A. Nosov, Tables of Some Riemann Functions, Integrals and Series, Kuybyshev. Gos. Ped. Inst., Kuybyshev, 1982 (in Russian)

[56] V. F. Volkodavov, N. Ya. Nikolaev, Boundary–Value Problems for the Euler–Poisson–Darboux equation, Izd. Kuybyshev. Gos. Ped. Inst., Kuybyshev, 1984 (in Russian) | MR

[57] V. F. Volkodavov, N. Ya. Nikolaev, Integral Equations of Volterra of the First Kind with Some Kernel Special Functions and Applications, Samara Univ., Samara, 1992 (in Russian)

[58] V. N. Vragov, Boundary-Value Problems for Nonclassical Equations of Mathematical Physics, NGU, Novosibirsk, 1983 (in Russian)

[59] F. R. Gantmacher, Matrix Theory, Nauka, Moscow, 1988 (in Russian) | MR

[60] I. M. Gel'fand, M. I. Graev, N. Ya. Vilenkin, Integral Geometry and Connected Problems of Representation Theory, Gos. Fiz.-Mat. Lit., Moscow, 1962 (in Russian) | MR

[61] A. V. Glushak, “On perturbation of abstract Euler–Poisson–Darboux equation”, Mat. zametki, 60:3 (1996), 363–369 (in Russian) | DOI | MR | Zbl

[62] A. V. Glushak, “On stabilization of solution of Dirichlet problem for an elliptic equation in the Banach space”, Diff. uravn., 33:4 (1997), 510–514 (in Russian) | MR | Zbl

[63] A. V. Glushak, “Bessel Operator Function”, Dokl. RAN, 352:5 (1997), 587–589 (in Russian) | MR | Zbl

[64] A. V. Glushak, “Operator Bessel function and connected semigroups and modified Hilbert transform”, Diff. uravn., 35:1 (1999), 128–130 (in Russian) | MR | Zbl

[65] A. V. Glushak, “Regular and singular perturbations for an abstract Euler–Poisson–Darboux equations”, Mat. zametki, 66:3 (1999), 364–371 (in Russian) | DOI | MR | Zbl

[66] A. V. Glushak, “Legendre operator function”, Izv. RAN. Ser. Mat., 65:6 (2001), 3–14 (in Russian) | DOI | MR | Zbl

[67] A. V. Glushak, “A problem of Cauchy-type for abstract equation with fractional derivatives”, Mat. zametki, 77:1 (2005), 28–41 (in Russian) | DOI | MR | Zbl

[68] A. V. Glushak, “On connection of integrated cosine operator function with Bessel operator function”, Diff. uravn., 42:5 (2006), 583–589 (in Russian) | MR | Zbl

[69] A. V. Glushak, “Initial problem for weakly loaded Euler–Poisson–Darboux equation”, Proc. “Actual problems in the theory of partial differential equations”, MGU, Moscow, 2016, 101 (in Russian)

[70] A. V. Glushak, “Nonlocal problem for abstract Euler–Poisson–Darboux equation”, Izv. vuzov. Ser. Mat., 2016, no. 6, 27–35 (in Russian) | MR | Zbl

[71] A. V. Glushak, “Abstract problem for the Bessel–Struve equation”, Diff. uravn., 53:7 (2017), 891–905 (in Russian) | DOI | MR | Zbl

[72] A. V. Glushak, V. I. Kononenko, S. D. Shmulevich, “On a singular abstract Cauchy problem”, Izv. vuzov. Ser. Mat., 1986, no. 6, 55–56 (in Russian) | MR | Zbl

[73] A. V. Glushak, O. A. Pokruchin, “Criterion of solvability for Cauchy problem for abstract Euler–Poisson–Darboux equations”, Diff. uravn., 52:1 (2016), 41–59 (in Russian) | DOI | MR | Zbl

[74] A. V. Glushak, T. G. Romanchenko, “Connection formulas for solutions of abstract singular differential equations”, Nauch. vedom. Belgorod. gos. un-ta. Ser. Mat. Fiz., 42:6 (2016), 36–39 (in Russian)

[75] V. P. Glushko, Linear Degenerate Differential Equations, VGU, Voronezh, 1972 (in Russian)

[76] L. S. Gnoenskiy, G. A. Kamenskiy, L. E. El'sgol'ts, Mathematical Foundations of the Theory of Controllable Systems, Nauka, Moscow, 1969 (in Russian)

[77] G. A. Grinberg, Selected Problems of Mathematical Theory of Electric and Magnetic Phenomena, AN SSSR, Moscow, 1948 (in Russian)

[78] V. S. Guliev, Integral Operators, Functional Spaces and Approximation Problems for the Heisenberg Group, ELM, Baku, 1996 (in Russian)

[79] V. S. Guliev, Functional Spaces, Integral Operators and Two-Weighted Estimates on Homogeneous Groups. Some Applications, Chashyogly, Baku, 1999 (in Russian)

[80] M. I. Gurevich, Jet Theory for an Ideal Liquid, Nauka, Moscow, 1979 (in Russian)

[81] I. M. Guseynov, “On one transmutation operator”, Mat. zametki, 62:2 (1997), 206–215 (in Russian) | DOI | MR | Zbl

[82] I. M. Guseynov, A. A. Nabiev, R. T. Pashaev, “Transmutations and asymptotic formulas for eigenvalues of a polynomial pencil of Sturm–Liouville operators”, Sib. mat. zh., 41:3 (2000), 554–566 (in Russian) | MR | Zbl

[83] G. V. Dzhayani, Solution of Some Problems for a Degenerated Elliptic Equations and Applications to Prismatic Shells, Tbilisi Univ., Tbilisi, 1982 (in Russian)

[84] G. V. Dzhayani, Euler–Poisson–Darboux Equation, Tbilisi Univ., Tbilisi, 1984 (in Russian)

[85] C. Gini, Means, Russian translation, Statistika, Moscow, 1970

[86] M. M. Dzhrbashyan, Integral Transforms and Representations of Functions in the Complex Domain, Nauka, Moscow, 1966 (in Russian)

[87] Kh. A. Din', “Integral equations with Legendre function as kernels in special cases”, Dokl. AN Belorus. SSR, 33:7 (1989), 591–594 (in Russian) | MR | Zbl

[88] I. E. Egorov, S. G. Pyatkov, S. V. Popov, Nonclassical Differential-Operator Equations, Nauka, Novosibirsk, 2000 (in Russian) | MR

[89] I. E. Egorov, V. Evs. Fedorov, Nonclassical Higher Order Equations of Mathematical Physics, VTS SO RAN, Novosibirsk, 1995 (in Russian)

[90] V. I. Zhegalov, A. N. Mironov, Differential Equations with Highest Partial Derivatives, Kazan. Mat. Ob-vo, Kazan', 2001 (in Russian)

[91] V. I. Zhegalov, A. N. Mironov, E. A. Utkina, Equations with Dominating Partial Derivative, Kazan. Univ., Kazan', 2014 (in Russian)

[92] Ya. I. Zhitomirskiy, “A Cauchy problem for for systems of partial differential equations with Bessel-type differential operators”, Mat. sb., 36(78):2 (1955), 299–310 (in Russian) | MR | Zbl

[93] N. V. Zhukovskaya, S. M. Sitnik, “Differential equations of fractional order of Euler type”, Mat. zametki SVFU, 25:2 (2018), 27–39 (in Russian)

[94] V. M. Zhuravlev, Nonlinear Waves. Exactly Solvable Problems, Ul'yanovsk, 2001 (in Russian)

[95] V. A. Zaytsev, “On Huygens principle for some equations with singularities”, Dokl. AN SSSR, 242:1 (1978), 28–31 (in Russian) | MR | Zbl

[96] V. A. Zaytsev, “Weak lacunas for one-dimensional strictly hyperbolic equations with constant coefficients”, Sib. mat. zh., 25:4 (1984), 54–62 (in Russian) | MR

[97] V. E. Zakharov, S. V. Manakov, S. P. Novikov, L. P. Pitaevskiy, Soliton Theory: the Method of Inverse Problem, Nauka, Moscow, 1980 (in Russian) | MR

[98] A. M. Zverkin, G. A. Kamenskiy, S. B. Norkin, L. E. El'sgol'ts, “Differential Equations with Deviating Argument”, Usp. mat. nauk, 17:2 (1962), 77–164 (in Russian) | MR | Zbl

[99] A. I. Ibragimov, “On behavior near boundary points and theorems on removable singularities for the second order elliptic equations with continuous derivatives”, Dokl. AN SSSR, 250:1 (1980), 25–28 (in Russian) | MR | Zbl

[100] V. M. Ivakin, “Modified Dirichlet problem for equations and systems degenerated at the boundary”, Diff. uravn., 18:2 (1982), 319–324 (in Russian) | MR | Zbl

[101] L. A. Ivanov, “On Cauchy problem for operators disintegrated into products of Euler–Poisson–Darboux multipliers”, Diff. uravn., 14:4 (1978), 736–739 (in Russian) | MR | Zbl

[102] L. A. Ivanov, “A Cauchy problem for some operators with singularities”, Diff. uravn., 18:6 (1982), 1020–1028 (in Russian) | MR | Zbl

[103] V. A. Il'in, “Kernels of fractional order”, Mat. sb., 41(83):4 (1957), 459–480 (in Russian) | MR | Zbl

[104] K. S. Kazaryan, “On a Dirichlet problem in weighted metrics”, Application of Method of Function Theory and Functional Analysis to Problems of Mathematical Physics, EGU, Erevan, 1982, 134–136 (in Russian)

[105] G. A. Kamenskiy, A. L. Skubachevskiy, Linear Boundary-Value Problems for Differential–Difference equations, MAI, Moscow, 1992 (in Russian)

[106] O. V. Kaptsov, Methods of Integration of Partial Differential Equations, Fizmatlit, Moscow, 2009 (in Russian)

[107] N. K. Karapetyants, S. G. Samko, Equations with Involution Operators and Applications, Izd-vo Rostov. un-ta, Rostov-na-Donu, 1988 (in Russian) | MR

[108] Sh. T. Karimov, “Multidimensional Erdélyi–Kober operator and its applications to solving of Cauchy problem for three–dimensional hyperbolic equation with singular coefficients”, Uzb. mat. zh., 2013, no. 1, 70–80 (in Russian)

[109] Sh. T. Karimov, “On a method of solution of the Cauchy problem for a generalized Euler–Poisson–Darboux equation”, Uzb. mat. zh., 2013, no. 3, 57–69 (in Russian)

[110] Sh. T. Karimov, “Solution of the Cauchy problem for three-dimensional hyperbolic equation with singular coefficients and spectral parameter”, Uzb. mat. zh., 2014, no. 2, 55–65 (in Russian)

[111] Sh. T. Karimov, “On a method of solution of the Cauchy problem for one-dimensional polywave equation with singular Bessel operator”, Izv. vuzov. Ser. Mat., 2017, no. 8, 27–41 (in Russian) | Zbl

[112] D. B. Karp, Spaces with hypergeometric reproducing kernels and fractional Fourier transforms, PhD Thesis, Vladivostok, 2000 (in Russian)

[113] D. B. Karp, S. M. Sitnik, “Fractional Hankel transforms and its applications”, Abstr. of Voronezh. Spring Math. School (17–23 Apr. 1996), VGU, Voronezh, 1996, 92 (in Russian)

[114] V. V. Katrakhov, “On an eigenvalue problem for singular elliptic operators”, Dokl. AN SSSR, 207:2 (1972), 284–287 (in Russian) | MR | Zbl

[115] V. V. Katrakhov, “On the theory of partial differential equations with singular coefficients”, Dokl. AN SSSR, 218:1 (1974), 17–20 (in Russian) | MR | Zbl

[116] V. V. Katrakhov, “Spectral function for some singular differential operators”, Diff. uravn., 12:7 (1976), 1256–1266 (in Russian) | MR | Zbl

[117] V. V. Katrakhov, “Transmutations in the theory of one-dimensional pseudodifferential operators”, Application of Method of Function Theory and Functional Analysis to Problems of Mathematical Physics, IM SO AN USSR, Novosibirsk, 1979, 72–75 (in Russian)

[118] V. V. Katrakhov, “Transmutations and pseudodifferential operators”, Sib. mat. zh., 21:1 (1980), 86–97 (in Russian) | MR | Zbl

[119] V. V. Katrakhov, “Isometric transmutations and a spectral function for a class of one-dimensional singular pseudodifferential operators”, Dokl. AN SSSR, 251:5 (1980), 1048–1051 (in Russian) | MR | Zbl

[120] V. V. Katrakhov, “General boundary-value problems for a class of singular and degenerated equations”, Dokl. AN SSSR, 251:6 (1980), 1296–1300 (in Russian) | MR | Zbl

[121] V. V. Katrakhov, “General boundary-value problems for a class of singular and degenerated elliptic equations”, Mat. sb., 112:3 (1980), 354–379 (in Russian) | MR | Zbl

[122] V. V. Katrakhov, “On a boundary-value problem for the Poisson equation”, Dokl. AN SSSR, 259:5 (1981), 1041–1045 (in Russian) | MR | Zbl

[123] V. V. Katrakhov, “Singular boundary-value problems and transmutations”, Well-Posed Boundary-Value Problems for Nonclassical Equations of Mathematical Physics, IM SO AN USSR, Novosibirsk, 1981, 87–91 (in Russian)

[124] V. V. Katrakhov, “Transmutation method in the theory of general weighted boundary-value problems for singular and degenerated elliptic equations with a parameter”, Dokl. AN SSSR, 266:5 (1982), 1037–1040 (in Russian) | MR | Zbl

[125] V. V. Katrakhov, Singular elliptic boundary–value problems. Method of transmutations, Doctoral Thesis, Novosibirsk, 1989 (in Russian)

[126] V. V. Katrakhov, “On a boundary-value problem for the Poisson equation”, Mat. sb., 182:6 (1991), 849–876 (in Russian) | MR | Zbl

[127] V. V. Katrakhov, “Singular boundary-value problems for some elliptic equations in corner domains”, Dokl. AN SSSR, 316:5 (1991), 1047–1050 (in Russian) | MR | Zbl

[128] V. V. Katrakhov, A. A. Katrakhova, The Taylor formula with Bessel operator for functions in one or two variables, Dep. VINITI, Voronezh, 1982 (in Russian)

[129] V. V. Katrakhov, I. A. Kipriyanov, “Powers of a singular elliptic operator”, Theory of Cubature Formulas and Applications of Functional Analysis to Problems of Mathematical Physics, Novosibirsk, 1980, 60–80 (in Russian) | MR | Zbl

[130] V. V. Katrakhov, S. M. Sitnik, “Boundary-value problem for the steady-state Shrödinger equation with a singular potential”, Dokl. AN SSSR, 278:4 (1984), 797–799 (in Russian) | MR | Zbl

[131] V. V. Katrakhov, S. M. Sitnik, “Factorization method in the transmutation theory”, Memorial Digest to the Memory of Boris A. Bubnov: Nonclassical Equations and Mixed-Type Equations, Novosibirsk, 1990, 104–122 (in Russian) | Zbl

[132] V. V. Katrakhov, S. M. Sitnik, “Composition method for constructing $B$-elliptic, $B$-hyperbolic and $B$-parabolic transmutations”, Dokl. RAN, 337:3 (1994), 307–311 (in Russian) | MR | Zbl

[133] V. V. Katrakhov, S. M. Sitnik, “Estimates for Jost solutions to one–dimensional Shrödinger equation with a singular potential”, Dokl. RAN, 340:1 (1995), 18–20 (in Russian) | MR | Zbl

[134] A. P. Kachalov, Ya. V. Kurylev, “Transmutation method for inverse scattering problem, one-dimensional Stark effect”, Zap. nauch. sem. LOMI, 179, 1989, 73–87 (in Russian) | MR | Zbl

[135] B. V. Kvyadaras, “A solution of the Dirichlet problem for a degenerated elliptic equation”, Partial Differential Equations, Proceedings of the Conference on Differential Equations and Computational Mathematics, Nauka, Novosibirsk, 1980, 35–36 (in Russian)

[136] M. V. Keldysh, “On solvability and stability of the Dirichlet problem”, Usp. mat. nauk, 1941, no. 8, 171–231 (in Russian) | MR | Zbl

[137] M. V. Keldysh, “On some cases of degeneration of elliptic equations on the boundary of a domain”, Dokl. AN SSSR, 77:1 (1951), 181–183 (in Russian)

[138] A. A. Kilbas, M. Saygo, V. A. Zhuk, “On a composition of generalized fraction integration operators with a differential operator of axisymmetric potential theory”, Diff. uravn., 27:9 (1991), 1640–1642 (in Russian) | MR | Zbl

[139] A. A. Kilbas, O. V. Skoromnik, “Solution of the multidimensional integral equation of the first kind with Legendre function at a pyramidal domain”, Dokl. RAN, 429:4 (2009), 442–446 (in Russian) | Zbl

[140] I. A. Kipriyanov, “Fourier–Bessel transforms and embedding theorems for weighted classes”, Tr. MIAN, 89, 1967, 130–213 (in Russian) | Zbl

[141] I. A. Kipriyanov, “Boundary–value problems for singular elliptic partial differential operators”, Dokl. AN SSSR, 195:1 (1970), 32–35 (in Russian) | MR | Zbl

[142] I. A. Kipriyanov, “On a class of singular elliptic operators”, Diff. uravn., 7:11 (1971), 2066–2077 (in Russian) | MR | Zbl

[143] I. A. Kipriyanov, “On a class of singular elliptic equations”, Sib. mat. zh., 14:3 (1973), 560–568 (in Russian) | Zbl

[144] I. A. Kipriyanov, Singular Elliptic Boundary-Value Problems, Nauka, Fizmatlit, Moscow, 1997 (in Russian) | MR

[145] I. A. Kipriyanov, L. A. Ivanov, “On lacunas for some classes of singular equations”, Mat. sb., 110(152):2 (1979), 235–250 (in Russian) | MR | Zbl

[146] I. A. Kipriyanov, L. A. Ivanov, “Euler–Poisson–Darboux equation in the Riemann space”, Dokl. AN SSSR, 260:4 (1981), 790–794 (in Russian) | MR | Zbl

[147] I. A. Kipriyanov, L. A. Ivanov, “Cauchy problem for the Euler–Poisson–Darboux equation in homogeneous symmetric Riemann space. I”, Tr. MIAN, 170, 1984, 139–147 (in Russian) | MR | Zbl

[148] I. A. Kipriyanov, L. A. Ivanov, “Cauchy problem for the Euler–Poisson–Darboux equation in symmetric space”, Mat. sb., 124(166):1 (1984), 45–55 (in Russian) | MR | Zbl

[149] I. A. Kipriyanov, L. A. Ivanov, “The D'Alambert representation and energy equipartition”, Diff. uravn., 26:3 (1990), 458–464 (in Russian) | MR | Zbl

[150] I. A. Kipriyanov, V. V. Katrakhov, “On a class of multidimensional singular pseudodifferential operators”, Mat. sb., 104:1 (1977), 49–68 (in Russian) | MR | Zbl

[151] I. A. Kipriyanov, V. V. Katrakhov, “Boundary-value problem for elliptic equations of the second order with singularities at isolated boundary points”, Dokl. AN SSSR, 276:2 (1984), 274–276 (in Russian) | MR | Zbl

[152] I. A. Kipriyanov, V. V. Katrakhov, On a singular elliptic boundary-value problem at the sphere domain, Preprint, IPM DVO RAN, 1989 (in Russian)

[153] I. A. Kipriyanov, V. V. Katrakhov, Preprint, IPM DVO RAN, 1989 (in Russian)

[154] I. A. Kipriyanov, V. V. Katrakhov, “On a boundary-value problem for elliptic equations of the second order at the sphere domain”, Dokl. AN SSSR, 313:3 (1990), 545–548 (in Russian) | MR | Zbl

[155] I. A. Kipriyanov, M. I. Klyuchantsev, “On Poisson kernels for boundary-value problems with Bessel differential operator”, Partial Differential Equations, Moscow, 1970, 119–134 (in Russian) | Zbl

[156] I. A. Kipriyanov, V. I. Kononenko, “On fundamental solutions for partial differential equations with Bessel differential operator”, Dokl. AN SSSR, 170:2 (1966), 261–264 (in Russian) | MR | Zbl

[157] I. A. Kipriyanov, V. I. Kononenko, “Fundamental solutions for $B$-elliptic equations”, Diff. uravn., 3:1 (1967), 114–129 (in Russian) | MR | Zbl

[158] I. A. Kipriyanov, V. I. Kononenko, “On fundamental solutions for some singular partial differential equations”, Diff. uravn., 5:8 (1969), 1470–1483 (in Russian) | MR | Zbl

[159] I. A. Kipriyanov, A. A. Kulikov, “Fundamental solutions for $B$-hypoelliptic equations”, Diff. uravn., 27:8 (1991), 1387–1395 (in Russian) | MR | Zbl

[160] S. B. Klimentov, “Hardy classes of generalized analytic functions”, Izv. vuzov. Sev.-Kavkaz. reg. Ser. Estestv. nauki, 2003, no. 3, 6–10 (in Russian) | Zbl

[161] S. B. Klimentov, “Smirnov classes of generalized analytic functions”, Izv. vuzov. Sev.-Kavkaz. reg. Ser. Estestv. nauki, 2005, no. 1, 13–17 (in Russian)

[162] S. B. Klimentov, “BMO classes of generalized analytic functions”, Vladikavkaz. mat. zh., 8:1 (2006), 27–39 (in Russian) | MR | Zbl

[163] S. B. Klimentov, Boundary Properties of Generalized Analytic Functions, Izd. Yuzhnogo mat. in-ta VNTS RAN i RSO-A, Vladikavkaz, 2014 (in Russian) | MR

[164] M. I. Klyuchantsev, “On construction of $r$-even solutions to singular differential equations”, Dokl. AN SSSR, 224:5 (1975), 1004–1007 (in Russian) | MR | Zbl

[165] M. I. Klyuchantsev, “Integrals of fractional order and singular boundary-value problems”, Diff. uravn., 12:6 (1976), 983–990 (in Russian) | MR | Zbl

[166] A. N. Kolmogorov, S. V. Fomin, Elements of Function Theory and Functional Analysis, Nauka, Moscow, 1981 (in Russian) | MR

[167] D. Colton, R. Kress, Integral Equation Methods in Scattering Theory, Russian translation, Mir, Moscow, 1987

[168] Yu. F. Korobeynik, Shift Operators on Numerical Families, Izd-vo Rostov. Univ., Rostov-na-Donu, 1983 (in Russian) | MR

[169] Yu F. Korobeynik, On Solvability in Complex Domain of Some General Classes of Linear Integral Equations, Izd-vo Rostov. Univ., Rostov-na-Donu, 2005 (in Russian)

[170] A. N. Kochubey, “Cauchy problem of evolutionary equations of fractional order”, Diff. uravn., 25:8 (1989), 1359–1369 (in Russian) | MR

[171] A. N. Kochubey, “Diffusion of fractional order”, Diff. uravn., 26:4 (1990), 660–770 (in Russian) | MR | Zbl

[172] N. S. Koshlyakov, E. B. Gliner, M. M. Smirnov, Partial Differential Equations of Mathematical Physics, Vysshaya shkola, Moscow, 1962 (in Russian)

[173] V. F. Kravchenko (ed.), Digital Signal and Image Processing in Radio Physics Applications, Fizmatlit, Moscow, 2007 (in Russian)

[174] V. F. Kravchenko, V. L. Rvachev, Algebra of Logic, Atomistic Functions and Wavelets in Physical Applications, Fizmatlit, Moscow, 2006 (in Russian)

[175] S. G. Kreyn, Yu. I. Petunin, E. M. Semenov, Interpolation of Linear Operators, Nauka, Moscow, 1978 (in Russian)

[176] L. D. Kudryavtsev, Direct and Inverse Embedding Theorems. Applications to Solving Elliptic Equations by Variational Method, Nauka, Moscow, 1959 (in Russian)

[177] L. D. Kudryavtsev, S. M. Nikol'skiy, “Spaces of multivariate differentiable functions and embedding theorems”, Itogi nauki i tekhn. Sovrem. probl. mat. Fundam. napravl., 26, VINITI, Moscow, 1988, 5–157 (in Russian) | MR | Zbl

[178] N. V. Kuznetsov, “On eigenfunctions of one integral equation”, Zap. nauch. sem. LOMI, 17, 1970, 66–150 (in Russian) | MR | Zbl

[179] N. V. Kuznetsov, “Peterson's hypothesis for parabolic forms of zero weight and Linnik's hypothesis. Kloostermann sums”, Mat. sb., 111(153):3 (1980), 334–383 (in Russian) | MR | Zbl

[180] N. V. Kuznetsov, Trace Formulas and Some of Its Applications in Number Theory, Dal'nauka, Vladivostok, 2003 (in Russian)

[181] R. Courant, Partial Differentail Equations, Russian translation, Mir, Moscow, 1979

[182] A. G. Kusraev, Majorizable Operators, Nauka, Moscow, 2003 (in Russian) | MR

[183] M. M. Lavrent'ev, One-Dimensional Inverse Problems of Mathematical Physics, Nauka, Novosibirsk, 1982 (in Russian)

[184] O. A. Ladyzhenskaya, N. N. Ural'tseva, Linear and Quasilinear Equations of Elliptic Type, Nauka, Moscow, 1973 (in Russian) | MR

[185] P. Lax, R. Phillips, Scattering Theory, Russian translation, Mir, Moscow, 1971

[186] P. Lax, R. Phillips, A Scattering Theory for Automorphic Functions, Russian translation, Mir, Moscow, 1979

[187] E. M. Landis, The Second Order Equations of Elliptic and Parabolic Types, Nauka, Moscow, 1971 (in Russian)

[188] E. M. Landis, “Problems of E. M. Landis”, Usp. mat. nauk, 37:6 (1982), 278–282 (in Russian)

[189] N. S. Landkof, Foundations of Modern Potential Theory, Nauka, Moscow, 1966 (in Russian) | MR

[190] A. A. Larin, “On spectral expansions corresponding to self-adjoint extensions of some singular elliptic operators”, Dokl. AN SSSR, 293:2 (1987), 309–312 (in Russian) | MR | Zbl

[191] A. A. Larin, “On eigenfunction properties of some singular elliptic operators”, Diff. uravn., 27:5 (1991), 849–856 (in Russian) | MR | Zbl

[192] A. A. Larin, “On boundedness of powers of self-adjoint extensions of singular elliptic operators acting on weighted classes”, Diff. uravn., 28:3 (1992), 528–529 (in Russian) | MR | Zbl

[193] A. A. Larin, “On representation of solutions to a singular elliptic equation of the second order nearby the corner point”, Diff. uravn., 36:4 (2000), 566–568 (in Russian) | MR | Zbl

[194] A. A. Larin, “On one boundary-value problem in a plane corner for a second-order singular elliptic equation”, Diff. uravn., 36:12 (2000), 1687–1694 (in Russian) | MR | Zbl

[195] A. A. Larin, “On a restriction to spherical surface theorem for Fourier–Bessel transform”, Dokl. Adygskoy (Cherkesskoy) mezhd. akad. nauk, 16:3 (2014), 22–29 (in Russian)

[196] B. Ya. Levin, “Fourier and Laplace type transforms by means of solutions to the second order differential equations”, Dokl. AN SSSR, 106:2 (1956), 187–190 (in Russian) | Zbl

[197] B. Ya. Levin, Distribution of Zeroes of Entire Functions, GITTL, Moscow, 1956 (in Russian)

[198] B. M. Levitan, Expansions in Eigenfunctions Of Second-Order Differential Equations, Gostekhizdat, Moscow, 1950 (in Russian) | MR

[199] B. M. Levitan, “Expansions in Bessel function series and Fourier integrals”, Usp. mat. nauk, 6:2 (1951), 102–143 (in Russian) | MR | Zbl

[200] B. M. Levitan, Almost Periodic Functions, GITTL, Moscow, 1953 (in Russian) | MR

[201] B. M. Levitan, Generalized Translations and Some Applications, GIFML, Moscow, 1962 (in Russian)

[202] B. M. Levitan, Theory of Generalized Translation Operators, Nauka, Moscow, 1973 (in Russian)

[203] B. M. Levitan, Inverse Sturm–Lioville Problems, Nauka, Moscow, 1984 (in Russian) | MR

[204] B. M. Levitan, A. Ya. Povzner, “Sturm–Lioville differential equations on semiaxis and the Plancherel theorem”, Dokl. AN SSSR, 52:6 (1946), 483–486 (in Russian)

[205] B. M. Levitan, I. S. Sargsyan, Sturm–Lioville and Dirac Operators, Nauka, Moscow, 1988 (in Russian) | MR

[206] M. A. Leyzin, “On Embedding theorems for a class of singular differential equations in a half-space”, Diff. uravn., 12:6 (1976), 1073–1083 (in Russian) | MR | Zbl

[207] M. A. Leyzin, “On embedding of some weighted classes”, Methods of Solution for Operator Equations, VGU, Voronezh, 1978, 96–103 (in Russian)

[208] A. F. Leont'ev, “Growth estimate of one differential equation for large in moduli values and its application to some function theory problems”, Sib. mat. zh., 1:3 (1960), 456–487 (in Russian) | Zbl

[209] M. E. Lerner, Maximum Principles for Hyperbolic Type Equations and New Properties of the Riemann Function, Samar. gos. tekh. un-t, Samara, 2001 (in Russian)

[210] P. I. Lizorkin, “The generalized Liouville differentiation and multiplier method in the theory of class embeddings of differentiable functions”, Tr. MIAN, 105, 1969, 89–167 (in Russian) | MR | Zbl

[211] P. I. Lizorkin, “Function classes constructed by means over spheres. Case of Sobolev spaces”, Tr. MIAN, 192, 1990, 122–139 (in Russian) | MR | Zbl

[212] P. I. Lizorkin, S. M. Nikol'skiy, “An elliptic degenerated equation. Variational method”, Dokl. AN SSSR, 257:1 (1981), 42–45 (in Russian) | MR | Zbl

[213] P. I. Lizorkin, S. M. Nikol'skiy, “An elliptic degenerated equation. Differential properties of solutions”, Dokl. AN SSSR, 257:2 (1981), 278–282 (in Russian) | MR | Zbl

[214] P. I. Lizorkin, S. M. Nikol'skiy, “Coercive properties of elliptic equations with strong degeneracy (case of generalized solutions)”, Dokl. AN SSSR, 259:1 (1981), 28–30 (in Russian) | MR | Zbl

[215] J.-L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Russian translation, Mir, Moscow, 1971

[216] G. S. Litvinchuk, Boundary value problems and singular integral equations with shift, Nauka, Moscow, 1977 (in Russian)

[217] Yu. Luke, Mathematical Functions and Their Approximations, Russian translation, Mir, Moscow, 1980

[218] L. N. Lyakhov, “Inversion of $B$-potentials”, Dokl. AN SSSR, 321:3 (1991), 466–469 (in Russian) | MR | Zbl

[219] L. N. Lyakhov, Weighted Spherical Functions and Riesz Potentials Generated by Generalized Translations, VGTA, Voronezh, 1997 (in Russian)

[220] L. N. Lyakhov, $B$-Hypersingular Integrals and Their Applications to Description of Kipriyanov's Functional Classes and to Integral Equations with $B$-potential Kernels, LGPU, Lipetsk, 2007 (in Russian)

[221] L. N. Lyakhov, I. P. Polovinkin, E. L. Shishkina, “On a problem of I. A. Kipriyanov for a singular ultrahyperbolic equation”, Diff. uravn., 50:4 (2014), 516–528 (in Russian) | DOI | MR | Zbl

[222] L. N. Lyakhov, I. P. Polovinkin, E. L. Shishkina, “Formulas for solution to Cauchy problem for singular wave equation with Bessel operators in time”, Dokl. RAN, 459:5 (2014), 533–538 (in Russian) | DOI | Zbl

[223] L. N. Lyakhov, E. L. Shishkina, Fractional Derivatives and Integrals and Their Applications, VGU, Voronezh, 2011 (in Russian)

[224] M. M. Malamud, “On transmutations for ordinary differential equations of higher orders”, Mathematical Analysis and Probability Theory, Naukova dumka, Kiev, 1978, 108–111 (in Russian)

[225] M. M. Malamud, “Necessary conditions for existence of transmutations for equations of higher orders”, Funkts. analiz i ego prilozh., 16:3 (1982), 74–75 (in Russian) | MR | Zbl

[226] M. M. Malamud, On a question of transmutations, Preprint, IM AN USSR, Kiev, 1984 (in Russian)

[227] M. M. Malamud, “Transmutations for equations of higher orders”, Mat. fiz. i nelin. mekh., 1986, no. 6, 108–111 (in Russian)

[228] M. M. Malamud, “On a question of transmutations for ordinary differential equations”, Tr. Mosk. mat. ob-va, 53, 1990, 68–97 (in Russian) | MR | Zbl

[229] O. I. Marichev, Method of Calculation of Integrals of Special Functions, Nauka i Tekhnika, Minsk, 1978 (in Russian)

[230] O. I. Marichev, A. A. Kilbas, O. A. Repin, Boundary-Value Problems for Partial Differential Equations with Discontinuous Coefficients, Izd-vo Samar. Gos. Ekonom. Univ., Samara, 2008 (in Russian)

[231] V. A. Marchenko, “Some questions of the theory of second-order differential operators”, Dokl. AN SSSR, 72:3 (1950), 457–460 (in Russian) | Zbl

[232] V. A. Marchenko, “Transmutation operators”, Dokl. AN SSSR, 74:2 (1950), 185–188 (in Russian) | Zbl

[233] V. A. Marchenko, “On inversion formulas generated by the second order differential operator”, Dokl. AN SSSR, 74:4 (1950), 657–660 (in Russian) | MR | Zbl

[234] V. A. Marchenko, “Some questions of the theory of one-dimensional second-order differential operators. I”, Tr. Mosk. mat. ob-va, 1, 1952, 327–420 (in Russian) | MR | Zbl

[235] V. A. Marchenko, “Some questions of the theory of one-dimensional second-order differential operators. II”, Tr. Mosk. mat. ob-va, 2, 1953, 3–83 (in Russian) | MR | Zbl

[236] V. A. Marchenko, Spectral Theory of Sturm–Liouville Operators, Naukova Dumka, Kiev, 1972 (in Russian) | MR

[237] V. A. Marchenko, Sturm–Liouville Operators and Their Applications, Naukova Dumka, Kiev, 1977 (in Russian)

[238] V. A. Marchenko, Nonlinear Equations and Operator Algebras, Naukova Dumka, Kiev, 1986 (in Russian) | MR

[239] V. A. Marchenko, “Generalized translation, transmutations and inverse problems”, Mathematical Events in XX Century, Fazis, Moscow, 2003 (in Russian)

[240] M. I. Matiychuk, Parabolic Singular Boundary-Value Problems, In-t matematiki NAN Ukraïni, Kiïv, 1999 (in Ukraine)

[241] M. I. Matiychuk, Parabolic and Elliptic Boundary-Value Problems with Singularities, Prut, Chernivtsi, 2003 (in Ukraine)

[242] V. I. Matsaev, “On an existence of transmutations for higher order differential equations”, Dokl. AN SSSR, 130:3 (1960), 499–502 (in Russian) | MR | Zbl

[243] Kh. Mekhrez, S. M. Sitnik, “Monotonicity of ratios for some hypergeometric functions”, Sib. elektron. mat. izv., 13 (2016), 260–268 (in Russian) | DOI | MR | Zbl

[244] V. Z. Meshkov, “Weighted differential inequalities and its applications for estimates of decay rate for second-order elliptic equations”, Tr. MIAN, 190, 1989, 139–158 (in Russian) | MR | Zbl

[245] V. Z. Meshkov, “On a possible rate of decay at infinity for solutions to partial differential second order equations”, Mat. sb., 182:3 (1991), 364–383 (in Russian) | MR | Zbl

[246] V. V. Meshcheryakov, Differential-difference operators associated with kernel systems of Coxeter type, PhD Thesis, Kolomna, 2008 (in Russian)

[247] S. Mizohata, The Theory of Partial Differential Equations, Russian translation, Mir, Moscow, 1977

[248] C. Miranda, Partial Differential Equations of Elliptic Type, Russian translation, Mir, Moscow, 1957

[249] E. I. Moiseev, Mixed Type Equations with Spectral Parameter, MGU, Moscow, 1988 (in Russian)

[250] A. B. Muravnik, “On stabilization of solutions of some singular quasilinear parabolic problems”, Mat. zametki, 74:6 (2003), 858–865 (in Russian) | DOI | MR | Zbl

[251] A. B. Muravnik, “On stabilization of solutions of singular elliptic equations”, Fundam. i prikl. mat., 12:4 (2006), 169–186 (in Russian) | MR | Zbl

[252] A. B. Muravnik, “Functional-differential parabolic equations: integral representations and quantitative properties of solutions to Cauchy problems”, Sovrem. mat. Fundam. napravl., 52, 2014, 3–141 (in Russian)

[253] A. D. Myshkis, Linear Differential Equations with Delayed Argument, Gostekhizdat, Moscow–Leningrad, 1951 (in Russian) | MR

[254] M. A. Naymark, Linear Differential Operators, Nauka, Moscow, 1969 (in Russian)

[255] R. Narasimhan, Analysis on Real and Complex Manifolds, Russian translation, Mir, Moscow, 1971

[256] F. Natterer, The Mathematics of Computerized Tomography, Russian translation, Mir, Moscow, 1990

[257] A. M. Nakhushev, Equations of Mathematical Biology, Vysshaya Shkola, Moscow, 1995 (in Russian)

[258] A. M. Nakhushev, Elements of Fractional Calculus and Their Application, Nal'chik, 2000 (in Russian)

[259] A. M. Nakhushev, Fractional Calculus and Its Applications, Fizmatlit, Moscow, 2003 (in Russian)

[260] A. M. Nakhushev, Loaded Equations and Their Applications, Nauka, Moscow, 2012 (in Russian)

[261] L. P. Nizhnik, Inverse Nonstationary Scattering Theory Problem, Naukova Dumka, Kiev, 1973 (in Russian) | MR

[262] L. P. Nizhnik, Inverse Scattering Problems for Hyperbolic Equations, Naukova dumka, Kiev, 1990 (in Russian)

[263] S. M. Nikol'skiy, Approximation of Multivariate Functions and Embedding Theorems, Nauka, Moscow, 1977 (in Russian)

[264] S. M. Nikol'skiy, “Variational problem for an elliptic equation with degeneration at the boundary”, Tr. MIAN, 150, 1979, 212–238 (in Russian) | MR | Zbl

[265] S. M. Nikol'skiy, P. I. Lizorkin, “On some inequalities for functions from weighted classes and boundary-value problems with strong degeneracy at the boundary”, Dokl. AN SSSR, 159:3 (1964), 512–515 (in Russian) | MR | Zbl

[266] O. G. Novozhenova, Biography and Scientific Works of Alexey Nikiforovich Gerasimov. On Linear Operatos, Viscoelasticity, Eleutherosis and Fractional Derivatives, Pero, Moscow, 2018 (in Russian)

[267] A. A. Novruzov, “On Dirichlet problems for second-order elliptic equations”, Dokl. AN SSSR, 246:1 (1979), 11–14 (in Russian) | MR | Zbl

[268] V. A. Nogin, E. V. Sukhinin, “Inversion and description of hyperbolic potentials with $L_p$-densities”, Dokl. RAN, 329:5 (1993), 550–552 (in Russian) | MR | Zbl

[269] A. Newell, Solitons in Mathematics and Physics, Russian translation, Mir, Moscow, 1989 | MR

[270] O. A. Oleynik, E. V. Radkevich, Equations with Nonnegative Characteristic Form, MGU, Moscow, 2010 (in Russian)

[271] A. V. Omel'chenko, Methods of Integral Transforms in Problems of Mathematical Physics, MTSNMO, Moscow, 2010 (in Russian)

[272] V. F. Osipov, Almost Periodic Bohr–Fresnel Functions, S.-Peterb. Univ., Saint-Petersburg, 1992 (in Russian) | MR

[273] A. E. Pasenchuk, Abstract Singular Operators, Novocherkassk, 1993 (in Russian)

[274] S. S. Platonov, “Generalized Bessel shifts and some problems of function approximation theory in metric $L_2$. 1”, Tr. PetrGU. Ser. Mat., 7, 2000, 70–82 (in Russian)

[275] S. S. Platonov, “Generalized Bessel shifts and some problems of function approximation theory in metric $L_2$. 2”, Tr. PetrGU. Ser. Mat., 8, 2001, 20–36 (in Russian) | Zbl

[276] S. S. Platonov, “Bessel harmonic analysis and approximation of functions on a semiaxis”, Izv. RAN. Ser. Mat., 71:5 (2007), 149–196 (in Russian) | DOI | MR | Zbl

[277] A. Ya. Povzner, “On Sturm–Liouville differential equations on a semiaxis”, Mat. sb., 23(65):1 (1948), 3–52 (in Russian) | MR | Zbl

[278] G. N. Polozhiy, Equations of Mathematical Physics, Vysshaya shkola, Moscow, 1964 (in Russian) | MR

[279] G. N. Polozhiy, A Generalization of the Theory of Analytic Functions of Complex Variable. $P$-Analytic and $(P,Q)$-Analytic Functions and Some Their Applications, KGU, Kiev, 1965 (in Russian)

[280] G. N. Polozhiy, Theory and Applications of $p$-Analytic Functions, Naukova dumka, Kiev, 1973 (in Russian)

[281] A. D. Polyanin, V. F. Zaytsev, Handbook on Nonlinear Equations of Mathematical Physics: Exact Solutions, Fizmatlit, Moscow, 2002 (in Russian)

[282] V. T. Polyatskiy, “On properties of solution of some equation”, Usp. mat. nauk, 17:4 (1965), 119–126 (in Russian) | MR | Zbl

[283] Z. Presdorf, Some Classes of Singular Equations, Russian translation, Mir, Moscow, 1979

[284] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, v. 1, 2, 3, Nauka, Moscow, 1981, 1983, 1986 (in Russian) | MR

[285] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, “Calculation of integrals and the Mellin transform”, Itogi nauki i tekhn. Mat. analiz, 27, 1989, 3–146 (in Russian) | MR | Zbl

[286] A. V. Pskhu, “Integral transforms with the Wright function in the kernel”, Dokl. Adygskoy (Cherkesskoy) mezhd. akad. nauk, 6:1 (2002), 35–47 (in Russian)

[287] A. V. Pskhu, Boundary-Value Problems for Differential Equations with Partial Derivatives of Fractional and Continual Order, Nal'chik, 2005 (in Russian)

[288] S. P. Pul'kin, “Some boundary-value problems for the equation $u_{xx}\pm u_{yy}+\frac pxu_x$”, Uch. zap. Kuybyshev. ped. in-ta, 21, 1958, 3–54 (in Russian)

[289] S. P. Pul'kin, Selected Works, Univers Grupp, Samara, 2007 (in Russian)

[290] L. S. Pul'kina, “On one nonclassical problem for degenerating hyperbolic equation”, Izv. vuzov. Ser. Mat., 1991, no. 11, 48–51 (in Russian) | MR | Zbl

[291] A. G. Ramm, Multidimensional Inverse Scattering Problems, Russian translation, Mir, Moscow, 1994

[292] V. A. Rvachev, “Finite solutions of functional differential equations and their applications”, Usp. mat. nauk, 45:1 (1990), 77–103 (in Russian) | MR | Zbl

[293] V. L. Rvachev, V. A. Rvachev, Approximation Theory and Atomic Functions, Znanie, Moscow, 1978 (in Russian)

[294] V. L. Rvachev, V. A. Rvachev, Nonclassical Methods of Approximation Theory in Boundary-Value Problems, Naukova dumka, Kiev, 1979 (in Russian)

[295] O. A. Repin, Boundary-Value Problems with Shift for hyperbolic and mixed equations, Samara, 1992 (in Russian)

[296] F. Riesz, B. Sz.-Nagy, Functional Analysis, Russian translation, Mir, Moscow, 1979

[297] Ya. A. Roytberg, Z. G. Sheftel', “On general elliptic problems with strong degeneration”, Dokl. AN SSSR, 254:6 (1980), 1336–1341 (in Russian) | MR

[298] L. E. Rossovskiy, “Elliptic functional differential equations with contractions and extensions of independent variables of the unknown function”, Sovrem. mat. Fundam. napravl., 54, 2014, 3–138 (in Russian)

[299] L. E. Rossovskiy, L. E. Skubachevskiy, “Solvability and regularity of solutions for some classes of elliptic functional differential equations”, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril., 66, 1999, 114–192 (in Russian) | MR | Zbl

[300] S. Rutkauskas, “Dirichlet problems with asymptotic conditions for an elliptic system degenerated at a point. I”, Diff. uravn., 38:3 (2002), 385–392 (in Russian) | MR | Zbl

[301] S. Rutkauskas, “Dirichlet problems with asymptotic conditions for an elliptic system degenerated at a point. II”, Diff. uravn., 38:5 (2002), 681–686 (in Russian) | MR | Zbl

[302] S. Rutkauskas, “On the Dirichlet-type problem for elliptic systems with degeneration on an axis”, Mat. zametki, 100:2 (2016), 270–278 (in Russian) | DOI | MR | Zbl

[303] V. S. Ryko, Compositional structure of the Fourier series, discrete Fourier and Mellin transforms, and calculation of their sums, Dep. v VINITI AN SSSR No 2826-V 87, Minsk, 1987 (in Russian)

[304] V. S. Ryko, A method of summation and improvement of convergence of functional series, Dep. v VINITI AN SSSR No 3542-V 88, Vologda, 1988 (in Russian)

[305] K. B. Sabitov, R. R. Il'yasov, “Solution of the Tricomi problem for a mixed-type equation with a singular coefficient by a spectral method”, Izv. vuzov. Ser. Mat., 2004, no. 2, 64–71 (in Russian) | MR | Zbl

[306] S. G. Samko, A. A. Kilbas, O. I. Marichev, Integrals and Derivatives of Fractional Order and Some Their Applications, Nauka i Tekhnika, Minsk, 1987 (in Russian)

[307] L. A. Sakhnovich, “Spectral analysis of the Volterra operators and inverse problems”, Dokl. AN SSSR, 115:4 (1957), 666–669 (in Russian) | MR | Zbl

[308] L. A. Sakhnovich, “Inverse problem for differential operators of order $n>2$ with analytic coefficients”, Mat. sb., 46(88):1 (1958), 61–76 (in Russian) | MR | Zbl

[309] L. A. Sakhnovich, “Necessary conditions of existence of transmutations for a fourth-order equation”, Usp. mat. nauk, 16:5 (1961), 199–204 (in Russian) | MR | Zbl

[310] S. M. Sitnik, “On the decay rate of solutions of steady-state Schrödinger equation with a potential depending on one variable”, Boundary-Value Problems for Nonclassical Equations of Mathematical Physics, Novosibirsk, 1985, 139–147 (in Russian)

[311] S. M. Sitnik, On the decay rate of solutions of some elliptic and ultraelliptic equations, Dep. v VINITI No 7771-V86, VGU, Voronezh, 13.11.1986 (in Russian)

[312] S. M. Sitnik, On unitary transmutations, Dep. v VINITI No 7770-V86, VGU, Voronezh, 13.11.1986 (in Russian)

[313] S. M. Sitnik, Transmutations for the Bessel differential expression, Dep. v VINITI No 535-V87, VGU, Voronezh, 23.01.1987 (in Russian)

[314] S. M. Sitnik, “On one pair of transmutations”, Boundary-Value Problems for Nonclassical Equations of Mathematical Physics, Novosibirsk, 1987, 168–173 (in Russian) | MR

[315] S. M. Sitnik, Transmutation method for the steady-state Schrödinger equation, PhD Thesis, Voronezh, 1987 (in Russian)

[316] S. M. Sitnik, “On the decay rate of solutions of some elliptic and ultraelliptic equations”, Diff. uravn., 24:3 (1988), 538–539 (in Russian) | MR | Zbl

[317] S. M. Sitnik, “Transmutations for singular differential equations with the Bessel operator”, Boundary-Value Problems for Nonclassical Equations of Mathematical Physics, Novosibirsk, 1989, 179–185 (in Russian) | MR

[318] S. M. Sitnik, Unitarity and boundedness of the Buschman–Erdélyi operators of zero order of smoothness, Preprint, Inst. Automat. Control Proc. RAS, Vladivostok, 1990 (in Russian)

[319] S. M. Sitnik, “Factorization and estimates of norms of the Buschman–Erdélyi operators in weighted Lebesgue spaces”, Dokl. AN SSSR, 320:6 (1991), 1326–1330 (in Russian) | MR | Zbl

[320] S. M. Sitnik, Transmutation and Jost representation for an equation with singular potential, Preprint, Inst. Automat. Control Proc. RAS, Vladivostok, 1993 (in Russian)

[321] S. M. Sitnik, Inequalities for full elliptic Legendre integrals, Preprint, Inst. Automat. Control Proc. RAS, Vladivostok, 1994 (in Russian)

[322] S. M. Sitnik, “Inequalities for Bessel functions”, Dokl. RAN, 340:1 (1995), 29–32 (in Russian) | MR | Zbl

[323] S. M. Sitnik, “A method of consecutive refinements of the Cauchy–Bunyakovskii inequality and its application to estimates of special functions”, Abstr. of Voronezh. Vesen. Mat. Shkola “Sovremennye metody v teorii kraevykh zadach. Pontryaginskie chteniya-VII”, VGU, Voronezh, 1996, 164 (in Russian)

[324] S. M. Sitnik, “Taylor formula for Bessel-type operators”, Abst. of Voronezh. Vesen. Mat. Shkola “Sovremennye metody v teorii kraevykh zadach. Pontryaginskie chteniya-VII”, VGU, Voronezh, 1996, 102 (in Russian)

[325] S. M. Sitnik, “On some generalizations of fractional integrodifferentiation”, Proc. Int. Rus.-Uzb. Symp. “Uravneniya smeshannogo tipa i rodstvennye problemy analiza i informatiki”, Nal'chik, 2003, 86–88 (in Russian)

[326] S. M. Sitnik, “Fractional integrodifferentiation for the Bessel differential operator”, Proc. Int. Rus.-Kazakh. Symp. “Uravneniya smeshannogo tipa i rodstvennye problemy analiza i informatiki”, Nal'chik, 2004, 163–167 (in Russian)

[327] S. M. Sitnik, “Generalization of the Cauchy–Bunyakovskii inequalities by the means method and their applications”, Chernozemnyy al'manakh nauch. issl. Ser. Fundam. mat., 2005, no. 1(1), 3–42 (in Russian)

[328] S. M. Sitnik, “On a generalization of the Hille–Tamarkin formula for the resolvent in the case of Bessel fractional differentiation operators”, Abstr. of III Int. Conf. “Nelokal'nye kraevye zadachi i rodstvennye problemy matematicheskoy biologii, informatiki i fiziki”, Nal'chik, 2006, 269–270 (in Russian)

[329] S. M. Sitnik, “Operators of fractional integrodifferentiation for the Bessel differential operator”, Proc. Fourth All-Rus. Sci. Conf. “Matematicheskoe modelirovanie i kraevye zadachi”, Part 3, Samara, 2007, 158–160 (in Russian)

[330] S. M. Sitnik, “Construction of Vekua–Erdélyi–Lowndes transmutations”, Abstr. of Int. Conf. decicated to 100th anniversary since birthday of Acad. I. N. Vekua “Differentsial'nye uravneniya, teoriya funktsiy i prilozheniya”, Novosibirsk, 2007, 469–470 (in Russian)

[331] S. M. Sitnik, “Transmutations and their applications”, Studies in Contemporary Analysis and Mathematical Modelling, Vladikavkaz Nauch. Tsentr RAN i RSO-A, Vladikavkaz, 2008, 226–293 (in Russian)

[332] S. M. Sitnik, “The method of factorization of transmutations in the theory of differential equations”, Vestn. Samar. gos. un-ta. Estestvennonauch. ser., 2008, no. 8/1(67), 237–248 (in Russian)

[333] S. M. Sitnik, “Refinements and generalizations of classic inequalities”, Issl. po mat. anal., Itogi nauki. Yuzhnyy federal'nyy okrug. Ser. Mat. forum., 3, Yuzhnyy Mat. Inst. VNTS RAN i RSO Alaniya, Vladikavkaz, 2009, 221–266 (in Russian)

[334] S. M. Sitnik, “On explicit realizations of fractional powers of the Bessel differential operator and their applications to differential equations”, it Dokl. Adygskoy (Cherkesskoy) mezhd. akad. nauk, 12:2 (2010), 69–75 (in Russian)

[335] S. M. Sitnik, “On integral representation of solutions of one differential equation with singularities in coefficietns”, Vladikavkaz. mat. zh., 12:4 (2010), 73–78 (in Russian) | MR | Zbl

[336] S. M. Sitnik, “Transmutation of special form for a differential operator with singular at zero potential”, Nonclassical Equations of Mathematical Physics, Inst. Mat. im. S. L. Soboleva SO RAN, Novosibirsk, 2010, 264–278 (in Russian)

[337] S. M. Sitnik, “Different classes of Buschman–Erdélyi transmutations”, Abstr. of Int. Conf. “Differentsial'nye uravneniya i smezhnye voprosy”, MGU, Moscow, 2011, 344–345 (in Russian)

[338] S. M. Sitnik, “New boundary-value problems with $K$-trace for solutions with strong singularities in works by V. V. Katrakhov”, Abstr. of Int. Conf. “Obratnye i nekorrektnye zadachi matematicheskoy fiziki”, Sib. nauch. izd-vo, Novosibirsk, 2012, 439 (in Russian)

[339] S. M. Sitnik, “Works by V. V. Katrakhov on the transmutation theory”, Proc. of the Second Rus.-Uzb. Symp. “Uravneniya smeshannogo tipa i rodstvennye problemy analiza i informatiki”, KBR, Nal'chik, 2012, 241–243 (in Russian)

[340] S. M. Sitnik, “Survey of basic properties of Buschman–Erdélyi transmutations”, Chelyabinsk. fiz.-mat. zh., 1:4 (2016), 63–93 (in Russian) | MR

[341] S. M. Sitnik, Application of Buschman–Erdélyi transmutations and their generalizations in the theory of differential equations with singularities in coefficients, Doctoral Thesis, Voronezh, 2016 (in Russian)

[342] S. M. Sitnik, D. B. Karp, Composition formulas for integral transforms with Bessel functions in kernels, Preprint, Inst. Automat. Control Proc. RAS, Vladivostok, 1993 (in Russian)

[343] S. M. Sitnik, D. B. Karp, Fractional Hankel transform and its applications in mathematical physics, Preprint, Inst. Automat. Control Proc. RAS, Vladivostok, 1994 (in Russian)

[344] S. M. Sitnik, G. V. Lyakhovetskiy, Composition formulas for Buschman–Erdélyi transmutations, Preprint, Inst. Automat. Control Proc. RAS, Vladivostok, 1991 (in Russian)

[345] S. M. Sitnik, G. V. Lyakhovetskiy, Vekua–Erdélyi–Lowndes transmutations, Preprint, Inst. Automat. Control Proc. RAS, Vladivostok, 1994 (in Russian)

[346] S. M. Sitnik, E. L. Shishkina, “On one identity for an iterated weighted spherical mean and its applications”, Sib. elektron. mat. izv., 13 (2016), 849–860 (in Russian) | DOI | MR | Zbl

[347] S. M. Sitnik, E. L. Shishkina, The Transmutation Method for Differential Equations with Bessel Operators, Fizmatlit, Moscow, 2018 (in Russian)

[348] S. M. Sitnik, E. L. Shishkina, “On fractional powers of the Bessel operator on a semiaxis”, Sib. elektron. mat. izv., 15 (2018), 1–10 (in Russian) | DOI | MR | Zbl

[349] A. L. Skubachevskiy, “Nonclassical boundary-value problems. I”, Sovrem. mat. Fundam. napravl., 26, 2007, 3–132 (in Russian) | MR | Zbl

[350] A. L. Skubachevskiy, Nonlocal Boundary-Value Problems and Their Applications to Study of Multidimensional Diffusion Processes and Processes of Thermoregulation in Living Cells, Textbook, RUDN, Moscow, 2008 (in Russian)

[351] A. L. Skubachevskiy, “Nonclassical boundary-value problems. II”, Sovrem. mat. Fundam. napravl., 33, 2009, 3–179 (in Russian) | MR

[352] A. L. Skubachevskiy, P. L. Gurevich, Application of Methods of Nonlinear Functional Analysis to Nonlocal Problems of Heat Conduction, Textbook, RUDN, Moscow, 2008 (in Russian)

[353] M. M. Smirnov, Degenerating Elliptic and Hyperbolic Equations, Nauka, Moscow, 1966 (in Russian)

[354] M. M. Smirnov, Equations of Mixed Type, Nauka, Moscow, 1970 (in Russian)

[355] S. L. Sobolev, Introduction to Theory of Cubature Formulas, Nauka, Moscow, 1974 (in Russian) | MR

[356] S. L. Sobolev, Equations of Mathematical Physics, Nauka, Moscow, 1992 (in Russian) | MR

[357] A. P. Soldatov, One-Dimensional Singular Operators and Boundary-Value Problems of Function Theory, Vysshaya shkola, Moscow, 1991 (in Russian)

[358] N. Ya. Sonin, Investigations on Cylinder Functions and Special Polynomials, Gostekhteoretizdat, Moscow, 1954 (in Russian)

[359] A. S. Sokhin, “On one class of transmutations”, Tr. fiz.-tekh. in-ta nizk. temp. AN USSR, 1, 1969, 117–125 (in Russian) | MR

[360] A. S. Sokhin, “Inverse scattering problems for equations with a singularity”, Tr. fiz.-tekh. in-ta nizk. temp. AN USSR, 2, 1971, 182–233 (in Russian) | MR

[361] A. S. Sokhin, “Inverse scattering problems for equations with singularities of special form”, Teor. funktsiy, funkts. analiz i ikh prilozh., 17 (1973), 36–64 (in Russian)

[362] A. S. Sokhin, “On transforms of operators for equations with singularity of special form”, Vestn. Khar'kov. un-ta, 113 (1974), 36–42 (in Russian) | MR | Zbl

[363] V. V. Stashevskaya, “The transmutation method”, Dokl. AN SSSR, 113:3 (1953), 409–412 (in Russian) | MR

[364] V. V. Stashevskaya, “On an inverse problem of spectral analysis for differential operator with singularity at zero”, Uch. zap. Khar'kov. mat. ob-va, 1957, no. 5, 49–86 (in Russian)

[365] E. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Russian translation, Mir, Moscow, 1974

[366] S. A. Tersenov, Introduction to Theory of Equations Degenerating at the Boundary, NGU, Novosibirsk, 1973 (in Russian)

[367] E. Titchmarsh, Introduction to the Theory of Fourier Integrals, Russian translation, GITTL, Moscow–Leningrad, 1948

[368] A. N. Tikhonov, A. A. Samarskiy, Equations of Mathematical Physics, Nauka, Moscow, 1972 (in Russian) | MR

[369] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Russian translation, Mir, Moscow, 1980

[370] F. Tricomi, Lectures on Partial Differential Equations, Russian translation, Inostr. lit., Moscow, 1957

[371] E. E. Tyrtyshnikov, Matrix Analysis and Linear Algebra, Fizmatlit, Moscow, 2007 (in Russian)

[372] E. Whittaker, G. Watson, A Course of Modern Analysis, Russian translation, v. 2, Transcendental Functions, GIFML, Moscow, 1963

[373] S. V. Uspenskiy, “On embedding theorems for weight classes”, Tr. MIAN, 60, 1961, 282–303 (in Russian) | MR | Zbl

[374] M. K. Fage, “Construction of transmutations and solution of one problem on moments for ordinary linear differential equations of arbitrary order”, Usp. mat. nauk, 12:1 (1957), 240–245 (in Russian) | MR | Zbl

[375] M. K. Fage, “Operator-analytic functions of one independent variable”, Dokl. AN SSSR, 112:6 (1957), 1008–1011 (in Russian) | MR | Zbl

[376] M. K. Fage, “Integral representations of operator-analytic functions of one independent variable”, Dokl. AN SSSR, 115:5 (1957), 874–877 (in Russian) | MR | Zbl

[377] M. K. Fage, “Operator-analytic functions of one independent variable”, Tr. Mosk. mat. ob-va, 7, 1958, 227–268 (in Russian) | MR | Zbl

[378] M. K. Fage, “Integral representations of operator-analytic functions of one independent variable”, Tr. Mosk. mat. ob-va, 8, 1958, 3–48 (in Russian) | MR | Zbl

[379] M. K. Fage, Operator-Analytic Functions of One Independent Variable, Izd-vo L'vov Univ., L'vov, 1959 (in Ukraine)

[380] D. K. Fage, N. I. Nagnibida, Problem of Equivalency of Ordinary Differential Operators, Nauka, Novosibirsk, 1977 (in Russian)

[381] L. D. Faddeev, “Inverse problem of quantum scattering theory. 1”, Usp. mat. nauk, 14:4 (1959), 57–119 (in Russian) | MR | Zbl

[382] L. D. Faddeev, “Expansion in eigenfunctions of the Laplace operator on a fundamental domain of a discrete group on the Lobachevskiy plane”, Tr. Mosk. mat. ob-va, 17, 1967, 323–350 (in Russian) | MR | Zbl

[383] L. D. Faddeev, “Inverse problem of quantum scattering theory. 2”, Itogi nauki i tekhn. Sovrem. probl. mat., 3, 1974, 93–180 (in Russian) | MR | Zbl

[384] V. E. Fedorov, “Nonlocal problem for degenerated evolution equations on semiaxis”, Mat. zametki SVFU, 22:1 (2015), 35–43 (in Russian) | Zbl

[385] V. E. Fedorov, D. M. Gordievskikh, M. V. Plekhanova, “Equations in Banach spaces with a degenerated operator under fractional differentiation symbol”, Diff. uravn., 51:10 (2015), 1367–1375 (in Russian) | DOI | Zbl

[386] V. G. Fetisov, “Operators and equations in locally bounded domains”, Investigations on Functional Analysis and Its Applications, Nauka, Moscow, 2005, 249–292 (in Russian)

[387] M. K. Fishman, “On equivalence of some linear operators in analytic space”, Mat. sb., 68:1 (1965), 63–74 (in Russian) | MR | Zbl

[388] K. Friedrichs, Perturbation of Spectra in Hilbert Space, Russian translation, Mir, Moscow, 1968

[389] A. Kh. Khanmamedov, “Transmutations for perturbed Hill difference equation and one their application”, Sib. mat. zh., 44:4 (2003), 926–937 (in Russian) | MR | Zbl

[390] G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Russian translation, Inostr. lit., Moscow, 1948

[391] I. G. Khachatryan, “On transmutations for higher-order differential equations”, Izv. AN Armen. SSR. Ser. Mat., 13:3 (1978), 215–236 (in Russian) | MR

[392] I. G. Khachatryan, “On transmutations for higher-order differential equations preserving asymptotics of solutions”, Izv. AN Armen. SSR. Ser. Mat., 14:6 (1979), 424–445 (in Russian) | MR | Zbl

[393] Kan Cher Khe, Singular boundary-value problems for equations of mathematical physics with Bessel operators, Doctoral Thesis, Khabarovsk, 1991 (in Russian)

[394] J. Hale, Theory of Functional Differential Equations, Russian translation, Mir, Moscow, 1984

[395] S. Helgason, The Radon Transform, Russian translation, Mir, Moscow, 1983 | MR

[396] S. Helgason, Groups and Geometric Analysis, Russian translation, Mir, Moscow, 1987 | MR

[397] L. Hörmander, Linear Partial Differential Operators, Russian translation, Mir, Moscow, 1965

[398] R. Horn, Matrix Analysis, Russian translation, Mir, Moscow, 1989

[399] A. P. Khromov, “Finite-dimensional perturbations of Volterra operators”, Sovrem. mat. Fundam. napravl., 10, 2004, 3–163 (in Russian) | MR | Zbl

[400] V. A. Chernyatin, Substantiation of the Fourier Method in Mixed Problem for Partial Differential Equations, MGU, Moscow, 1991 (in Russian)

[401] K. Chadan, P. Sabatier, Inverse Problems in Quantum Scattering Theory, Russian translation, Mir, Moscow, 1980

[402] V. P. Shatskiy, “On hyperbolic systems with two planes of singularities”, Dokl. AN SSSR, 242:1 (1978), 56–59 (in Russian) | MR

[403] V. P. Shatskiy, “On one boundary-value problem for singular symmetric systems of odd order”, Dokl. AN SSSR, 243:4 (1979), 806–809 (in Russian) | MR

[404] V. P. Shatskiy, “On some degenerating first-order systems in domains with characteristic part of the boundary”, Dokl. AN SSSR, 262:6 (1982), 1332–1335 (in Russian) | MR

[405] E. L. Shishkina, “Generalized weight function $r^\gamma$”, Vestn. VGU. Ser. Fiz. Mat., 2006, no. 1, 215–221 (in Russian) | Zbl

[406] E. L. Shishkina, “An identity for iterated weighted spherical means generated by generalized shift”, Proc. Sci. Conf. “Gertsenovskie chteniya-2013”, 66, RGPU im. A. I. Gertsena, Saint-Petersburg, 2013, 143–145 (in Russian)

[407] E. L. Shishkina, “On properties of one averaging kernel in a weighted Lebesgue class”, Nauch. vedom. Belgorod. gos. un-ta. Ser. Mat. Fiz., 42:6 (2016), 12–19 (in Russian)

[408] E. L. Shishkina, “Weighted generalized functions corresponding to quadratic form with complex coefficients”, Chelyabinsk. fiz.-mat. zh., 2:1 (2017), 88–98 (in Russian) | MR

[409] I. A. Shishmarev, Introduction to Theory of Elliptic Equations, MGU, Moscow, 1979 (in Russian)

[410] R. Ya. Shostak, “Alexey Vasilievich Letnikov”, Historical-Mathematical Investigations, Trudy seminara MGU po istorii matematiki, 5, GITTL, Moscow–Leningrad, 1952, 167–238 (in Russian) | MR

[411] L. E. El'sgol'ts, Introduction to the Theory of Differential Equations with Deviating Agrument, Nauka, Moscow, 1964 (in Russian)

[412] L. E. El'sgol'ts, S. B. Norkin, Introduction to the Theory of Differential Equations with Deviating Agrument, Nauka, Moscow, 1971 (in Russian)

[413] T. G. Ergashev, “Fourth potential of double layer for generalized two-axial-symmetric Helmholtz equation”, Vestn. Tomsk. gos. un-ta. Mat. i mekh., 2017, no. 50, 45–56 (in Russian) | DOI

[414] V. A. Yurko, Introduction to Theory of Inverse Spectral Problems, Nauka, Moscow, 2007 (in Russian)

[415] G. N. Yakovlev, “Unbounded solutions of degenerating elliptic equations”, Tr. MIAN, 117, 1978, 312–320 (in Russian) | MR | Zbl

[416] A. I. Yanushauskas, Analytical Theory of Elliptic Equations, Nauka, Novosibirsk, 1979 (in Russian)

[417] O. E. Yaremko, The Transmutations Method in Problems of Mathematical Modelling, Penz. Gos. un-t, Penza, 2012 (in Russian)

[418] V. Ya. Yaroslavtseva, “On one class of transmutations and its application to differential equations”, Dokl. AN SSSR, 227:4 (1976), 816–819 (in Russian) | MR

[419] V. Ya. Yaroslavtseva, Nonhomogeneous boundary-value problem in a half-space for one class of singular equations, Submitted to Diff. uravn., 1989 (in Russian)

[420] Ali I., Kiryakova V., Kalla S. L., “Solutions of fractional multi-order integral and differential equations using a Poisson-type transform”, J. Math. Anal. Appl., 269:1 (2002), 172–199 | DOI | MR | Zbl

[421] Almalki F., Kisil V., Geometric dynamics of a harmonic oscillator, non-admissible mother wavelets and squeezed states, 2018, arXiv: 1805.01399v1

[422] Andrews G. E., Askey R., Roy R., Special functions, Cambridge University Press, Cambridge, 1999 | MR | Zbl

[423] Antimirov M. Ya., Kolyshkin A. A., Vaillancourt R., Applied integral transforms, Am. Math. Soc., Providence, 1993 | MR | Zbl

[424] Appell P., Kampe de Feriet J., Fonctions hypergeometriques et hyperspheriques; polynomes d'Hermite, Gauthier-Villars, Paris, 1926 | Zbl

[425] Connett W. C., Gebuhrer M.-O., Schwartz F. L. (eds.), Applications of hypergroups and related measure algebras, Am. Math. Soc., Providence, 1995 | MR | Zbl

[426] Baccar C., Hamadi N. B., Achdi L. T., “Inversion formulas for Riemann–Liouville transform and its dual associated with singular partial differential operators”, Int. J. Math. Math. Sci., 2006 (2006), Art. ID 86238, 26 pp. | DOI | MR

[427] Bailey W. N., Generalized hypergeometric series, Stechert-Hafner Service Agency, New York–London, 1964 | MR

[428] Bajlekova E. G., “Subordination principle for fractional evolution equations”, Fract. Calc. Appl. Anal., 3:3 (2000), 213–230 | MR

[429] Bajlekova E. G., Fractional evolution equations in Banach spaces, Thesis, Technische Universiteit Eindhoven, 2001 | MR | Zbl

[430] Bers L., “On a class of differential equations in mechanics of continua”, Quart. Appl. Math., 5:1 (1943), 168–188 | DOI | MR

[431] Bers L., “A remark on an applications of pseudo-analytic functions”, Amer. J. Math., 78:3 (1956), 486–496 | DOI | MR | Zbl

[432] Bers L., Gelbart A., “On a class of functions defined by partial differential equations”, Trans. Am. Math. Soc., 56 (1944), 67–93 | DOI | MR | Zbl

[433] Bourgain J., Kenig C. E., “On localization in the continuous Anderson–Bernoulli model in higher dimension”, Invent. Math., 161:2 (2005), 389–426 | DOI | MR | Zbl

[434] Bozhinov N., Convolution representations of commutants and multipliers, Publishing House Bulgarian Acad. Sci., Sofia, 1988 | MR

[435] Braaksma B. L. J., Asymptotic expansions and analytic continuation for a class of Barnes-integrals, Noordhoff, Groningen, 1963

[436] Bragg Z. R., Dettman J. W., “An operator calculus for related partial differential equations”, J. Math. Anal. Appl., 22:2 (1968), 261–271 | DOI | MR | Zbl

[437] Bragg Z. R., Dettman J. W., “Related problems in partial differential equations”, Bull. Am. Math. Soc., 74:2 (1968), 375–378 | DOI | MR | Zbl

[438] Bragg Z. R., Dettman J. W., “A class of related Dirichlet and initial value problems”, Proc. Am. Math. Soc., 21:1 (1969), 50–56 | DOI | MR | Zbl

[439] Bresters D. W., “On a generalized Euler–Poisson–Darboux equation”, SIAM J. Math. Anal., 9:5 (1978), 924–934 | DOI | MR | Zbl

[440] Bullen P. S., Handbook of means and their inequalities, Kluwer Academic Publishers, Dordrecht–London, 2003 | MR | Zbl

[441] Bullen P. S., Mitrinović D. S., Vasić P. M., Means and their inequalities, D. Reidel, Dordrecht, 1988 | MR | Zbl

[442] Buschman R. G., “An inversion integral for a Legendre transformation”, Amer. Math. Monthly, 69:4 (1962), 288–289 | DOI | MR | Zbl

[443] Buschman R. G., “An inversion integral for a general Legendre transformation”, SIAM Review, 5:3 (1963), 232–233 | DOI | MR | Zbl

[444] Campos H., Kravchenko V. V., Torba S. M., “Transmutations, $L$-bases and complete families of solutions of the stationary Schrödinger equation in the plane”, J. Math. Anal. Appl., 389:2 (2012), 1222–1238 | DOI | MR | Zbl

[445] Carroll R. W., Transmutation and operator differential equations, North Holland, Amsterdam–New York–Oxford, 1979 | MR | Zbl

[446] Carroll R. W., Transmutation, scattering theory and special functions, North Holland, Amsterdam–New York–Oxford, 1982 | MR | Zbl

[447] Carroll R. W., Transmutation theory and applications, North Holland, Amsterdam–New York, 1986 | MR

[448] Carroll R. W., Topics in soliton theory, North Holland, 1991 | MR | Zbl

[449] Carroll R. W., Calculus revisited, Springer, Dordrecht–Boston–London, 2002 | MR

[450] Carroll R. W., Boumenir A., Toward a general theory of transmutation, 1995, arXiv: funct-an/9501006

[451] Carroll R. W., Showalter R. E., Singular and degenerate Cauchy problems, Academic Press, N.Y., 1976 | MR

[452] Castillo-Pérez C. R., Kravchenko V. V., Torba S. M., “Spectral parameter power series for perturbed Bessel equations”, Appl. Math. Comput., 220 (2013), 676–694 | MR | Zbl

[453] Chadan K., Cotton D., Paivarinta L., Rundell W., An introduction to inverse scattering and inverse spectral problems, SIAM, 1997 | MR | Zbl

[454] Chebli H., “Opérateurs de translation généralises et semigroupes de convolution”, Springer Lect. Notes, 404 (1974), 35–59 | MR | Zbl

[455] Chebli H., Positivité des opérateurs de “translation généralises” associé à un opérateur de Sturm–Liouville et quelques applications á l'analyse harmonique, Thesis, Strasbourg, 1974 | MR

[456] Chebli H., “Sur un thèoréme de Paley–Winer associé à la décomposition spectrale d'un opérateur de Sturm–Liouville sur $(0,\infty)$”, J. Funct. Anal., 17 (1974), 447–461 | DOI | MR | Zbl

[457] Chebli H., “Thèoréme de Paley–Winer associé à un opérateur différentiel singulier sur $(0,\infty)$”, J. Math. Pures Appl., 58 (1979), 1–19 | MR | Zbl

[458] Chebli H., Fitouhi A., Hamza M. M., “Expansion in series of Bessel functions and transmutations for perturbed Bessel operators”, J. Math. Anal. Appl., 181:3 (1994), 789–802 | DOI | MR | Zbl

[459] Cheikh B., “Relations between harmonic analysis associated with two differential operators of different orders”, J. Comput. Appl. Math., 153:1 (2003), 61–71 | DOI | MR | Zbl

[460] Colton D., Solution of boundary value problems by the method of integral operators, Pitman Press, London, 1976 | MR | Zbl

[461] Colton D., Analytic theory of partial differential equations, Pitman Press, London, 1980 | MR | Zbl

[462] Copson E. T., “On a singular boundary value problem for an equation of hyperbolic type”, Arch. Ration. Mech. Anal., 1 (1957), 349–356 | DOI | MR

[463] Copson E. T., “On the Riemann–Green function”, Arch. Ration. Mech. Anal., 1 (1957/58), 324–348 | DOI | MR

[464] Copson E. T., Partial differential equations, Cambridge University Press, London–New York–Melbourne, 1975 | MR | Zbl

[465] Copson E. T., Erdélyi A., “On a partial differential equation with two singular lines”, Arch. Ration. Mech. Anal., 2:1 (1958), 76–86 | DOI | MR | Zbl

[466] Darboux G., Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, v. 2, Gauthier-Villars, Paris, 1915 | MR

[467] Davey B., Kenig C., Wang J.-N., “The Landis conjecture for variable coeffcient second-order elliptic PDEs”, Trans. Am. Math. Soc., 369:11 (2017), 8209–8237 | DOI | MR | Zbl

[468] Deans S. R., The Radon transform and some of its applications, Dover, Mineola–New York, 1990 | MR

[469] Delsarte J., “Sur certaines transformation fonctionnelles relative aux équations linéares aux dèrivées partielles du seconde ordre”, C. R. Acad. Sci., 206 (1938), 1780–1782

[470] Delsarte J., “Sur une extension de la formule de Taylor”, J. Math. Pures Appl., 17 (1938), 217–230

[471] Delsarte J., “Une extension nouvelle de la théory de fonction presque périodiques de Bohr”, Acta Math., 69 (1939), 259–317 | DOI | MR

[472] Delsarte J., “Hypergroupes et operateurs de permutation et de transmutation”, Colloq. Internat. Centre Nat. Rech. Sci., 71 (1956), 29–44 | MR

[473] Delsarte J., Lectures on topics in mean periodic functions and the two-radius theorem, Tata Inst. Fundam. Research, Bombay, 1961

[474] Delsarte J., Lions J. L., “Transmutations d'opérateurs différentiels dans le domaine complexe”, Comment. Math. Helv., 32 (1957), 113–128 | DOI | MR | Zbl

[475] Delsarte J., Lions J. L., “Moyennes généralisées”, Comment. Math. Helv., 34 (1959), 59–69 | DOI | MR

[476] Diaz J. B., Weinberger H. F., “A solution of the singular initial value problem for the Euler–Poisson equation”, Proc. Am. Math. Soc., 4 (1953), 703–715 | MR | Zbl

[477] Dimovski I., Convolutional calculus, Kluwer Acad. Publ., Dordrecht, 1990 | MR | Zbl

[478] Dimovski I., Hristov V., Sifi M., “Commutants of the Dunkl operators in $C(\mathbb R)$”, Fract. Calc. Appl. Anal., 9:3 (2006), 195–213 | MR | Zbl

[479] Dimovski I., Kiryakova V. S., “Transmutations, convolutions and fractional powers of Bessel-type operators via Meijer $G$-functions”, Proc. conf. “Complex Anal. and Appl.” (1983, Sofia), Varna, 1985, 45–66 | MR

[480] Dimovski I., Kiryakova V. S., “Generalized Poisson transmutations and corresponding representations of hyper-Bessel functions”, C. R. Acad. Bulgare Sci., 39:10 (1986), 29–32 | MR | Zbl

[481] Djrbashian M. M., Harmonic analysis and boundary value problems in the complex domain, Birkhäuser, Basel–Boston–Berlin, 1993 | MR | Zbl

[482] Drabek P., Kufner A., Nicolosi F., Quasilinear elliptic equations with degenerations and singularities, Walter de Gruyter, Berlin–New York, 1997 | MR | Zbl

[483] Dunkl Ch., “Differential-difference operators associated to reflection groups”, Trans. Am. Math. Soc., 311 (1989), 167–183 | DOI | MR | Zbl

[484] Dunkl Ch., “Intertwining operators associated to the group S3”, Trans. Am. Math. Soc., 347 (1995), 3347–3374 | MR | Zbl

[485] Dunkl Ch., An Intertwining operator for the group B2, 2006, arXiv: math.CA/0607823 | MR

[486] Dwork B., Generalized hypergeometric functions, Oxford, 1990 | MR

[487] Dwork B., Gerotto S., Sullivan F. J., Introduction to $G$-functions, Princeton, 1994 | MR

[488] Dzrbashian M. M., Harmonic analysis and boundary value problems in the complex domain, Birkhäuser, Basel–Boston–Berlin, 1993 | MR

[489] Eidelman S. D., Ivasyshen S. D., Kochubei A. N., Analytic methods in the theory of differential and pseudo-differential equations of parabolic type, Springer, Basel, 2004 | MR

[490] Erdélyi A., “On fractional integration and its application to the Hankel transforms”, Quart. J. Math. (Oxford), 11 (1940), 293–303 | DOI | MR

[491] Erdélyi A., Some applications of fractional integration, Math. Note, 316, D1-82-0286, Boeing Sci. Res. Labor., 1963

[492] Erdélyi A., “An integral equation involving Legendre functions”, SIAM Rev., 12:1 (1964), 15–30 | MR | Zbl

[493] Erdélyi A., “An application of fractional integrals”, J. Analyse Math., 14 (1965), 113–126 | DOI | MR | Zbl

[494] Erdélyi A., “Some integral equations involving finite parts of divergent integrals”, Glasgow Math. J., 8:1 (1967), 50–54 | DOI | MR | Zbl

[495] Erdélyi A., “On the Euler–Poisson–Darboux equation”, J. Analyse Math., 23 (1970), 89–102 | DOI | MR | Zbl

[496] Estrada R., Rubin B., Null spaces of Radon transforms, 2015, arXiv: 1504.03766v1 | MR

[497] Euler L., “Institutiones calculi integralis”, Opera Omnia., 1:13 (1914), 212–230

[498] Exton H., Multiple hypergeometric functions and applications, John Wiley and Sons, New York, 1976 | MR | Zbl

[499] Fitouhi A., Hamza M. M., “A uniform expansion for the eigenfunction of a singular second-order differential operator”, SIAM J. Math. Anal., 21:6 (1990), 1619–1632 | DOI | MR | Zbl

[500] Fitouhi A., Jebabli I., Shishkina E., Sitnik S. M., “Applications of integral transforms composition method to wave-type singular differential equations and index shift transmutations”, Electron. J. Differ. Equ., 2018:130 (2018), 1–27 | MR

[501] Fox D. N., “The solution and Huygens' principle for a singular Cauchy problem”, J. Math. Mech., 8 (1959), 197–219 | MR | Zbl

[502] Gadjiev A. D., Selected works, ELM, Baku, 2003 | MR | Zbl

[503] Gallardo L., Triméche Kh., “Un analogue d'un theoreme de Hardy pour la transformation de Dunkl”, C. R. Math. Acad. Sci. Paris, 334 (2002), 849–854 | DOI | MR | Zbl

[504] Gallardo L., Triméche Kh., “A version of Hardy's theorem for the Dunkl transform”, J. Aust. Math. Soc., 77 (2004), 371–385 | DOI | MR | Zbl

[505] Ghergu M., Radulescu V., Singular elliptic problems. Bifurcation and asymptotic analysis, Oxford University Press, New York, 2008 | MR | Zbl

[506] Gilbert R., Function theoretic methods in partial differential equations, Academic Press, N.Y., 1969 | MR | Zbl

[507] Gilbert R., Constructive methods for elliptic equations, Springer, Berlin–Heidelberg, 1974 | MR | Zbl

[508] Gilbert R., Begehr H., Transformations, transmutations and kernel functions, v. 1–2, Longman, Harlow, 1992 | MR | Zbl

[509] Glaeske H. J., Prudnikov A. P., Skornik K. A., Operational calculus and related topics, Chapman Hall/CRC, Boca Raton, 2006 | MR | Zbl

[510] Golenia J., Samoilenko A. M., Prykarpatsky Ya. A., Prykarpatsky A. K., The general differential-geometric structure of multidimensional Delsarte transmutation operators in parametric functional spaces and their applications in soliton theory/ Part 2, 2004, arXiv: math-ph/0404016

[511] Gorenflo R., Kilbas A. A., Mainardi F., Rogosin S. V., Mittag-Leffler functions, related topics and applications, Springer, Heidelberg–New York–Dordrecht–London, 2014 | MR | Zbl

[512] Gorenflo R., Mainardi F., “Fractional calculus: integral and differential equations of fractional order”, Fractals and fractional calculus in continuum mechanics, Springer, Wien–New York, 1997, 223–278 | DOI | MR

[513] Hasanov A., Karimov E. T., “Fundamental solutions for a class of three-dimensional elliptic equations with singular coefficients”, Appl. Math. Lett., 22 (2009), 1828–1832 | DOI | MR | Zbl

[514] Higgins T. P., “A hypergeometric function transform”, J. SIAM, 12:3 (1964), 601–612 | MR

[515] Hille E., Tamarkin J. D., “On the theory of linear integral equations”, Ann. Math., 31 (1930), 479–528 | DOI | MR | Zbl

[516] Holzleitner M., Transformation operators for spherical Schrödinger operators, 2018, arXiv: 1805.10526v1

[517] Jafford K., “Inversion of the Lions transmutation operators using generalized wavelets”, Appl. Comput. Harmon. Anal., 4:1 (1997), 97–112 | DOI | MR

[518] Kamoun L., Sifi M., “Bessel–Struve intertwining operator and generalized Taylor series on the real line”, Integral Transforms Spec. Funct., 16:1 (2005), 39–55 | DOI | MR | Zbl

[519] Karimov S. T., “Multidimensional generalized Erdélyi–Kober operator and its application to solving Cauchy problems for differential equations with singular coefficients”, Fract. Calc. Appl. Anal., 18:4 (2015), 845–861 | DOI | MR | Zbl

[520] Karlsson P. W., Srivastava H. M., Multiple Gaussian hypergeometric series, Ellis Horwood, New York, 1985 | MR | Zbl

[521] Karp D., López J. L., “Representations of hypergeometric functions for arbitrary values of the parameters and their use”, J. Approx. Theory, 218 (2017), 42–70 | DOI | MR | Zbl

[522] Karp D., López J. L., A class of Meijer's $G$ functions and further representations of the generalized hypergeometric functions, 2018, arXiv: 1801.08670v1

[523] Karp D., Prilepkina E., “Hypergeometric functions as generalized Stieltjes transforms”, J. Math. Anal. Appl., 393:2 (2012), 348–359 | DOI | MR | Zbl

[524] Karp D., Prilepkina E., “Completely monotonic gamma ratio and infinitely divisible $H$-function of Fox”, Comput. Methods Funct. Theory, 16:1 (2016), 135–153 | DOI | MR | Zbl

[525] Karp D., Prilepkina E., “Hypergeometric differential equation and new identities for the coefficients of Nørlund and Bühring”, SIGMA Symmetry Integrability Geom. Methods Appl., 12 (2016), 052 | DOI | MR | Zbl

[526] Karp D., Prilepkina E., “Some new facts concerning the delta neutral case of Fox's $H$-function”, Comput. Methods Funct. Theory., 17:2 (2017), 343–367 | DOI | MR | Zbl

[527] Karp D., Savenkova A., Sitnik S. M., “Series expansions for the third incomplete elliptic integral via partial fraction decompositions”, J. Comput. Appl. Math., 207:2 (2007), 331–337 | DOI | MR | Zbl

[528] Karp D., Sitnik S. M., “Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity”, J. Comput. Appl. Math., 205:1 (2007), 186–206 | DOI | MR | Zbl

[529] Karp D., Sitnik S. M., “Inequalities and monotonicity of ratios for generalized hypergeometric function”, J. Approx. Theory., 161:1 (2009), 337–352 | DOI | MR | Zbl

[530] Karp D., Sitnik S. M., “Log-convexity and log-concavity of hypergeometric-like functions”, J. Math. Anal. Appl., 364:2 (2010), 384–394 | DOI | MR | Zbl

[531] Kenig C. E., “Some recent quantitative unique continuation theorems”, Sémin. Équ. Dériv. Partielles. Éc. Polytech. Cent. Math. Palaiseau, 2005–2006 (2006), XX1–XX10 | MR | Zbl

[532] Kenig C. E., Silvestre L., Wang J. N., “On Landis' conjecture in the plane”, Commun. Part. Differ. Equ., 40:4 (2015), 766–789 | DOI | MR | Zbl

[533] Kilbas A. A., Saigo M., $H$-transforms. Theory and applications, Chapman Hall, CRC, Boca Raton, 2004 | MR | Zbl

[534] Kilbas A. A., Skoromnik O. V., “Integral transforms with the Legendre function of the first kind in the kernels on $\mathcal L_{\nu,r}$-spaces”, Integral Transforms Spec. Funct., 20:9 (2009), 653–672 | DOI | MR | Zbl

[535] Kilbas A. A., Srivastava H. M., Truhillo J. J., Theory and applications of fractional differential equations, Elsevier, Amsterdam, etc., 2006 | MR | Zbl

[536] Kilbas A. A., Truhillo J. J., “Differential equations of fractional order: methods, results and problems. Part I”, Appl. Anal., 78:1–2 (2001), 153–192 | DOI | MR | Zbl

[537] Kilbas A. A., Truhillo J. J., “Differential equations of fractional order: methods, results and problems. Part II”, Appl. Anal., 81:2 (2001), 435–493 | DOI | MR

[538] Kilbas A. A., Zhukovskaya N. V., “Euler-type non-homogeneous differential equations with three Liouville fractional derivatives”, Fract. Calc. Appl. Anal., 12:2 (2009), 205–234 | MR | Zbl

[539] Kiryakova V., “An explanation of Stokes phenomenon by the method of transmutations”, Proc. conf. Diff. Equations and Appl., Rousse, 1982, 349–353 | Zbl

[540] Kiryakova V., Generalized fractional calculus and applications, Longman, Harlow, 1994 | MR | Zbl

[541] Kiryakova V., “Applications of the generalized Poisson transformation for solving hyper-Bessel differential equations”, Godishnik VUZ. Appl. Math., 22:4 (1986), 129–140 (in Bulgarian) | MR

[542] Kiryakova V., “All the special functions are fractional differintegrals of elementary functions”, J. Phys. A Math. Gen., 30:14 (1997), 5085–5103 | DOI | MR | Zbl

[543] Kiryakova V., “Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus”, J. Comput. Appl. Math., 118:1–2 (2000), 241–259 | DOI | MR | Zbl

[544] V. Kiryakova, “The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus”, Computers Math. Appl., 59:5 (2010), 1885–1895 | DOI | MR | Zbl

[545] H. Kober, “On fractional integrals and derivatives”, Quart. J. Math. (Oxford), 11 (1940), 193–211 | DOI | MR | Zbl

[546] H. Kober, “On a theorem of Schur and on fractional integrals of purely imaginary order”, Trans. Am. Math. Soc., 50:1 (1941), 160–174 | MR | Zbl

[547] W. Koepf, Hypergeometric summation. An algorithmic approach to summation and special function identities, Vieweg, Wiesbaden, 1998 | MR | Zbl

[548] V. Kokilashvili, A. Meskhi, H. Rafeiro, S. Samko, Integral operators in non-standard function spaces, v. 1–2, Birkhäuser/Springer, Basel, 2016 | MR | Zbl

[549] T. H. Koornwinder, “Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators”, SIGMA Symmetry Integrability Geom. Methods Appl., 11 (2015), 074 | DOI | MR | Zbl

[550] V. V. Kravchenko, Applied pseudoanalytic function theory, Birkhäuser, Basel, 2009 | MR | Zbl

[551] V. V. Kravchenko, “Construction of a transmutation for the one-dimensional Schrödinger operator and a representation for solutions”, Appl. Math. Comput., 328 (2018), 75–81 | DOI | MR

[552] V. V. Kravchenko, L. J. Navarro, S. M. Torba, “Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions”, Appl. Math. Comput., 314:1 (2017), 173–192 | MR

[553] V. V. Kravchenko, J. A. Otero, S. M. Torba, “Analytic approximation of solutions of parabolic partial differential equations with variable coefficients”, Adv. Math. Phys., 2017 (2017), 2947275 | DOI | MR | Zbl

[554] V. V. Kravchenko, S. M. Torba, “Transmutations for Darboux transformed operators with applications”, J. Phys. A Math. Theor., 45:7 (2012), 075201 | DOI | MR | Zbl

[555] V. V. Kravchenko, S. M. Torba, “Analytic approximation of transmutation operators and applications to highly accurate solution of spectral problems”, J. Comput. Appl. Math., 275 (2015), 1–26 | DOI | MR | Zbl

[556] V. V. Kravchenko, S. M. Torba, “Construction of transmutation operators and hyperbolic pseudoanalytic functions”, Complex Anal. Oper. Theory, 9:2 (2015), 379–429 | DOI | MR | Zbl

[557] V. V. Kravchenko, S. M. Torba, “Asymptotics with respect to the spectral parameter and Neumann series of Bessel functions for solutions of the one-dimensional Schrödinger equation”, J. Math. Phys., 58:12 (2017), 122107 | DOI | MR | Zbl

[558] V. V. Kravchenko, S. M. Torba, “A Neumann series of Bessel functions representation for solutions of Sturm–Liouville equations”, Calcolo, 55:1 (2018), Paper No. 11 | DOI | MR | Zbl

[559] V. V. Kravchenko, S. M. Torba, R. Castillo-Pérez, “A Neumann series of Bessel functions representation for solutions of perturbed Bessel equations”, Appl. Anal., 97:5 (2018), 677–704 | DOI | MR | Zbl

[560] V. V. Kravchenko, S. M. Torba, K. V. Khmelnytskaya, “Transmutation operators: construction and applications”, Proc. of the 17th Int. Conf. on Comput. and Math. Methods in Sci. and Eng. CMMSE-2017, Cadiz, Andalucia, España, 2017, 1198–1206 | MR

[561] A. Kufner, L. Maligranda, L.-E. Persson, The Hardy inequality. About its history and some related results, Vydavatelský Servis, Pilsen, 2007 | MR | Zbl

[562] B. M. Levitan, “Transmutation operators and the inverse spectral problem”, Contemp. Math., 183 (1995), 237–244 | DOI | MR

[563] P. Lindqvist, Notes on the $p$-Laplace equation, Report, 102, Univ. of Jyväskylä. Dep. of Math. and Stat., 2006 | MR | Zbl

[564] J. L. Lions, “Opérateurs de Delsarte et problème mixte”, Bull. Soc. Math. France, 84 (1956), 9–95 | DOI | MR | Zbl

[565] J. L. Lions, “Quelques applications d'opérateurs de transmutations”, Colloques Internat., Nancy, 1956, 125–142 | MR

[566] J. L. Lions, Equations differentielles operationnelles et problémes aux limites, Springer, Berlin–Göttingen–Heidelberg, 1961 | MR

[567] E. R. Love, “Some integral equations involving hypergeometric functions”, Proc. Edinb. Math. Soc., 15:3 (1967), 169–198 | DOI | MR | Zbl

[568] E. R. Love, “Two more hypergeometric integral equations”, Proc. Cambridge Phil. Soc., 63:4 (1967), 1055–1076 | DOI | MR | Zbl

[569] J. S. Lowndes, “An application of some fractional integrals”, Glasg. Math. J., 20:1 (1979), 35–41 | DOI | MR | Zbl

[570] J. S. Lowndes, “On some generalizations of Riemann–Liouville and Weil fractional integrals and their applications”, Glasg. Math. J., 22:2 (1981), 73–80 | DOI | MR

[571] J. S. Lowndes, “Cauchy problems for second order hyperbolic differential equations with constant coefficients”, Proc. Edinb. Math. Soc., 26:3 (1983), 97–105 | DOI | MR

[572] D. Ludwig, “The Radon transform on Euclidean space”, Math. Methods Appl. Sci., 19 (1966), 49–81 | MR | Zbl

[573] Y. L. Luke, The special functions and their approximations, v. 1, Academic Press, New York–London, 1969 | MR | Zbl

[574] Y. L. Luke, Mathematical functions and their approximations, Academic Press, New York–San Francisco–London, 1975 | MR | Zbl

[575] V. B. Matveev, “Intertwining relations between the Fourier transform and discrete Fourier transform, the related functional identities and beyond”, Inverse Problems, 17 (2001), 633–657 | DOI | MR | Zbl

[576] V. B. Matveev, V. B. Salle, Darboux–Backlund transformations and applications, Springer, N.Y., 1991 | MR

[577] A. C. McBride, Fractional calculus and integral transforms of generalized functions, Pitman, San Francisco–London–Melbourne, 1979 | MR | Zbl

[578] Kh. Mehrez, S. M. Sitnik, “On monotonicity of ratios of some $q$-hypergeometric functions”, Mat. Vesn., 68:3 (2016), 225–231 | MR | Zbl

[579] Kh. Mehrez, S. M. Sitnik, “Proofs of some conjectures on monotonicity of ratios of Kummer, Gauss and generalized hypergeometric functions”, Analysis (Munich), 36:4 (2016), 263–268 | MR | Zbl

[580] Kh. Mehrez, S. M. Sitnik, “Functional Inequalities for the Mittag-Leffler Functions”, Results Math., 72:1–2 (2017), 703–714 | DOI | MR | Zbl

[581] E. Mitidieri, J. Serrin, V. Radulescu (eds.), Recent trends in nonlinear partial differential equations I: Evolution problems, Am. Math. Soc., Providence, 2013 | Zbl

[582] E. Mitidieri, J. Serrin, V. Radulescu (eds.), Recent trends in nonlinear partial differential equations, v. II, Stationary Problems, Am. Math. Soc., Providence, 2013 | Zbl

[583] D. S. Mitrinović, J. E. Pečarić, A. M. Fink, Classical and new inequalities in analysis, Kluwer, Dordrecht, 1993 | MR | Zbl

[584] A. B. Muravnik, “On weighted norm estimates for the mixed Fourier–Bessel transforms on non-negative functions”, Integral methods in science and engineering, v. 1, Analytic methods, Longman, Harlow, 1997, 119–123 | MR | Zbl

[585] A. B. Muravnik, “Fourier–Bessel transformation of measures and singular differential operators”, Paul Erdős and his mathematics, J'anos Bolyai Math. Soc., Budapest, 1999, 182–184 | MR | Zbl

[586] A. B. Muravnik, “Fourier–Bessel transformation of measures with several special variables and properties of singular differential equations”, J. Korean Math. Soc., 37:6 (2000), 1043–1057 | MR | Zbl

[587] A. B. Muravnik, “Fourier–Bessel transformation of compactly supported non-negative functions and estimates of solutions of singular differential equations”, Funct. Differ. Equ., 8:3–4 (2001), 353–363 | MR | Zbl

[588] A. B. Muravnik, “Fourier–Bessel transformation of measures and singular differential equations”, \it Recent progress in functional analysis, Elsevier, Amsterdam, 2001, 335–345 | MR | Zbl

[589] A. B. Muravnik, “Nonclassical Cauchy problem for singular parabolic integro-differential equations”, Russ. J. Math. Phys., 9:3 (2002), 300–314 | MR | Zbl

[590] A. B. Muravnik, “On stabilization of solutions of elliptic equations containing Bessel operators”, Integral methods in science and engineering. Analytic and numerical techniques, Birkhäuser, Boston, 2004, 157–162 | DOI | MR | Zbl

[591] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, NIST handbook of mathematical functions, Cambridge University Press, Cambridge, 2010 | MR | Zbl

[592] B. Opic, A. Kufner, Hardy-type inequalities, Longman, Harlow, 1990 | MR | Zbl

[593] H. Ozaktas, Z. Zalevsky, M. Kutay, The fractional Fourier transform: with applications in optics and signal processing, John Wiley Sons, Chishester ets., 2001

[594] J. Paneva-Konovska, From Bessel to multi-index Mittag-Leffler functions, World Scientific, London, 2016 | MR

[595] M. Petkovšek, H. S. Wilf, D. Zeilberger, $A=B$, A. K. Peters, Wellesley, 1996 | MR | Zbl

[596] S. Pike, P. Sabatier, Scattering. Scattering and inverse scattering in pure and applied science, v. 1–2, Academic Press, San Diego, 2002 | MR | Zbl

[597] S. D. Poisson, “Mémoire sur l'intégration des équations linéaires aux différences partielles”, J. Éc. Roy. Polytech., 19:12 (1823), 215–248

[598] A. D. Poularicas (ed.), The transforms and applications handbook, CRC Press, Boca Raton, 2010 | MR

[599] J. Prüss, Evolutionary integral equations and applications, Birkhäuser, Basel, 2012 | MR | Zbl

[600] S. G. Pyatkov, Operator theory. Nonclassical problems, VSP, Utrecht, 2002 | MR | Zbl

[601] V. Radulescu, D. Repovs, Partial differential equations with variable exponents: Variational methods and qualitative analysis, CRC Press, Boca Raton, 2015 | MR | Zbl

[602] M. Riesz, “L'intégrale de Riemann–Liouville et le problème de Cauchy”, Acta Math., 81 (1949), 1–223 | DOI | MR | Zbl

[603] B. Riemann, “On the propagation of flat waves of finite amplitude”, Ouvres, OGIZ, Moscow–Leningrad, 1948, 376–395

[604] J. Rodrigues, “Operational calculus for the generalized Bessel operator”, Serdica Math. J., 15 (1989), 179–186 | MR

[605] M. Rösler, “Positivity of Dunkl's intertwining operator”, Duke Math. J., 98 (1999), 445–463 | DOI | MR | Zbl

[606] M. Rösler, “Dunkl operators: theory and applications”, Orthogonal polynomials and special functions, Springer, Berlin, 2003, 93–135 | DOI | MR | Zbl

[607] L. Rossi, The Landis conjecture with sharp rate of decay, 2018, arXiv: 1807.00341v1

[608] B. Rubin, “On the Funk–Radon–Helgason inversion method in integral geometry”, Contemp. Math., 599 (2013), 175–198 | DOI | MR | Zbl

[609] B. Rubin, Gegenbauer–Chebyshev integrals and Radon transforms, 2015, arXiv: 1410.4112v2 | MR

[610] B. Rubin, “Radon transforms and Gegenbauer–Chebyshev integrals. I”, Anal. Math. Phys., 7:2 (2017), 117–150 | DOI | MR | Zbl

[611] M. S. Salakhitdinov, A. Hasanov, “The Dirichlet problem for the generalized bi-axially symmetric Helmholtz equation”, Eurasian Math. J., 3:4 (2012), 99–110 | MR | Zbl

[612] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: theory and applications, Gordon Breach, N.Y., 1993 | MR | Zbl

[613] A. M. Samoilenko, Ya. A. Prykarpatsky, A. K. Prykarpatsky, “The generalized de Rham–Hodge theory aspects of Delsarte–Darboux type transformations in multidimension”, Cent. Eur. J. Math., 3:3 (2005), 529–557 | DOI | MR | Zbl

[614] R. T. Seely, “Extensions of $C^\infty$ functions defined in a half space”, Proc. Am. Math. Soc., 15 (1964), 625–626 | MR

[615] R. T. Seely, “Complex powers of an elliptic operator”, Proc. Sympos. Pure Math., 10 (1967), 288–307 | DOI | MR

[616] E. L. Shishkina, “Inversion of integral of $B$-potential type with density from $\Phi_\gamma$”, J. Math. Sci., 160:1 (2009), 95–102 | DOI | MR | Zbl

[617] E. L. Shishkina, “On the boundedness of hyperbolic Riesz $B$-potential”, Lith. Math. J., 56:4 (2016), 540–551 | DOI | MR | Zbl

[618] E. L. Shishkina, “Generalized Euler–Poisson–Darboux equation and singular Klein–Gordon equation”, J. Phys. Conf. Ser., 973 (2018), 1–21 | DOI

[619] E. L. Shishkina, “Properties of mixed hyperbolic $B$-potential”, Progr. Fract. Differ. Appl., 4:2 (2018), 1–16 | DOI

[620] E. L. Shishkina, “Singular Cauchy problem for the general Euler–Poisson–Darboux equation”, Open Math., 16 (2018), 23–31 | DOI | MR | Zbl

[621] E. L. Shishkina, S. M. Sitnik, “General form of the Euler–Poisson–Darboux equation and application of the transmutation method”, Electron. J. Differ. Equ., 177 (2017), 1–20 | MR

[622] E. L. Shishkina, S. M. Sitnik, “On fractional powers of Bessel operators”, J. Inequal. Spec. Funct., 8:1 (2017), 49–67 | MR

[623] J. Siersma, Thesis, Groningen, 1979

[624] S. M. Sitnik, Generalized Young and Cauchy–Bunyakowsky inequalities with applications: a survey, 2010, arXiv: 1012.3864[math.CA]

[625] S. M. Sitnik, Transmutations and applications: a survey, 2010, arXiv: 1012.3741[math.CA]

[626] S. M. Sitnik, “Some problems in the modern theory of transmutations”, Spectral theory and differential equations (STDE-2012), Int. Conf. in honor of V. A. Marchenko's 90th birthday, Kharkiv, 2012, 101–102

[627] S. M. Sitnik, “Buschman–Erdélyi transmutations, classification and applications”, Analytic methods of analysis and differential equations, Cambridge Scientific Publishers, Cambridge, 2013, 171–201

[628] S. M. Sitnik, “A short survey of recent results on Buschman–Erdélyi transmutations”, J. Inequal. Spec. Funct., 8:1 (2017), 140–157 | MR

[629] S. M. Sitnik, “Buschman–Erdélyi transmutations and applications”, Abstr. of the 8th Int. Conf. Transform Methods and Special Functions (Bulgaria, Sofia, 27–31 Aug. 2017), Inst. Math. Inf. Bulg. Acad. Sci., 2017, 59

[630] A. L. Skubachevskii, Elliptic functional differential equations and applications, Birkhäuser, Basel, 1997 | MR | Zbl

[631] L. J. Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966 | MR | Zbl

[632] I. G. Sprinkhuizen-Kuyper, “A fractional integral operator corresponding to negative powers of a certain second-order differential operator”, J. Math. Anal. Appl., 72 (1979), 674–702 | DOI | MR | Zbl

[633] G. A. Sviridyuk, V. E. Fedorov, Linear Sobolev type equations and degenerate semigroups of operators, VSP, Utrecht, 2003 | MR | Zbl

[634] Ta Li, “A new class of integral transform”, Proc. Am. Math. Soc., 11:2 (1960), 290–298 | DOI | MR | Zbl

[635] Ta Li, “A note on integral transform”, Proc. Am. Math. Soc., 12:6 (1961), 556 | MR | Zbl

[636] Kh. Triméche, “Transformation intégrale de Riemann–Liouville généralises et convergence des series de Taylor généralisées au sens de Delsarte”, Rev. Fac. Sci. Tunis, 1981, no. 1, 7–14

[637] Kh. Triméche, “Transformation intégrale de Weil et thèoréme de Paley–Winer associés à un opérateur différentiel singulier sur $(0,\infty)$”, J. Math. Pures Appl., 60 (1981), 51–98 | MR | Zbl

[638] Kh. Triméche, “Transmutation operators and mean-periodic functions associated with differential operators”, Math. Rep., 4:1 (1988), 1–14 | MR

[639] Kh. Triméche, “Inversion of the Lions transmutation operators using generalized wavelets”, Appl. Comput. Harmon. Anal., 4:1 (1997), 97–112 | DOI | MR | Zbl

[640] Kh. Triméche, Generalized harmonic analysis and wavelet packets, Gordon and Breach, Amsterdam, 2001 | MR | Zbl

[641] Kh. Triméche, “Inversion of the Dunkl intertwining operator and its dual using Dunkl wavelets”, Rocky Mountain J. Math., 32:2 (2002), 889–895 | DOI | MR

[642] N. Virchenko, “On some generalized symmetric integral operators of Buschman–Erdélyi's type”, J. Nonlinear Math. Phys., 3:3–4 (1996), 421–425 | DOI | MR | Zbl

[643] N. Virchenko, I. Fedotova, Generalized associated Legendre functions and their applications, World Scientific, Singapore, 2001 | MR | Zbl

[644] V. V. Volchkov, Integral geometry and convolution equations, Kluwer, Dordrecht, 2003 | MR | Zbl

[645] N. Wiener, “The Dirichlet problem”, J. Math. Phys. Mass. Inst. Techn., 3 (1924), 127–146 | MR | Zbl

[646] A. Weinstein, “Discontinuous integrals and generalized theory of potential”, Trans. Am. Math. Soc., 63:2 (1948), 342–354 | DOI | MR | Zbl

[647] A. Weinstein, “Generalized axially symmetric potential theory”, Bull. Am. Math. Soc., 59 (1953), 20–38 | DOI | MR | Zbl

[648] A. Weinstein, Selecta, Pitman, London–San Francisco, 1978 | MR