A survey of Buschman~--- Erdelyi transmutations

S. M. Sitnik

Čelâbinskij fiziko-matematičeskij žurnal, Tome 1 (2016) no. 4, pp. 63-93.

Résumé

This is a survey paper. In introduction some historical and priority information is given. The author's convenient classification of Buschman — Erdelyi transmutations is proposed. Based on this classification results on Buschman — Erdelyi transmutations of the first, second and third kinds are surveyed, including unitarian Sonine — Katrakhov and Poisson — Katrakhov transmutations. In the last section applications of Buschman — Erdelyi transmutations are considered.

Détail

Keywords: transmutation, integral transform, Buschman — Erdelyi transmutation, Erdelyi — Kober operator, Sonine — Poisson operator, fractional integral.

Bibliographie

[1] R. W. Carroll, Transmutation and Operator Differential Equations, North Holland, Amsterdam–New York–Oxford, 1979, 244 pp. | MR | Zbl

[2] R. W. Carroll, Transmutation, Scattering Theory and Special Functions, North Holland, Amsterdam–New York–Oxford, 1982, 457 pp. | MR | Zbl

[3] R. W. Carroll, Transmutation Theory and Applications, North Holland, Amsterdam–New York–Oxford, 1985, 351 pp. | MR | Zbl

[4] M. K. Fage, N. I. Nagnibida, Equivalence problem for ordinary linear differential operators, Nauka Publ., Novosibirsk, 1977, 280 pp. (In Russ.) | MR

[5] H. Begehr, R. P. Gilbert, Transformations, Transmutations and Kernel Functions, v. 1, Longman Sci. Technical, Harlow, 1993, 272 pp. | Zbl

[6] K. Triméche, Transmutation Operators and Mean–Periodic Functions Associated with Differential Operators, Harwood Academic Publ., Chur–London–Paris–New York–Melbourne, 1988, 283 pp. | MR

[7] S. M. Sitnik, Transmutations and Applications : a survey, arXiv: (accessed 12.12.2015) 1012.3741

[8] B. M Levitan, “Expansions into Fourier series and integrals of Bessel functions”, Russian Mathematical Surveys, 6:2 (1951), 102–143 (In Russ.) | MR | Zbl

[9] R. W. Carroll, R. E. Showalter, Singular and Degenerate Cauchy problems, Academic Press, New York–San Francisco–London, 1976, 333 pp. | MR

[10] I. H. Dimovski, Convolutional Calculus, Kluwer Academic Publ., Dordrecht–Boston–London, 1990, 184 pp. | MR | Zbl

[11] V. S. Kiryakova, Generalized Fractional Calculus and Applications, Longman Sci. Technical, Harlow, 1994, 360 pp. | Zbl

[12] V. V. Kravchenko, Applied Pseudoanalytic Function Theory, Birkhäuser Verlag, Basel–Boston–Berlin, 2009, 182 pp. | MR | Zbl

[13] J. L. Lions, Equations Differentielles Operationnelles: Et problémes aux Limites, Springer–Verlag, Berlin–Heidelberg, 1961, 291 pp. | MR

[14] I. N. Vekua, Generalized Analytic Functions, v. 25, International Series of Monographs in Pure and Applied Mathematics, Pergamon Press, Oxford, New York, 1962, xxix+668 pp. | MR | MR | Zbl

[15] I. A. Kipriyanov, Singular elliptic boundary value problems, Nauka Publ., Moscow, 1997, 198 pp. (In Russ.) | MR

[16] B. M. Levitan, Generalized translation operators and some of their applications, Israel Program for Scientific Translations, Jerusalem, 1964, 200 pp. | MR | MR | Zbl

[17] B. M. Levitan, Inverse Sturm — Liouville Problems, VNU Science Press, Utrecht, 1987, 245 pp. | MR | MR | Zbl

[18] V. A. Marchenko, Spectral theory of Sturm — Liouville operators, Naukova Dumka Publ., Kiev, 1972, 220 pp. (In Russ.) | MR

[19] V. A. Marchenko, Sturm — Liouville Operators and Applications, Rev. ed., AMS Chelsea Publishing, Providence–Rhode Island, 2011, 393 pp. | MR | MR | Zbl

[20] C. Shadan, P. Sabatier, Inverse problems in quantum scattering theory, Texts and Monographs in Physics, Springer–Verlag, New York–Berlin–Heidelberg–London–Paris–Tokyo, 1989, 499 pp. | MR | MR

[21] A. P. Khromov, “Finite-dimensional perturbations of Volterra operators”, Journal of Mathematical Sciences, 138:5 (2006), 5893–6066 | DOI | MR | Zbl

[22] I. A. Kipriyanov, “Fourier — Bessel transforms and imbedding theorems for weighted classes”, Proceedings of the Steklov Institute of Mathematics, 89, 1967, 130–213 (In Russ.) | Zbl

[23] R. G. Buschman, “An inversion integral for a Legendre transformation”, American Mathematics Month., 69:4 (1962), 288–289 | DOI | MR | Zbl

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

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

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

[27] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: theory and applications, Gordon and Breach Sci. Publ., London–New York, 1993, 1012 pp. | MR | Zbl

[28] S.M. Sitnik, Unitary and Boundedness of Buschman —Erdelyi operators with zero order of smoothness, Preprint, Institute of Automation and Control Processes of FEB of USSR Academy of Sciences, 1990, 44 pp. (In Russ.)

[29] V. V. Katrakhov, S. M. Sitnik, “Factorization method in transmutation theory”, Non-classical and mixed type equations, ed. V. N. Vragov, Sobolev Institute of Mathematics of SB of USSR Academy of Sciences, Novosibirsk, 1990, 104–122 (In Russ.) | MR | Zbl

[30] G. V. Lyakhovetskii, S. M. Sitnik, Compositions formulas for Buschman — Erdelyi operators, Preprint, Institute of Automation and Control Processes of FEB of USSR Academy of Sciences, Vladivostok, 1991, 11 pp. (In Russ.)

[31] S. M. Sitnik, “Factorization and estimates of the norms of Buschman — Erdelyi operators in weighted Lebesgue spaces”, Doklady Mathematics, 44:2 (1992), 641–646 | MR | Zbl

[32] S. M. Sitnik, “Transmutations and their applications”, Studies in contemporary analysis and mathematical modeling, Vladikavkaz Scientific Center of RAS and North Ossetia — Alania Republic Publ., Vladikavkaz, 2008, 226–293 (In Russ.)

[33] V. V. Katrakhov, S. M. Sitnik, “A boundary-value problem for the steady-state Schrödinger equation with a singular potential”, Soviet Mathematics. Doklady, 30:2 (1984), 468–470 | MR | MR | Zbl

[34] V. V. Katrakhov, S. M. Sitnik, “Composition method for constructing $B$-elliptic, $B$-hyperbolic, and $B$-parabolic transformation operators”, Doklady Mathematics, 50:1 (1995), 70–77 | MR

[35] V. V. Katrakhov, S. M. Sitnik, “Estimates of the Jost solution to a one-dimensional Schrödinger equation with a singular potential”, Doklady Mathematics, 51:1 (1995), 14–16 | MR | Zbl

[36] S. M. Sitnik, “Factorization method for transmutations in the theory of differential equations”, Bulletin of Samara State University, 67:8–1 (2008), 237–248 (In Russ.)

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

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

[39] S. M. Sitnik, “A solution to the problem of unitary generalization of Sonine–Poisson transmutations”, Scientific statements of Belgorod State University. Series: Mathematics. Physics, 5(76) (2010), 135–153 (In Russ.)

[40] S. M. Sitnik, “Boundedness of Buschman–Erdelyi Transmutations”, Analytical Methods of Analysis and Differential Equations, Proceedings of 5th International Conference, v. 1, Mathematical Analysis, Belorussian National Academy of Sciences, Minsk, 2010, 120–125 (In Russ.)

[41] S. M. Sitnik, “On representation in integral form of solutions to a differential equation with singularities in coefficients”, Vladikavkaz mathematical journal, 12:4 (2010), 73–78 (In Russ.) | MR | Zbl

[42] S. M. Sitnik, “On explicit realizations of fractional powers for Bessel differential operator and its applications to differential equations”, Reports of Adyghe (Circassian) International Academy of Sciences, 12:2 (2010), 69–75 (In Russ.)

[43] S. M. Sitnik, “Transmutation operator of special form for a differential operator with singular at origin potential”, Non-Classical Equations of Mathematical Physics, ed. A.I. Kozhanov, Sobolev Institute of Mathematics of SB RAS, Novosibirsk, 2010, 264–278 (In Russ.)

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

[45] A. A. Kilbas, O. V. Skoromnik, “Solution of a multidimensional integral equation of the first kind with the Legendre function in the kernel over a pyramidal domain.”, Doklady Mathematics, 80:3 (2009), 847–851 | DOI | MR | Zbl

[46] N. O. Virchenko, I. Fedotova, Generalized Associated Legendre Functions and Their Applications, World Sci., Singapore, 2001, 195 pp. | MR | Zbl

[47] O. I. Marichev, Method of calculating for integrals of special functions, Nauka i tekhnika Publ., Minsk, 1978 (In Russ.) | MR

[48] S. Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators and Spherical Functions, Academic Press, 1984, 667 pp. | MR | MR | Zbl

[49] A. Kufner, L.-E. Persson, Weighted Inequalities of Hardy Type, World Sci., New Jersey–Lonodon–Singapore–Hong Kong, 2003, 360 pp. | MR | Zbl

[50] H. Bateman, A. Erdelyi, Higher transcendental functions, v. 1, McGraw–Hill, New York–Toronto–London, 1953, 302 pp. | MR

[51] A. V. Pskhu, Boundary value problems for partial differential equations of fractional and continual order, KBNTs RAN Publ., Nal'chik, 2005, 186 pp. (In Russ.)

[52] A. V. Glushak, O. A. Pokruchin, “On properties of weighted Cauchy problems for abstract Malmsten equation”, Scientific statements of Belgorod State University. Series: Mathematics. Physics, 17(112) (2011), 102–110 (In Russ.)

[53] Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 2009, Special Issue on Dunkl Operators and Related Topics (accessed 05.10.2016) http://www.emis.de/journals/SIGMA/Dunkl_operators.html

[54] D. Ludwig, “The Radon Transform on Euclidean Space”, Communications on Pure and Applied Mathematics, XIX (1996), 49–81 | MR

[55] S. R. Deans, The Radon Transform and Some of Its Applications, Dover Publ., N.Y., 2007, 297 pp. | MR | Zbl

[56] I. A. Aliev, B. Rubin, “Spherical harmonics associated to the Laplace–Bessel operator and generalized spherical convolutions”, Analysis and Applications (Singap.), 1 (2003), 81–109 | DOI | MR | Zbl

[57] B. Rubin, Weighted spherical harmonics and generalized spherical convolutions, Preprint no. 2., The Hebrew Univ. of Jerusalem, 1999/2000, 38 pp. | MR

[58] V. V. Katrakhov, “On a boundary value problem for the Poisson equation”, Soviet Mathematics. Doklady, 24 (1981), 152–156 | MR | Zbl

[59] V. V. Katrakhov, “On a singular boundary value problem for the Poisson equation”, Sbornik: Mathematics, 73:1 (1992), 231–256 | DOI | MR | Zbl | Zbl

[60] I. A. Kipriyanov, V. V. Katrakhov, “On a class of one-dimensional singular pseudodifferential operators”, Mathematics of the USSR — Sbornik, 33:1 (1977), 43–61 | MR | Zbl