Geodesic
On multipliers for Fourier series in Sobolev orthogonal polynomials
Sbornik. Mathematics, Tome 213 (2022) no. 8, pp. 1058-1095
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Multipliers are studied for Fourier series in polynomials orthogonal in continuous-discrete Sobolev spaces. For the multiplier operator, existence results and norm estimates are obtained. The proofs are based on a representation of the Fejér kernel, the construction of a ‘humpbacked majorant’ and estimates for the norms of maximal functions. Bibliography: 45 titles.
Keywords: Fourier series, Sobolev polynomials, norm of an operator.
Mots-clés : orthogonal polynomials, multipliers 
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B. P. Osilenker. On multipliers for Fourier series in Sobolev orthogonal polynomials. Sbornik. Mathematics, Tome 213 (2022) no. 8, pp. 1058-1095. http://geodesic.mathdoc.fr/item/SM_2022_213_8_a2/

[1] R. Courant and D. Hilbert, Methoden der mathematischen Physik, Grundlehren Math. Wiss., I, 2. verb. Aufl., J. Springer, Berlin, 1931, xiv+469 pp. | MR | Zbl

[2] H. L. Krall, “Certain differential equations for Tchebycheff polynomials”, Duke Math. J., 4:4 (1938), 705–718 | DOI | MR | Zbl

[3] A. M. Krall, “Orthogonal polynomials satisfying fourth order differential equations”, Proc. Roy. Soc. Edinburgh Sect. A, 87:3–4 (1981), 271–288 | DOI | MR | Zbl

[4] D. C. Lewis, “Polynomial least square approximations”, Amer. J. Math., 69:2 (1947), 273–278 | DOI | MR | Zbl

[5] F. Marcellán and Y. Xu, “On Sobolev orthogonal polynomials”, Expo. Math., 33:3 (2015), 308–352 | DOI | MR | Zbl

[6] I. I. Sharapudinov, “Approximation properties of Fourier series of Sobolev orthogonal polynomials with Jacobi weight and discrete masses”, Mat. Zametki, 101:4 (2017), 611–629 ; English transl. in Math. Notes, 101:4 (2017), 718–734 | DOI | MR | Zbl | DOI

[7] I. I. Sharapudinov, “Sobolev orthogonal polynomials generated by Jacobi and Legendre polynomials, and special series with the sticking property for their partial sums”, Mat. Sb., 209:9 (2018), 142–170 ; English transl. in Sb. Math., 209:9 (2018), 1390–1417 | DOI | MR | Zbl | DOI

[8] I. I. Sharapudinov, “Sobolev-orthogonal systems of functions and some of their applications”, Uspekhi Mat. Nauk, 74:4(448) (2019), 87–164 ; English transl. in Russian Math. Surveys, 74:4 (2019), 659–733 | DOI | MR | Zbl | DOI

[9] A. S. Kostenko and M. M. Malamud, “Schrödinger operators with $\delta'$-interactions and the Krein-Stieltjes string”, Dokl. Ross. Akad. Nauk, 432:1 (2010), 12–17 ; English transl. in Dokl. Math., 81:3 (2010), 342–347 | MR | Zbl | DOI

[10] B. P. Osilenker, “On Fourier series in generalized eigenfunctions of a discrete Sturm-Liouville operator”, Funktsional. Anal. i Prilozhen., 52:2 (2018), 90–93 ; English transl. in Funct. Anal. Appl., 52:2 (2018), 154–157 | DOI | MR | Zbl | DOI

[11] A. N. Tikhonov and A. A. Samarskii, Equations of mathematical physics, 7th ed., Moscow State University publishing house, Moscow; Nauka, 2004, 800 pp. ; English transl. of 2nd ed., The Macmillan Co., New York, 1963, xvi+765 pp. | MR | MR | Zbl

[12] H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978, 528 pp. | MR | Zbl

[13] A. N. Tikhonov and A. A. Samarskii, Equations of mathematical physics, Izv. Ross. Akad. Nauk Ser. Mat., 83, no. 2, 2019 ; English transl. in Izv. Math., 83:2 (2019), 391–412 | DOI | MR | Zbl | DOI

[14] S. Albeverio, Z. Brzeźniak and L. Dabrowski, “Fundamental solution of the heat and Schrödinger equations with point interaction”, J. Funct. Anal., 130:1 (1995), 220–254 | DOI | MR | Zbl

[15] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Math. Monogr., The Clarendon Press, Oxford Univ. Press, New York, 1993, vi+363 pp. | MR | Zbl

[16] T. Kilpeläinen, “Weighted Sobolev spaces and capacity”, Ann. Acad. Sci. Fenn. Ser. A I Math., 19:1 (1994), 95–113 | MR | Zbl

[17] A. Kufner, Weighted Sobolev spaces, Teubner-Texte Math., 31, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1980, 151 pp. | MR | Zbl

[18] A. Kufner and A.-M. Sändig, Some applications of weighted Sobolev spaces, Teubner-Texte Math., 100, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1987, 268 pp. | MR | Zbl

[19] W. D. Evans, L. L. Littlejohn, F. Marcellan, C. Markett and A. Ronveaux, “On recurrence relations for Sobolev orthogonal polynomials”, SIAM J. Math. Anal., 26:2 (1995), 446–467 | DOI | MR | Zbl

[20] I. A. Rocha, F. Marcellán and L. Salto, “Relative asymptotics and Fourier series of orthogonal polynomials with a discrete Sobolev inner product”, J. Approx. Theory, 121:2 (2003), 336–356 | DOI | MR | Zbl

[21] B. P. Osilenker, “Multipliers of Fourier series with respect to orthogonal polynomials in continuous-discrete Sobolev spaces”, Current problems in mathematics and mechanics, MAKS Press, Moscow, 2019, 500–503 (Russian)

[22] B. P. Osilenker, Multiplier theorem for Fourier series in continuous-discrete Sobolev orthogonal polynomials, arXiv: 2012.00550

[23] B. Muckenhoupt, “Poisson integrals for Hermite and Laguerre expansions”, Trans. Amer. Math. Soc., 139 (1969), 231–242 | DOI | MR | Zbl

[24] B. Osilenker, Fourier series in orthogonal polynomials, World Sci. Publ., River Edge, NJ, 1999, vi+287 pp. | DOI | MR | Zbl

[25] E. M. Dyn'kin and B. P. Osilenker, “Weighted estimates for singular integrals and their applications”, Itogi Nauki Tekhn. Ser. Mat. Anal., 21, VINITI, Moscow, 1983, 42–129 ; English transl. in J. Math. Sci. (N.Y.), 30:3 (1985), 2094–2154 | MR | Zbl | DOI

[26] M. I. D'yachenko and P. L. Ul'yanov, Measure and integral, Faktorial Press, Moscow, 2002, 160 pp. (Russian)

[27] N. Dunford and J. T. Schwartz, Linear operators, v. I, Pure Appl. Math., 7, General theory, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958, xiv+858 pp. | MR | Zbl

[28] G. Freud, Orthogonal polynomials, Akad. Kiado, Budapest; Pergamon Press, Oxford, 1971, 294 pp. | DOI

[29] J. M. Rodríguez, V. Álvarez, E. Romera and D. Pestana, “Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials. I”, Acta Appl. Math., 80:3 (2004), 273–308 | DOI | MR | Zbl

[30] J. M. Rodríguez, E. Romera, D. Pestana and V. Álvarez, “Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials. II”, Approx. Theory Appl. (N.S.), 18:2 (2002), 1–32 | DOI | MR | Zbl

[31] J. M. Rodríguez, “Approximation by polynomials and smooth functions in Sobolev spaces with respect to measures”, J. Approx. Theory, 120:2 (2003), 185–216 | DOI | MR | Zbl

[32] R. E. Edwards, Fourier series. A modern introduction, v. 1, Grad. Texts in Math., 64, 2nd ed., Springer-Verlag, New York–Berlin, 1979, xii+224 pp. | MR | Zbl

[33] B. Muckenhoupt and E. M. Stein, “Classical expansions and their relation to conjugate harmonic functions”, Trans. Amer. Math. Soc., 118 (1965), 17–92 | DOI | MR | Zbl

[34] B. P. Osilenker, “An extremal problem for algebraic polynomials in the symmetric discrete Gegenbauer-Sobolev space”, Mat. Zametki, 82:3 (2007), 411–425 ; English transl. in Math. Notes, 82:3 (2007), 366–379 | DOI | MR | Zbl | DOI

[35] B. P. Osilenker, “On linear summability methods of Fourier series in polynomials orthogonal in a discrete Sobolev space”, Sibirsk. Mat. Zh., 56:2 (2015), 420–435 ; English transl. in Siberian Math. J., 56:2 (2015), 339–351 | MR | Zbl | DOI

[36] B. P. Osilenker, “Generalized trace formula and asymptotics of the averaged Turan determinant for polynomials orthogonal with a discrete Sobolev inner product”, J. Approx. Theory, 141:1 (2006), 70–97 | DOI | MR | Zbl

[37] H. Bavinck, “Differential operators having Sobolev-type Gegenbauer polynomials as eigenfunctions”, J. Comput. Appl. Math., 118:1–2 (2000), 23–42 | DOI | MR | Zbl

[38] H. Bavinck and H. G. Meijer, “Orthogonal polynomials with respect to a symmetric inner product involving derivatives”, Appl. Anal., 33:1–2 (1989), 103–117 | DOI | MR | Zbl

[39] H. Bavinck and H. G. Meijer, “On orthogonal polynomials with respect to an inner product involving derivatives: zeros and recurrence relations”, Indag. Math. (N.S.), 1:1 (1990), 7–14 | DOI | MR | Zbl

[40] A. F. Moreno, F. Marcellán and B. P. Osilenker, “Estimates for polynomials orthogonal with respect to some Gegenbauer-Sobolev type inner product”, J. Inequal. Appl., 3:4 (1999), 401–419 | MR | Zbl

[41] R. Koekoek, “Differential equations for symmetric generalized ultraspherical polynomials”, Trans. Amer. Math. Soc., 345:1 (1994), 47–72 | DOI | MR | Zbl

[42] F. Marcellán, B. P. Osilenker and I. A. Rocha, “On Fourier series of Jacobi-Sobolev orthogonal polynomials”, J. Inequal. Appl., 7:5 (2002), 673–699 | MR | Zbl

[43] F. Marcellán, B. P. Osilenker and I. A. Rocha, “On Fourier series of a discrete Jacobi-Sobolev inner product”, J. Approx. Theory, 117:1 (2002), 1–22 | DOI | MR | Zbl

[44] Ó. Ciaurri and J. Mínguez, “Fourier series of Gegenbauer-Sobolev polynomials”, SIGMA, 14 (2018), 024, 11 pp. | DOI | MR | Zbl

[45] A. Díaz-González, F. Marcellán-Español, H. Pijeira-Cabrera and W. Urbina-Romero, Discrete-continuous Jacobi-Sobolev spaces and Fourier series, arXiv: 1911.12746v1