Abstract
We consider the properties of systems of functions \(\Phi_1\) orthogonal with respect to a discrete-continuous Sobolev inner product of the form \(\langle f,g \rangle_S = f(a)g(a)+f(b)g(b)+\int_a^b f'(t)g'(t)dt\). In particular, we study the completeness of systems \(\Phi_1\) in the Sobolev space \(W^1_{L^2}\). Additionally, we analyze the properties of Fourier series with respect to systems \(\Phi_1\), and prove that these series converge uniformly to functions from \(W^1_{L^2}\).
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REFERENCES
Marcellan F., Xu Y. "On Sobolev orthogonal polynomials", Expositiones Math. 33, 308-352 (2015).
Marcellan F., Quintana Y., Urieles A. "On the Pollard decomposition method applied to some Jacobi–Sobolev expansions", Turk. J. Math. 37 (6), 934-948 (2013).
Ciaurri O., Minguez J. "Fourier series of Jacobi–Sobolev polynomials", Integral Transfand. Spec. Funct. 30, 334-346 (2019).
Ciaurri O., Minguez J. Fourier Series for Coherent Pairs of Jacobi Measures, to appear.
Fejzullahu B.Xh. "Asymptotic properties and Fourier expansions of orthogonal polynomials with a non-discrete Gegenbauer–Sobolev inner product", J. Approxim. Theory 162 (2), 397-406 (2010), DOI: https://doi.org/10.1016/j.jat.2009.07.002.
Fejzullahu B.Xh., Marcellan F., Moreno-Balcazar J.J. "Jacobi–Sobolev orthogonal polynomials: asymptotics and a Cohen type inequality", J. Approxim. Theory 170, 78-93 (2013), DOI: https://doi.org/10.1016/j.jat.2012.05.015.
Iserles A., Koch P.E., Norsett S.P., Sanz-Serna J.M. "On polynomials orthogonal with respect to certain Sobolev inner product", J. Approx. Theory 65 (2), 151-175 (1991).
Marcellan F., Osilenker B.P., Rocha I.A. "On Fourier series of a discrete Jacobi–Sobolev inner product", J. Approx. Theory 117 (1), 1-22 (2002), DOI: https://doi.org/10.1006/jath.2002.3681.
Rocha I., Marcellan F., Salto L. "Relative asymptotics and Fourier series of orthogonal polynomials with a discrete Sobolev inner product", J. Approx. Theory 121 (2), 336-356 (2003).
Osilenker B.P. "Convergence and summability of Fourier–Sobolev series", Vestn. MGSU 5 (2012) [in Russian], URL: https://cyberleninka.ru/article/n/shodimost-i-summiruemost-ryadov-furie-soboleva-1.
Osilenker B.P. "On linear summability methods of fourier series in polynomials orthogonal in a discrete Sobolev space", Sib. Math. J. 56 (2), 339-351 (2015).
Fejzullahu B.Xh., Marcellan F. "On convergence and divergence of Fourier expansions with respect to some Gegenbauer–Sobolev type inner product", Commun. Anal. Theory Cont. Fractions 16, 1-11 (2009).
Ciaurri O., Minguez J. "Fourier series of Gegenbauer–Sobolev polynomials", SIGMA Symm. Integrabi. Geom. Methods Appl. 14, article 024 (2018).
Sharapudinov I.I. "Sobolev-orthogonal systems of functions and some of their applications", Russian Math. Surveys 74 (4), 659-733 (2019).
Sharapudinov I.I. "Sobolev-orthogonal systems of functions associated with an orthogonal system", Izv. Math. 82 (1), 225-258 (2018).
Magomed-Kasumov M.G. "A Sobolev orthogonal system of functions generated by a Walsh system", Math. Notes 105 (4), 543-549 (2019).
Gadzhimirzaev R.M. "Sobolev-orthonormal system of functions generated by the system of Laguerre functions", Issues Anal. 8 (26) (1), 32-46 (2019).
Diaz-Gonzalez A., Marcellan F., Pijeira-Cabrera H., Urbina W. "DiscreteЦcontinuous Jacobi–Sobolev spaces and Fourier series", Bull. Malays. Math. Sci. Soc. 44, 571-598 (2020), DOI: https://doi.org/10.1007/s40840-020-00950-7.
Sharapudinov I.I. "Sobolev orthogonal polynomials associated with Chebyshev polynomials of the first kind and the Cauchy problem for ordinary differential equations", Differ. Equ. 54 (12), 1602-1619 (2018).
Sharapudinov I.I. "Sobolev-orthogonal systems of functions and the Cauchy problem for ODEs", Izv. Math. 83 (2), 391-412 (2019).
Szegö G. Orthogonal Polynomials (GIFML, Moscow, 1962) [in Russian].
Kwon K.H., Littlejohn L.L. "Sobolev orthogonal polynomials and second-order differential equations", Rocky Mountain J. Math. 28 (2), 547-594 (1998).
Kashin B.S., Saakyan A.A. Orthogonal Series, 2nd edition (Izd-vo AFTs, Moscow, 1999) [in Russian].
Golubov B.I., Efimov A.V., Skvortsov V.A. Walsh Series and Transformations: Theory and Applications (Nauka, Moscow, 1987) [in Russian].
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Russian Text © The Author(s), 2021, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2021, No. 12, pp. 56–66.
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Magomed-Kasumov, M.G. Sobolev Orthogonal Systems with Two Discrete Points and Fourier Series. Russ Math. 65, 47–55 (2021). https://doi.org/10.3103/S1066369X21120057
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DOI: https://doi.org/10.3103/S1066369X21120057