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Inner product and Gegenbauer polynomials in Sobolev space

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In this paper we consider the system of functions G_(r,n)^α (x) (r∈N,n=0,1,…) which is orthogonal with respect to the Sobolev-type inner product on (-1,1) and generated by orthogonal Gegenbauer polynomials. The main goal of this work is to study some properties related to the system {φ_(k,r) (x)}_(k≥0) of the functions generated by the orthogonal system {G_(r,n)^α (x)} of Gegenbauer functions. We study the conditions on a function f(x) given in a generalized Gegenbauer orthogonal system for it to be expandable into a generalized mixed Fourier series of the form f(x)~∑_(k=0)^(r-1)▒〖f^((k) ) (-1) (x+1)^k/k!+∑_(k=r)^∞▒〖G_(r,k)^α (f) 〗〗 φ_(r,k)^α (x), as well as the convergence of this Fourier series. The second result of this paper is the proof of a recurrence formula for the system {φ_(k,r) (x)}_(k≥0). We also discuss the asymptotic properties of these functions, and this represents the latter result of our contribution.

Keywords

inner product, Sobolev space, Gegenbauer polynomials

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DOI

10.20310/2686-9667-2022-27-138-150-163

UDC

517.518.36

Pages

150-163

References

[1] R.M. Gadzhimirzaev, “Sobolev-orthonormal system of functions generated by the system of Laguerre functions”, Probl. Anal. Issues Anal., 8(26):1 (2019), 32-46. [2] I.I. Sharapudinov, “Approximation of functions of variable smoothness by Fourier-Legendre sums”, Sb. Math., 191:5 (2000), 759-777. [3] I. Sharapudinov, Mixed Series of Orthogonal Polynomials, Daghestan Sientific Centre Press, Makhachkala, 2004. [4] I. I. Sharapudinov, “Approximation properties of mixed series in terms of Legendre polynomials on the classes W^r”, Sb. Math., 197:3 (2006), 433-452. [5] I.I. Sharapudinov, “Sobolev orthogonal systems of functions associated with an orthogonal system”, Izv. Math., 82:1 (2018), 212-244. [6] I.I. Sharapudinov, T.I. Sharapudinov, “Polynomials orthogonal in the Sobolev sens, generated by Chebychev polynomials orthogonal on a mesh”, Russian Math. (Iz. VUZ), 61:8 (2017), 59-70. [7] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, USA, 1964. [8] G. Szegiö, Orthogonal Plynomials. V. 23, American Mathematical Society, Providence, Rhode Island, 1975. [9] A.F. Nikiforov, V.B. Uvarov, Special Functions of Mathematical Physics, Birkhauser Veriag Basel, Springer Basel AG., 1988.

Received

2022-02-17

Section of issue

Scientific articles

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