On On the approximation of functions by Jacobi – Dunkl expansion in the weighted space $\mathbb{L}_{2}^{(\alpha,\beta)}$

  • O. Tyr Univ. Hassan II, Casablanca, Morocco
  • R. Daher Univ. Hassan II, Casablanca, Morocco
Keywords: JACOBI – DUNKL EXPANSION

Abstract

UDC 517.5

We prove some new estimates useful in applications  for the approximation of certain classes of functions characterized by the generalized continuity modulus from the space $\mathbb{L}_{2}^{(\alpha,\beta)}$ by partial sums of the Jacobi – Dunkl series. For this purpose, we use the generalized Jacobi – Dunkl translation operator obtained  by Vinogradov in the monograph [Theory of approximation of functions of real variable, Fizmatgiz, Moscow (1960) (in Russian)].

References

V. A. Abilov, F. V. Abilova, M. K. Kerimov, Some issues concerning approximations of functions by Fourier – Bessel sums, Comput. Math. and Math. Phys., 53, № 7, 867 – 873 (2013).

V. A. Abilov, F. V. Abilova, M. K. Kerimov, Some remarks concerning the Fourier transform in the space $L_{2}big(mathbb{R}^{n}big)$, Zh. Vychisl. Mat. i Mat. Fiz., 48, 939 – 945 (2008) ({it English transl.}: Comput. Math. and Math. Phys., 48, 885 – 891 (2008)).

R. Askey, S. Wainger, A convolution structure for Jacobi series, Amer. J. Math., 91, 463 – 485 (1969).

H. Bavinck, Approximation processes for Fourier – Jacobi expansions, Appl. Anal., 5, 293 – 312 (1976).

F. Chouchene, Bounds, asymptotic behavior and recurrence relations for the Jacobi – Dunkl polynomials, Int. J. Open Probl. Complex Anal., 6, № 1, 49 – 77 (2014).

F. Chouchene, I. Haouala, Dirichlet theorem for Jacobi – Dunkl expansions; https://hal.archives-ouvertes.fr/hal-02126595.

F. Chouchene, Harmonic analysis associated with the Jacobi – Dunkl operator on $left]-dfrac{pi}{2},dfrac{pi}{2}right[$, J. Comput. and Appl. Math., 178, 75 – 89 (2005).

G. Gasper, Positivity and the convolution structure for Jacobi series, Ann. Math., 93, 112 – 118 (1971).

S. S. Platonov, Some problems in the theory of approximation of functions on compact homogeneous manifolds, Mat. Sb., 200, № 6, 67 – 108 (2009) ({it English transl.}: Sb. Math., 200, № 6, 845 – 885 (2009)).

A. Sveshnikov, A. N. Bogolyubov, V. V. Kravtsov, Lectures on mathematical physics, Nauka, Moscow (2004) (in Russian).

A. N. Tikhonov, A. A. Samarskii, Equations of mathematical physics, Gostekhteorizdat, Moscow (1953) (Pergamon Press, Oxford (1964)).

O. L. Vinogradov, On the norms of generalized translation operators generated by Jacobi – Dunkl operators, Zap. Nauchn. Sem. POMI, 389, 34 – 57 (2011).

Published
27.11.2022
How to Cite
TyrO., and DaherR. “On On the Approximation of Functions by Jacobi – Dunkl Expansion in the Weighted Space $\mathbb{L}_{2}^{(\alpha,\beta)}$”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 10, Nov. 2022, pp. 1427 -40, doi:10.37863/umzh.v74i10.6275.
Section
Research articles