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On Orthogonal Expansions with Respect to the Generalized Jacobi Weight

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Abstract

In this paper we will prove a Cohen type inequality for orthogonal expansions with respect to the generalized Jacobi weight

$$(1-x)^\alpha(1+x)^\beta h(x)\prod_{i=1}^m |x_i-x|^{\nu_i},$$

where −1 < x 1 < · · ·< x m  < 1, α,β,ν i  > −1 (i = 1,. . .,m), and h is a positive continuous function on [−1, 1] and its modulus of continuity w(h,·) satisfies the condition \({\int_{0}^{2} t^{-1}w(h,t) dt <\infty}\). In particular, we investigate the asymptotic behaviour for the norm of the generalized Fourier-Jacobi expansions in the appropriate weighted space, the well known Lebesgue constants of the approximation theory literature. Finally, we prove that, for certain indices δ, there are functions whose Cesàro means of order δ in the Fourier expansions with respect to the generalized Jacobi weight are divergent a.e. on [−1, 1].

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Authors and Affiliations

  1. Faculty of Mathematics and Sciences, University of Prishtina, Mother Teresa 5, 10000, Prishtina, Kosovo

    Bujar Xh. Fejzullahu

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  1. Bujar Xh. Fejzullahu
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Correspondence to Bujar Xh. Fejzullahu.

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Fejzullahu, B.X. On Orthogonal Expansions with Respect to the Generalized Jacobi Weight. Results. Math. 63, 1177–1193 (2013). https://doi.org/10.1007/s00025-012-0261-y

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