Summability of Product Jacobi Expansions

Cesàro summability of Fourier orthogonal expansions on the cylinder

“…In fact, the sharpness of the bound for δ on the cube in [6], are, to the author's knowledge, not known. These questions are areas for further investigation.…”

“…While the orthogonal basis has a product structure, this structure does not translate into a product structure of the Cesàro means, so that the aforementioned results on the cube and ball do not provide a trivial result for the cylinder. Our proof of the result requires reducing to problem of convergence on the cylinder to a problem of convergence on the cube [−1, 1] m+1 , and then using the results proven by Xu and Li in [6] on the Cesàro summability of functions on the hypercube.…”

Triangular Cesàro summability of two dimensional Fourier series

“…Herriot [6], Berens, Li and Xu [1,2,7,12] and more recently Szili and Vértesi [10]). We will prove that σ α n f → f in B-norm, where B is a homogeneous Banach space, which includes the norm convergence in L p T 2 (1 p < ∞) and in C T 2 .…”

“…Li and Xu [7] proved for Jacobi polynomials that the L 1 -norms of the kernel functions are uniformly bounded.…”

Dirichlet Theorem for Jacobi-Dunkl Expansions

“…In particular, by using the properties of Jacobi polynomials ( [15]), the Fourier-Jacobi series has been studied extensively by many authors and several results concerning the approximation of functions by partial sums of these series are proved (see e.g. [1,11,13,14,16,17,18]). In this paper, we also discuss this subject.…”