On multivariate projection operators

Abstract

This paper deals with multivariate Fourier series considering triangular type partial sums. Among others we give the exact order of the corresponding operator norm. Moreover, a generalization of the so-called Faber-Marcinkiewicz-Berman theorem has been proved.

Bivariate Lagrange interpolation at the Padua points: Computational aspects

“…Such points allow us to give a simple, geometric and explicit construction of the interpolation formula, since the Lagrange polynomials are written in terms of the reproducing kernel corresponding to the product Chebyshev measure. Moreover, the Padua points have a Lebesgue constant with minimal order of growth O(log 2 (n)), as has been rigorously proved in [4] for the upper bound and in [12,16] for the exact order of growth.…”

“…The Lebesgue constant of interpolation at the Padua points has optimal order of growth Λ Pad s n = L Pad s n = O(log 2 (n)), s = 1, 2, 3, 4, as has been rigorously proved in [4,12,16]. In view of the multivariate extension of Jackson's theorem (cf., e.g., [1] and references therein), we have that for…”

Triangular Cesàro summability of two dimensional Fourier series

“…It is easy to see that |D k | Ck 2 . Recently Szili and Vértesi [10] verified that D k 1 ∼ (log k) 2 . Defining Zygmund [13,I.…”

“…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 .…”

Trivariate polynomial approximation on Lissajous curves

“…Concerning polynomial interpolation in the cube by sampling on the Lissajous curve, we resort to the approximate versions of Fekete points (points that maximize the absolute value of the Vandermonde determinant) studied in several recent papers [4,6,32]. By (24), it makes sense to start from a WAM, namely the Chebyshev lattice A n in (25), by the corresponding Vandermonde-like matrix…”