Stability Properties of Trigonometric Interpolation Operators

Abstract

Consider the space C of all 2π-periodic continuous real functions, and the subspace π of all n-th order trigonometric polynomials. The index n is held fixed, and the spaces are endowed with the usual supremum norm. Any operator L: C → π which can be written in the form $Lx = \sum\limits_1^m {x({s_k}){y_k}}$ with 0 ≤ s_k < 2≤ and y_k ε π is said to be carried by the point set {s₁,...,s_m}. If Lx = x for all x ε π, then L is a projection of C onto π. The uniform grid is defined to be the set of points t_k = kπ(2n + 1)^-1 for k = 0,...,2n.