Convolution operators with trigonometric spline kernels

Abstract

The Bernstein polynomials are algebraic polynomial approximation operators which possess shape preserving properties. These polynomial operators have been extended to spline approximation operators, the Bernstein-Schoenberg spline approximation operators, which are also shape preserving like the Bernstein polynomials [8].

Approximation and spectral properties of periodic spline operators

“…, where we define 1,3,14]). An extension of (5.11) to convolution operators with trigonometric B-spline kernels was studied in [7]. Let ^n,*: = {s£C 2m " 1 (IR):s| (O _ 1/ 2)ft,(j + i/2H) e Q u a l s a trigonometric polynomial of degree m}.…”

“…In Section 5 we show that the corresponding set of orthonormal trigonometric splines of degree m contains the finite section {e' vx : -m^vm } of the orthonormal Fourier system. In this case, the corresponding operator t%] k f, with a = 0, is a discrete analogue of the convolution operator with trigonometric Bspline kernel which was studied in [7].…”