Best Approximation and Cyclic Variation Diminishing Kernels

✍ Scribed by Oleg Davydov; Allan Pinkus

• Book ID: 102969301
• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 534 KB
• Volume: 89
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1996.3059

✦ Synopsis

We study best uniform approximation of periodic functions from { | 2? 0 K(x, y) h( y) dy : |h( y)| 1 = , where the kernel K(x, y) is strictly cyclic variation diminishing, and related problems including periodic generalized perfect splines. For various approximation problems of this type, we show the uniqueness of the best approximation and characterize the best approximation by extremal properties of the error function. The results are proved by using a characterization of best approximants from quasi-Chebyshev spaces and certain perturbation results.

1997 Academic Press

1. Introduction

This paper is about some approximation problems related to cyclic variation diminishing (CVD) kernels. CVD kernels are the periodic analogues of totally positive (TP) kernels. CVD kernels were introduced and discussed in two papers by Schoenberg and coauthors [5, 8] in 1958 and 1959. A more comprehensive consideration is to be found in the book of Karlin [4, Chaps. 5 and 9]. We first define the relevant concepts. We will later return to a general discussion of CVD kernels.