Nearly Comonotone Approximation

✍ Scribed by D. Leviatan; I.A. Shevchuk

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 370 KB
• Volume: 95
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1998.3194

✦ Synopsis

We discuss the degree of approximation by polynomials of a function f that is piecewise monotone in [&1, 1]. We would like to approximate f by polynomials which are comonotone with it. We show that by relaxing the requirement for comonotonicity in small neighborhoods of the points where changes in monotonicity occur and near the endpoints, we can achieve a higher degree of approximation. We show here that in that case the polynomials can achieve the rate of | 3 . On the other hand, we show in another paper, that no relaxing of the monotonicity requirements on sets of measures approaching 0 allows | 4 estimates.

1998 Academic Press

1. Introduction

Let I :=[&1, 1], and for s 1 let Y :=[ y i ] s i=0 , &1= y s < } } } < y 1 < y 0 =1. Finally let 2 (1) (Y ) be the set of continuous functions f on I, such that f is nondecreasing on [ y i , y i&1 ], when i is odd and it is nonincreasing on [ y i , y i&1 ], when i is even, and set 6(x) := s&1 i=1 (x& y i ).