Polynomial approximation with doubling weights having finitely many zeros and singularities

JOURNAL OF APPROXIMATION THEORY

卷:198

Article

Kopotun, Kirill A.

关键词: Doubling weights with zeros and singularities;    Weighted moduli of smoothness;    Polynomial approximation;    Direct and inverse theorems;    Classical Jacobi weights;    Generalized Jacobi weights and generalized Ditzian-Totik weights;    Weighted realization functionals;   

DOI  :  10.1016/j.jat.2015.05.003

来源: Elsevier

【 摘 要 】

We prove matching direct and inverse theorems for (algebraic) polynomial approximation with doubling weights w having finitely many zeros and singularities (i.e., points where w becomes infinite) on an interval and not too rapidly changing away from these zeros and singularities. This class of doubling weights is rather wide and, in particular, includes the classical Jacobi weights, generalized Jacobi weights and generalized Ditzian Totik weights. We approximate in the weighted L-p (quasi) norm parallel to f parallel to(p,w) with 0 < p < infinity, where parallel to f parallel to(p,w) := (integral(1)(-1) vertical bar f (u) vertical bar(P) w(u)du)(1/p). Equivalence type results involving related realization functionals are also discussed. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free