Interpolatory pointwise estimates for monotone polynomial approximation

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:459

Article

Kopotun, K. A.¹  Leviatan, D.²  Shevchuk, I. A.³ 

[1] Univ Manitoba, Dept Math, Winnipeg, MB R3T 2N2, Canada

[2] Tel Aviv Univ, Raymond & Beverly Sackler Sch Math Sci, IL-6139001 Tel Aviv, Israel

[3] Taras Shevchenko Natl Univ Kyiv, Fac Mech & Math, UA-01601 Kiev, Ukraine

关键词: Monotone polynomial approximation;    Degree of approximation;    Jackson-type interpolatory estimates;   

DOI  :  10.1016/j.jmaa.2017.11.038

来源: Elsevier

【 摘 要 】

Given a nondecreasing function f on [-1,1], we investigate how well it can be approximated by nondecreasing algebraic polynomials that interpolate it at +/- 1. We establish pointwise estimates of the approximation error by such polynomials that yield interpolation at the endpoints (i.e., the estimates become zero at +/- 1). We call such estimates interpolatory estimates. In 1985, DeVore and Yu were the first to obtain this kind of results for monotone polynomial approximation. Their estimates involved the second modulus of smoothness omega(2)(f, .) of f evaluated at root 1 - x(2)/n and were valid for all n >= 1. The current paper is devoted to proving that if f is an element of C-r[-1,1], r >= 1, then the interpolatory estimates are valid for the second modulus of smoothness of f((r)), however, only for n >= N with N = N(f, r), since it is known that such estimates are in general invalid with N independent of f. Given a number alpha > 0, we write alpha = r + beta where r is a nonnegative integer and 0 < beta <= 1, and denote by Lip*alpha the class of all functions f on [-1, 1] such that omega(2)(f((r)),t) = O(t(beta)). Then, one important corollary of the main theorem in this paper is the following result that has been an open problem for alpha >= 2 since 1985: If alpha > 0, then a function f is nondecreasing and in Lip* alpha, if and only if, there exists a constant C such that, for all sufficiently large n, there are nondecreasing polynomials P-n, of degree n, such that vertical bar f (x) - P-n(x)vertical bar <= C (root 1 - x(2)/n)(alpha) , x is an element of[-1,1]. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free