Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8900229 | Journal of Mathematical Analysis and Applications | 2018 | 42 Pages |
Abstract
Given a nondecreasing function f on [â1,1], 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 Ï2(f,â
) of f evaluated at 1âx2/n and were valid for all nâ¥1. The current paper is devoted to proving that if fâCr[â1,1], râ¥1, then the interpolatory estimates are valid for the second modulus of smoothness of f(r), however, only for nâ¥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 α>0, we write α=r+β where r is a nonnegative integer and 0<βâ¤1, and denote by Lipâα the class of all functions f on [â1,1] such that Ï2(f(r),t)=O(tβ). Then, one important corollary of the main theorem in this paper is the following result that has been an open problem for αâ¥2 since 1985:
Keywords
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Physical Sciences and Engineering
Mathematics
Analysis
Authors
K.A. Kopotun, D. Leviatan, I.A. Shevchuk,