Nonlinear piecewise polynomial approximation and multivariate B V spaces of a Wiener-L. Young type. I

JOURNAL OF APPROXIMATION THEORY

卷:218

Article

Brudnyi, Yu.¹ 

[1] Technion, Haifa, Israel

关键词: N-term approximation;    Piecewise polynomials;    Dyadic cubes;    Spaces of q-integrable functions of bounded variation;    Sobolev spaces;    Besov spaces;   

DOI  :  10.1016/j.jat.2017.03.002

来源: Elsevier

【 摘 要 】

The named space denoted by V-pq(k) consists of L-q functions on [0, 1)(d) of bounded p-variation of order k is an element of N. It generalizes the classical spaces V-p(0, 1) (=V-p infinity(1)) and BV (= [0, 1)(d)) (V-1q(1) where q := d/d-1) and is closely related to several important smoothness spaces, e.g., to Sobolev spaces over L-p), BV and BM 0 and to Besov spaces. The main approximation result concerns the space V-pq(k) of smoothness s := d (1/p - 1/q) is an element of(0, k]. It asserts the following: Let f is an element of V-Pq(k) be of smoothness s is an element of(0, k], 1 <= p < q < infinity and N is an element of N. There exist a family Delta(N) of N dyadic subcubes of [0, 1)(d) and a piecewise polynomial g(N) over Delta N of degree k - 1 such that parallel to f - g(N) parallel to(q) <= CN-s/d vertical bar f vertical bar V-pq(k) This implies similar results for the above mentioned smoothness spaces, in particular, solves the going back to the 1967 Birman-Solomyak paper (Birman and Solomyak, 1967) problem of approximation of functions from W-P(k)([0,1)(d)) in Lq([0, 1)(d)) whenever k/d = 1/p - 1/q and q < infinity. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free