Nonlinear piecewise polynomial approximation and multivariate BV spaces of a Wiener–L. Young type. I
| Autor: | Yu. A. Brudnyi |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Discrete mathematics
Numerical Analysis Smoothness (probability theory) Degree (graph theory) Applied Mathematics General Mathematics Dyadic cubes 010102 general mathematics Space (mathematics) 01 natural sciences 010101 applied mathematics Sobolev space Order (group theory) Interpolation space Birnbaum–Orlicz space 0101 mathematics Analysis Mathematics |
| Zdroj: | Journal of Approximation Theory. 218:9-41 |
| ISSN: | 0021-9045 |
| DOI: | 10.1016/j.jat.2017.03.002 |
| Popis: | The named space denoted by V p q k consists of L q functions on [ 0 , 1 ) d of bounded p -variation of order k ∈ N . It generalizes the classical spaces V p ( 0 , 1 ) ( = V p ∞ 1 ) and B V ( = [ 0 , 1 ) d ) ( V 1 q 1 where q ≔ d d − 1 ) and is closely related to several important smoothness spaces, e.g., to Sobolev spaces over L p , B V and B M O and to Besov spaces. The main approximation result concerns the space V p q k of smoothness s ≔ d 1 p − 1 q ∈ ( 0 , k ] . It asserts the following: Let f ∈ V p q k be of smoothness s ∈ ( 0 , k ] , 1 ≤ p q ∞ and N ∈ N . There exist a family Δ N of N dyadic subcubes of [ 0 , 1 ) d and a piecewise polynomial g N over Δ N of degree k − 1 such that ‖ f − g N ‖ q ⩽ C N − s ∕ d | f | V p q 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 L q ( [ 0 , 1 ) d ) whenever k d = 1 p − 1 q and q ∞ . |
| Databáze: | OpenAIRE |
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