2015/11/12 by Yu. A. Brudnyi, Yu. Brudnyi, Brudnyi, Yu.
Mathematics · #41A15 #41A46 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical Approximation and Integration #math.CA #math.FA #msc:41A15 #msc:41A46
paper · pdf · doi:10.48550/arxiv.1511.03971
37 pages
arxiv created 2015/11/12 · openalex publication_date 2015/11/12 · arxiv updated 2015/11/13 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
The named space denoted by Vpqk consists of Lq functions on [0,1)d of bounded p-variation of order k∈\mathbb N. It generalizes the classical spaces Vp(0,1) (=Vp∞1) and BV([0,1)d) (V1q1 where q:=\frac dd-1) and closely relates to several important smoothness spaces, e.g., to Sobolev spaces over Lp, BV and BMO and to Besov spaces. The main approximation result concerns the space Vpqk of smoothness s:=d(\frac1p-\frac1q)∈(0,k]. It asserts the following: Let f∈ Vpqk are of smoothness s∈(0,k] and N∈\mathbb N. There exist a family ΔN of N dyadic subcubes of [0,1)d and a piecewise polynomial gN over ΔN of degree k-1 such that ‖f-gN‖q\leqslant CN-s/d|f|_Vpqk. This implies the similar results for the above mentioned smoothness spaces, in particular, solves the going back to the 1967 Birman--Solomyak paper \citeBS problem of approximation of functions from Wpk([0,1)d) in Lq([0,1)d) when ever \frac kd=\frac1p-\frac1q and q<∞.