2003/09/01 by P. Wojtaszczyk · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Applied mathematics #Bounded function #Combinatorics #Differential Equations and Boundary Problems #Discrete mathematics #Exponent #Haar #Mathematical Approximation and Integration #Mathematical analysis #Mathematics #Mathematics Subject Classification #Norm (philosophy) #Pure mathematics #Sobolev space #Space (punctuation) #Term (time) #Thresholding #Wavelet
paper · doi:10.1112/s0024611503014084
openalex publication_date 2003/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
The aim of this paper is to provide an analysis of non-linear approximation in the Lp-norm p = d / (d − 1) of functions of bounded variation on Rd with d > 1 by polynomials in the Haar system. The exponent p is the natural exponent as it is the correct exponent in the Sobolev inequality. The approximation schemes that we discuss in this paper are mostly related to Haar thresholding and m-term approximation. These problems for d = 2 are studied in detail in a paper by Cohen, DeVore, Petrushev and Xu. The main aim of this paper is to extend their results to the case d ⩾ 2. We obtain the optimal order of the m-term Haar approximation and prove the stability of Haar thresholding in the BV-norm. As one of the main tools, we establish the boundedness of certain averaging projections in BV. 2000 Mathematics Subject Classification 41A46, 41A63.