Beyond Besov spaces. II: Oscillation spaces
In Part I of this work [J. Fourier Anal. Appl. 10, No.~3, 221--246 (2004; Zbl 1075.42014)], the author studied the information that for a function can be obtained from the distributions of wavelet coefficients at each scale. In this second part, new oscillation spaces are introduced through the distributions of suprema of wavelet coefficients on wavelet subtrees for functions that belong to the Besov spaces \(B^{s,\infty}_p\). These extensions of Besov spaces are sensitive to correlations between large wavelet coefficients located along lines of discontinuity and that are not revealed by Besov spaces. It is shown that these oscillation spaces do not depend on the particular wavelets selected and embedding properties between them and Besov spaces are described. Also, implications on dimensions of Hölder singularities, robustness properties and several examples taken from image processing and multifractal analysis are presented in this interesting paper.
Besov spaces, Hausdorff dimension, oscillation spaces
scientific article; zbMATH DE number 2157496