Banach frames for multivariate α-modulation spaces

AbstractThe α-modulation spaces Mp,qs,α(Rd), α∈[0,1], form a family of spaces that include the Besov and modulation spaces as special cases. This paper is concerned with construction of Banach frames for α-modulation spaces in the multivariate setting. The frames constructed are unions of independent Riesz sequences based on tensor products of univariate brushlet functions, which simplifies the analysis of the full frame. We show that the multivariate α-modulation spaces can be completely characterized by the Banach frames constructed

Approximation with brushlet systems

AbstractWe consider an orthonormal basis for L2(R) consisting of functions that are well localized in the spatial domain and have compact support in the frequency domain. The construction is based on smooth local cosine bases and is inspired by Meyer and Coifman's brushlets, which are local exponentials in the frequency domain. For brushlet bases associated with an exponential-type partition of the frequency axis, we show that the system constitutes an unconditional basis for Lp(R), 1<p<∞, Bqs(Lp(R)), 1<p,q<∞, s>0, and that the norm in these spaces can be expressed by the expansion coefficients. In Lp(R), we construct greedy brushlet-type bases and derive Jackson and Bernstein inequalities. Finally, we investigate a natural bivariate extension leading to ridgelet-type bases for L2(R2)

On the equivalence of brushlet and wavelet bases

AbstractWe prove that the Meyer wavelet basis and a class of brushlet systems associated with exponential type partitions of the frequency axis form a family of equivalent (unconditional) bases for the Besov and Triebel–Lizorkin function spaces. This equivalence is then used to obtain new results on nonlinear approximation with brushlets in Triebel–Lizorkin spaces