Abstract. A new construction of tight frames for L 2 (R d ) with flexible time-frequency localization is considered. The frames can be adapted to form atomic decompositions for a large family of smoothness spaces on R d , a class of so-called decomposition spaces. The decomposition space norm can be completely characterized by a sparseness condition on the frame coefficients. As examples of the general construction, new tight frames yielding decompositions of Besov space, anisotropic Besov spaces, α-modulation spaces, and anisotropic α-modulation spaces are considered. Finally, curvelet-type tight frames are constructed on R d , d ≥ 2.
The α-modulation spaces M s,α p,q (R d ), 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.
The α-modulation spaces M s,α p,q (R d ), α∈[0, 1], form a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that a pseudodifferential operator σ(x, D) with symbol in the Hörmander class S b ρ,0 extends to a bounded oper-, and 1
International audienceWe study the approximation properties of wavelet bi-frame systems in Lp(R^d). For wavelet bi-frame systems the approximation spaces associated with best m-term approximation are completely characterized for a certain range of smoothness parameters limited by the number of vanishing moments of the generators of the dual frame. The approximation spaces turn out to be essentially Besov spaces, just as in the classical orthonormal wavelet case. We also prove that for smooth functions, the canonical expansion in the wavelet bi-frame system is sparse and one can reach the optimal rate of approximation by simply thresholding the canonical expansion. For twice oversampled MRA based wavelet frames, a characterization of the associated approximation space is obtained without any restrictions given by the number of vanishing moments, but at a price of replacing the canonical expansion by another linear expansion
A construction of Triebel-Lizorkin type spaces associated with flexible decompositions of the frequency spaceℝdis considered. The class of admissible frequency decompositions is generated by a one parameter group of (anisotropic) dilations onℝdand a suitable decomposition function. The decomposition function governs the structure of the decomposition of the frequency space, and for a very particular choice of decomposition function the spaces are reduced to classical (anisotropic) Triebel-Lizorkin spaces. An explicit atomic decomposition of the Triebel-Lizorkin type spaces is provided, and their interpolation properties are studied. As the main application, we consider Hörmander type classes of pseudo-differential operators adapted to the anisotropy and boundedness of such operators between corresponding Triebel-Lizorkin type spaces is proved.
The α-modulation spaces are a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that brushlet bases can be constructed to form unconditional and even greedy bases for the α-modulation spaces. We study m-term nonlinear approximation with brushlet bases, and give complete characterizations of the associated approximation spaces in terms of α-modulation spaces.
Abstract. We study tight wavelet frame systems in Lp (R d ) and prove that such systems (under mild hypotheses) give atomic decompositions of Lp(R d ) for 1 < p < ∞. We also characterize Lp(R d ) and Sobolev space norms by the analysis coefficients for the frame. We consider Jackson inequalities for best mterm approximation with the systems in Lp(R d ) and prove that such inequalities exist. Moreover, it is proved that the approximation rate given by the Jackson inequality can be realized by thresholding the frame coefficients. Finally, we show that in certain restricted cases, the approximation spaces, for best m-term approximation, associated with tight wavelet frames can be characterized in terms of (essentially) Besov spaces.
We consider the problem of recovering a structured sparse representation of a signal in an overcomplete time-frequency dictionary with a particular structure. For infinite dictionaries that are the union of a nice wavelet basis and a Wilson basis, sufficient conditions are given for the Basis Pursuit and (Orthogonal) Matching Pursuit algorithms to recover a structured representation of an admissible signal. The sufficient conditions take into account the structure of the wavelet/Wilson dictionary and allow very large (even infinite) support sets to be recovered even though the dictionary is highly coherent.
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