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Journals and Conferences
The wavelet dimension function for arbitrary real dilations is defined and used to address several questions involving the existence of MRA wavelets and well-localized wavelets for irrational dilations. The theory of quasi-affine frames for rational dilations and the existence of non-MSF wavelets for certain irrational dilations play an important role in… (More)
We give a characterization of all (quasi) aane frames in L 2 (R n) which have a (quasi) aane dual in terms of the two simple equations in the Fourier transform domain. In particular, if the dual frame is the same as the original system, i.e. it is a tight frame, we obtain the well known characterization of wavelets. Although these equations have already… (More)
We investigate Riesz wavelets in the context of generalized multiresolution analysis (GMRA). In particular, we show that Zalik’s class of Riesz wavelets obtained by an MRA is the same as the class of biorthogonal wavelets associated with an MRA. 2003 Elsevier Science (USA). All rights reserved.
We extend the notion of the spectral function of shift-invariant spaces introduced by the authors in [BRz] to the case of general lattices. The main feature is that the spectral function is not dependent on the choice of the underlying lattice with respect to which a space is shift-invariant. We also show that in general the spectral function is not… (More)
The Feichtinger conjecture is considered for three special families of frames. It is shown that if a wavelet frame satisfies a certain weak regularity condition, then it can be written as the finite union of Riesz basic sequences each of which is a wavelet system. Moreover, the above is not true for general wavelet frames. It is also shown that a… (More)
For weights in the matricial Muckenhoupt classes we investigate a number of properties analogous to properties which hold in the scalar Muckenhoupt classes. In contrast to the scalar case we exhibit for each p, 1 < p <∞, a matrix weight W ∈Ap,q \ ⋃ p′<pAp′,q′ . We also give a necessary and sufficient condition on W in Ap,q, a “reverse inverse volume… (More)
We define homogeneous classes of x-dependent anisotropic symbols Ṡ γ,δ(A) in the framework determined by an expansive dilation A, thus extending the existing theory for diagonal dilations. We revisit anisotropic analogues of Hörmander-Mihlin multipliers introduced by Rivière  and provide direct proofs of their boundedness on Lebesgue and Hardy spaces by… (More)
We establish the linear independence of time-frequency translates for functions f having one sided decay limx→∞ |f(x)|e log x = 0 for all c > 0. We also prove such results for functions with faster than exponential decay, i.e., limx→∞ |f(x)|e = 0 for all c > 0, under some additional restrictions.
Let A1 and A2 be expansive dilations, respectively, on R n and R. Let ~ A ≡ (A1, A2) and Ap( ~ A) be the class of product Muckenhoupt weights on R × R for p ∈ (1, ∞]. When p ∈ (1, ∞) and w ∈ Ap( ~ A), the authors characterize the weighted Lebesgue space L w (R × R) via the anisotropic Lusin-area function associated with ~ A. When p ∈ (0, 1], w ∈ A∞( ~ A),… (More)
It is shown that for any expansive, integer valued two by two matrix, there exists a (multi)-wavelet whose Fourier transform is compactly supported and smooth. A key step is showing that for almost every equivalence class of integrally similar matrices there is a representative A which is strictly expansive in the sense that there is a compact set K which… (More)