We provide conditions under which a generalized shift-invariant (GSI) system can be approximated by a GSI system for which the generators have compact support in the Fourier domain. The approximation quality will be measured in terms of the Bessel bound (upper frame bound) for the difference between the two GSI systems. In particular, this leads to easily verifiable conditions for a perturbation of a GSI system to preserve the frame property.

Gabardo and Nashed [‘Nonuniform multiresolution analyses and spectral pairs’, J. Funct. Anal. 158(1) (1998), 209–241] have introduced the concept of nonuniform multiresolution analysis (NUMRA), based on the theory of spectral pairs, in which the associated translated set $\Lambda =\{0,{r}/{N}\}+2\mathbb Z$ is not necessarily a discrete subgroup of $\mathbb{R}$ , and the translation factor is $2\textrm{N}$ . Here r is an odd integer with $1\leq r\leq 2N-1$ such that r and N are relatively prime. The nonuniform wavelets associated with NUMRA can be used in signal processing, sampling theory, speech recognition and various other areas, where instead of integer shifts nonuniform shifts are needed. In order to further generalize this useful NUMRA, we consider the set $\widetilde {\Lambda }=\{0,{r_1}/{N},{r_2}/{N},\ldots ,{r_q}/{N}\}+s\mathbb Z$ , where s is an even integer, $q\in \mathbb {N}$ , $r_i$ is an integer such that $1\leq r_i\leq sN-1,\,(r_i,N)=1$ for all i and $N\geq 2$ . In this paper, we prove that the concept of NUMRA with the translation set $\widetilde {\Lambda }$ is possible only if $\widetilde {\Lambda }$ is of the form $\{0,{r}/{N}\}+s\mathbb Z$ . Next we introduce $\Lambda _s$ -nonuniform multiresolution analysis ( $\Lambda _s$ -NUMRA) for which the translation set is $\Lambda _s=\{0,{r}/{N}\}+s\mathbb Z$ and the dilation factor is $sN$ , where s is an even integer. Also, we characterize the scaling functions associated with $\Lambda _s$ -NUMRA and we give necessary and sufficient conditions for wavelet filters associated with $\Lambda _s$ -NUMRA.

This paper establishes the mapping properties of pseudo-differential operators and the Fourier integral operators on the weighted Morrey spaces with variable exponents and the weighted Triebel–Lizorkin–Morrey spaces with variable exponents. We obtain these results by extending the extrapolation theory to the weighted Morrey spaces with variable exponents. This extension also gives the mapping properties of Calderón–Zygmund operators on the weighted Hardy–Morrey spaces with variable exponents and the wavelet characterizations of the weighted Hardy–Morrey spaces with variable exponents.

A sequence $\left \{g_k\right \}_{k=1}^{\infty }$ in a Hilbert space ${\cal H}$ has the expansion property if each $f\in \overline {\text {span}} \left \{g_k\right \}_{k=1}^{\infty }$ has a representation $f=\sum _{k=1}^{\infty } c_k g_k$ for some scalar coefficients $c_k.$ In this paper, we analyze the question whether there exist small norm-perturbations of $\left \{g_k\right \}_{k=1}^{\infty }$ which allow to represent all $f\in {\cal H};$ the answer turns out to be yes for frame sequences and Riesz sequences, but no for general basic sequences. The insight gained from the analysis is used to address a somewhat dual question, namely, whether it is possible to remove redundancy from a sequence with the expansion property via small norm-perturbations; we prove that the answer is yes for frames $\left \{g_k\right \}_{k=1}^{\infty }$ such that $g_k\to 0$ as $k\to \infty ,$ as well as for frames with finite excess. This particular question is motivated by recent progress in dynamical sampling.

We prove that the HRT (Heil, Ramanathan, and Topiwala) Conjecture is equivalent to the conjecture that co-central translates of square-integrable functions on the Heisenberg group are linearly independent.

We give a characterization of all Parseval wavelet frames arising from a given frame multiresolution analysis. As a consequence, we obtain a description of all Parseval wavelet frames associated with a frame multiresolution analysis. These results are based on a version of Oblique Extension Principle with the assumption that the origin is a point of approximate continuity of the Fourier transform of the involved refinable functions. Our results are written for reducing subspaces.

We prove that an $L^{\infty }$ potential in the Schrödinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace ${\mathcal{W}}$. As a corollary, we obtain a similar result for Calderón’s inverse conductivity problem. Lipschitz stability estimates and a globally convergent nonlinear reconstruction algorithm for both inverse problems are also presented. These are the first results on global uniqueness, stability and reconstruction for nonlinear inverse boundary value problems with finitely many measurements. We also discuss a few relevant examples of finite dimensional subspaces ${\mathcal{W}}$, including bandlimited and piecewise constant potentials, and explicitly compute the number of required measurements as a function of $\dim {\mathcal{W}}$.

This paper presents a framework for compressed sensing that bridges a gap between existing theory and the current use of compressed sensing in many real-world applications. In doing so, it also introduces a new sampling method that yields substantially improved recovery over existing techniques. In many applications of compressed sensing, including medical imaging, the standard principles of incoherence and sparsity are lacking. Whilst compressed sensing is often used successfully in such applications, it is done largely without mathematical explanation. The framework introduced in this paper provides such a justification. It does so by replacing these standard principles with three more general concepts: asymptotic sparsity, asymptotic incoherence and multilevel random subsampling. Moreover, not only does this work provide such a theoretical justification, it explains several key phenomena witnessed in practice. In particular, and unlike the standard theory, this work demonstrates the dependence of optimal sampling strategies on both the incoherence structure of the sampling operator and on the structure of the signal to be recovered. Another key consequence of this framework is the introduction of a new structured sampling method that exploits these phenomena to achieve significant improvements over current state-of-the-art techniques.

In this article, we give necessary and sufficient conditions on a function to be a low-pass filter on a local field $K$ of positive characteristic associated with the scaling function for multiresolution analysis of ${{L}^{2}}\left( K \right)$. We use probability and martingale methods to provide such a characterization.

In this article, we investigate the pointwise behaviors of functions on the Heisenberg group. We find wavelet characterizations for the global and local Hölder exponents. Then we prove some a priori upper bounds for the multifractal spectrum of all functions in a given Hölder, Sobolev, or Besov space. These upper bounds turn out to be optimal, since in all cases they are reached by typical functions in the corresponding functional spaces. We also explain how to adapt our proof to extend our results to Carnot groups.

The unitary extension principle $\left( \text{UEP} \right)$ by A. Ron and Z. Shen yields a sufficient condition for the construction of Parseval wavelet frames with multiple generators. In this paper we characterize the $\text{UEP}$-type wavelet systems that can be extended to a Parseval wavelet frame by adding just one $\text{UEP}$-type wavelet system. We derive a condition that is necessary for the extension of a UEP-type wavelet system to any Parseval wavelet frame with any number of generators and prove that this condition is also sufficient to ensure that an extension with just two generators is possible.

In this work we introduce a class of discrete groups containing subgroups of abstract translations and dilations, respectively. A variety of wavelet systems can appear as $\pi \left( \Gamma \right)\psi $, where $\pi $ is a unitary representation of a wavelet group and $\Gamma $ is the abstract pseudo-lattice $\Gamma $. We prove a sufficent condition in order that a Parseval frame $\pi \left( \Gamma \right)\psi $ can be dilated to an orthonormal basis of the form $\tau \left( \Gamma \right)\Psi $, where $\tau $ is a super-representation of $\pi $. For a subclass of groups that includes the case where the translation subgroup is Heisenberg, we show that this condition always holds, and we cite familiar examples as applications.

In this paper, the dimension function of a self-affine generalized scaling set associated with an $n\,\times \,n$ integral expansive dilation $A$ is studied. More specifically, we consider the dimension function of an $A$-dilation generalized scaling set $K$ assuming that $K$ is a self-affine tile satisfying $BK\,=\,\left( K\,+\,{{d}_{1}} \right)\,\cup \,\left( K\,+\,{{d}_{2}} \right)$, where $B\,=\,{{A}^{t}},\,A$ is an $n\,\times \,n$ integral expansive matrix with $\left| \det \,A \right|\,=\,2$, and ${{d}_{1}},\,{{d}_{2}}\,\in \,{{\mathbb{R}}^{n}}$. We show that the dimension function of $K$ must be constant if either $n\,=1$ or 2 or one of the digits is 0, and that it is bounded by $2\left| K \right|$ for any $n$.

We investigate the solutions of refinement equations of the form

$$\phi (x)\,=\,\sum\limits_{\alpha \in {{\mathbb{Z}}^{S}}}{a(\alpha )\,}\phi (Mx\,-\,\alpha ),$$

where the function $\phi $ is in ${{L}_{p}}({{\mathbb{R}}^{s}})(1\,\le \,p\,\le \,\infty )$, $a$ is an infinitely supported sequence on ${{\mathbb{Z}}^{s}}$ called a refinement mask, and $M$ is an $s\,\times \,s$ integer matrix such that ${{\lim }_{n\to \infty }}\,{{M}^{-n}}\,=\,0$. Associated with the mask $a$ and $M$ is a linear operator ${{\text{Q}}_{a,M}}$ defined on ${{L}_{p}}({{\mathbb{R}}^{s}})$ by ${{\text{Q}}_{a,M}}{{\phi }_{0}}\,:=\,{{\sum }_{\alpha \in {{\mathbb{Z}}^{s}}}}\,a(\alpha ){{\phi }_{0}}(M\,\cdot \,-\alpha )$. Main results of this paper are related to the convergence rates of ${{(\text{Q}_{a,M}^{n}{{\phi }_{o}})}_{n=1,2,\ldots }}$ in ${{L}_{p}}({{\mathbb{R}}^{s}})$ with mask $a$ being infinitely supported. It is proved that under some appropriate conditions on the initial function ${{\phi }_{0}}$, $\text{Q}_{a,M}^{n}{{\phi }_{0}}$ converges in ${{L}_{p}}({{\mathbb{R}}^{s}})$ with an exponential rate.

A theory of higher rank multiresolution analysis is given in the setting of abelian multiscalings. This theory enables the construction, from a higher rank MRA, of finite wavelet sets whose multidilations have translates forming an orthonormal basis in ${{L}^{2}}({{\mathbb{R}}^{d}})$. While tensor products of uniscaled MRAs provide simple examples we construct many nonseparable higher rank wavelets. In particular we construct Latin square wavelets as rank 2 variants of Haar wavelets. Also we construct nonseparable scaling functions for rank 2 variants of Meyer wavelet scaling functions, and we construct the associated nonseparable wavelets with compactly supported Fourier transforms. On the other hand we show that compactly supported scaling functions for biscaled MRAs are necessarily separable.

We give a further decay estimate for the Dziubański–Hernández wavelets that are band-limited and have subexponential decay. This is done by constructing an appropriate bell function and using the Paley–Wiener theorem for ultradifferentiable functions.

We show that a characterization of scaling functions for multiresolution analyses given by Hernández and Weiss and that a characterization of low-pass filters given by Gundy both hold for multivariable multiresolution analyses.

We show that if is a frame and {ψＱ}Ｑ∈Q ∈ ∩ Mα(ℝn) is its dual frame (for the definition of Mα(ℝn), see Definition 2.1), where Q is the collection of dyadic cubes, then for any f ∈ S′(ℝn), there exists a sequence of polynomials, PL,L′,L″, such that

(0.1)

in the topology of S′(ℝn), where δ(i) = max(2i, 1). We prove this result by explicitly constructing the polynomials PL,L′,L″. Furthermore, using the above result, we assert that the linear span of the one-dimensional wavelet system is dense in a function space if and only if the dual space of this function space has an trivial intersection with the set of polynomials. This is proved by using the annihilator of the one-dimensional wavelet system.

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 sup-adjoint Gabor frame can be written as the finite union of Riesz basic sequences. Finally, we show how existing techniques can be applied to determine whether frames of translates can be written as the finite union of Riesz basic sequences. We end by giving an example of a frame of translates such that any Riesz basic subsequence must consist of highly irregular translates.

We formulate a localization property of wavelet coefficients for processes with stationary increments, in the estimation problem associated with the processes. A general setting for the estimation is adopted and examples that fit this setting are given. An evaluation of wavelet coefficient decay with respect to shift k∈ℕ is explicitly derived (only the asymptotic behavior, for large k, was previously known). It is this evaluation that makes it possible to establish the localization property of the wavelet coefficients. In doing so, it turns out that the theory of positive-definite functions plays an important role. As applications, we show that, in the wavelet coefficient domain, estimators that use a simple moment method are nearly as good as maximum likelihood estimators. Moreover, even though the underlying process is long-range dependent and process domain estimates imply the validity of a noncentral limit theorem, for the wavelet coefficient domain estimates we always obtain a central limit theorem with a small prescribed error.