In this article we investigate the frame properties and closedness for the shift invariant spaceWe derive necessary and sufficient conditions for an indexed family {φ i (· − j) : 1 ≤ i ≤ r, j ∈ Z d } to constitute a p-frame for V p ( ), and to generate a closed shift invariant subspace of L p . A function in the L p -closure of V p ( ) is not necessarily generated by p coefficients. Hence we often hope that V p ( ) itself is closed, i.e., a Banach space. For p = 2, this issue is complicated, but we show that under the appropriate conditions on the frame vectors, there is an equivalence between the concept of p-frames, Banach frames, and the closedness of the space they generate. The relation between a function f ∈ V p ( ) and the coefficients of its representations is neither obvious, nor unique, in general. For the case of p-frames, we are in the context of normed linear spaces, but we are still able to give a characterization of p-frames in terms of the equivalence between the norm of f and an p -norm related to its representations. A Banach frame does not have a dual Banach frame in general, however, for the shift invariant spaces V p ( ), dual Banach frames exist and can be constructed.

Abstract. The classical Wiener lemma and its various generalizations are important and have numerous applications in numerical analysis, wavelet theory, frame theory, and sampling theory. There are many different equivalent formulations for the classical Wiener lemma, with an equivalent formulation suitable for our generalization involving commutative algebra of infinite matricesIn the study of spline approximation, (diffusion) wavelets and affine frames, Gabor frames on non-uniform grid, and non-uniform sampling and reconstruction, the associated algebras of infinite matrices are extremely non-commutative, but we expect those noncommutative algebras to have a similar property to Wiener's lemma for the commutative algebra W. In this paper, we consider two non-commutative algebras of infinite matrices, the Schur class and the Sjöstrand class, and establish Wiener's lemmas for those matrix algebras.

From an average (ideal) sampling/reconstruction process, the question arises whether the original signal can be recovered from its average (ideal) samples and, if so, how. We consider the above question under the assumption that the original signal comes from a prototypical space modeling signals with a finite rate of innovation, which includes finitely generated shift-invariant spaces, twisted shift-invariant spaces associated with Gabor frames and Wilson bases, and spaces of polynomial splines with nonuniform knots as its special cases. We show that the displayer associated with an average (ideal) sampling/reconstruction process, which has a well-localized average sampler, can be found to be well-localized. We prove that the reconstruction process associated with an average (ideal) sampling process is robust, locally behaved, and finitely implementable, and thus we conclude that the original signal can be approximately recovered from its incomplete average (ideal) samples with noise in real time. Most of our results in this paper are new even for the special case when the original signal comes from a finitely generated shift-invariant space.Here for each γ ∈ Γ, the functionψ γ , to be known as the display block at the location γ, reflects the characteristic of the display device at the sampling location γ. We call the above reconstruction process an average reconstruction process and the collection Ψ := {ψ γ , γ ∈ Γ} of display blocks an displayer.For the efficiency and stability of the reconstruction process (1.3), (1.4) to recover a function f in the space V from its averaging samples { f, ψ γ , γ ∈ Γ} or from its ideal samples {f (γ), γ ∈ Γ}, we require the corresponding displayerΨ := {ψ γ , γ ∈ Γ} to be well-localized, and the average sampling/reconstruction process (1.3), (1.4) to be robust, local-behaved, and finitely implementable. In this paper, we show that those natural requirements for the average (ideal) sampling/reconstruction process would be met when signals in the space V have a finite rate of innovation and the average sampler Ψ is well-localized; see section 2.3 for our reasons for considering sampling/reconstruction of signals with a finite rate of innovation. Here a signal is said to have a finite rate of innovation if it has a finite degree of freedom per unit of time; see [23,34,40,42,45,46,58].The paper is organized as follows. We divide section 2 into five subsections. In the first three subsections, we make some basic assumptions on the sampling set Γ, the average sampler Ψ = {ψ γ , γ ∈ Γ}, and the space V , which is where the original function f for the average sampling/reconstruction process comes from. Briefly, we assume that the sampling set Γ is a relatively separated subset of R d , the average sampler Ψ is well-localized in the sense that every average sampling functional ψ γ in the average sampler Ψ is essentially located in a neighborhood of γ ∈ Γ, and the space V is the space V q (Φ, Λ), that is, as originally introduced in [52] for modeling Downloaded 08/13/19 to 132.170.27.1...

In this article, we study three interconnected inverse problems in shift invariant spaces: 1) the convolution/deconvolution problem; 2) the uniformly sampled convolution and the reconstruction problem; 3) the sampled convolution followed by sampling on irregular grid and the reconstruction problem. In all three cases, we study both the stable reconstruction as well as ill-posed reconstruction problems. We characterize the convolutors for stable deconvolution as well as those giving rise to ill-posed deconvolution. We also characterize the convolutors that allow stable reconstruction as well as those giving rise to ill-posed reconstruction from uniform sampling. The connection between stable deconvolution, and stable reconstruction from samples after convolution is subtle, as will be demonstrated by several examples and theorems that relate the two problems.

We consider convolution sampling and reconstruction of signals in certain reproducing kernel subspaces of L p , 1 ≤ p ≤ ∞. We show that signals in those subspaces could be stably reconstructed from their convolution samples taken on a relatively separated set with small gap. Exponential convergence and error estimates are established for the iterative approximationprojection reconstruction algorithm.

We introduce a general distributional framework that results in a unifying description and characterization of a rich variety of continuous-time stochastic processes. The cornerstone of our approach is an innovation model that is driven by some generalized white noise process, which may be Gaussian or not (e.g., Laplace, impulsive Poisson or alpha stable). This allows for a conceptual decoupling between the correlation properties of the process, which are imposed by the whitening operator L, and its sparsity pattern which is determined by the type of noise excitation. The latter is fully specified by a Lévy measure. We show that the range of admissible innovation behavior varies between the purely Gaussian and super-sparse extremes. We prove that the corresponding generalized stochastic processes are well-defined mathematically provided that the (adjoint) inverse of the whitening operator satisfies some Lp bound for p ≥ 1. We present a novel operator-based method that yields an explicit characterization of all Lévy-driven processes that are solutions of constant-coefficient stochastic differential equations (SDE). When the underlying system is stable, we recover the family of stationary CARMA processes, including the Gaussian ones. The approach remains valid when the system is unstable and leads to the identification of potentially useful generalizations of the Lévy processes, which are sparse and non-stationary. Finally, we show that these processes admit a sparse representation in some matched wavelet domain and provide a full characterization of their transform-domain statistics.

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Signals with finite rate of innovation are those signals having finite degrees of freedom per unit of time that specify them. In this paper, we introduce a prototypical space V q ( , ) modeling signals with finite rate of innovation, such as stream of (different) pulses found in GPS applications, cellular radio and ultra wideband communication. In particular, the space V q ( , ) is generated by a family of well-localized molecules of similar size located on a relatively separated set using q coefficients, and hence is locally finitely generated. Moreover that space V q ( , ) includes finitely generated shift-invariant spaces, spaces of non-uniform splines, and the twisted shift-invariant space in Gabor (Wilson) system as its special cases. Use the well-localization property of the generator , we show that if the generator is a frame for the space V 2 ( , ) and has polynomial (sub-exponential) decay, then its canonical dual (tight) frame has the same polynomial (sub-exponential) decay. We apply the above result about the canonical dual frame to the study of the Banach frame property of the generator for the space V q ( , ) with q = 2, and of the polynomial (sub-exponential) decay property of the mask associated with a refinable function that has polynomial (sub-exponential) decay.Keywords frame · Banach frame · localized frame · signals with finite rate of innovation · space of homogenous type · matrix algebra · refinable function · wavelets Mathematics Subject Classifications (2000) Primary 42C40 · Secondary 47B37 · 46C07