General A-P iterative algorithm in shift-invariant spaces

A general A-P iterative algorithm in a shift-invariant space is presented. We use the algorithm to show reconstruction of signals from weighted samples and also show that the general improved algorithm has better convergence rate than the existing one. An explicit estimate for a guaranteed rate of convergence is given.     

Local reconstruction for sampling in shift-invariant spaces

The local reconstruction from samples is one of most desirable properties for many applications in signal processing, but it has not been given as much attention. In this paper, we will consider the local reconstruction problem for signals in a shift-invariant space. In particular, we consider finding sampling sets X such that signals in a shift-invariant space can be locally reconstructed from their samples on X. For a locally finite-dimensional shift-invariant space V we show that signals in V can be locally reconstructed from its samples on any sampling set with sufficiently large density. For a shift-invariant space V(? ₁, ..., ? _N ) generated by finitely many compactly supported functions ? ₁, ..., ? _N , we characterize all periodic nonuniform sampling sets X such that signals in that shift-invariant space V(? ₁, ..., ? _N ) can be locally reconstructed from the samples taken from X. For a refinable shift-invariant space V(?) generated by a compactly supported refinable function ?, we prove that for almost all \((x_0, x_1)\in [0,1]^2\), any signal in V(?) can be locally reconstructed from its samples from \(\{x_0, x_1\}+{\mathbb Z}\) with oversampling rate 2. The proofs of our results on the local sampling and reconstruction in the refinable shift-invariant space V(?) depend heavily on the linear independent shifts of a refinable function on measurable sets with positive Lebesgue measure and the almost ripplet property for a refinable function, which are new and interesting by themselves.     

Invariance of shift-invariant spaces

In this paper, we give a characterization of shift-invariant subspaces which are also invariant under additional non-integer translations. Both principal and finitely generated shift-invariant subspaces are studied. Our results improve some known ones.     

Approximation from locally finite-dimensional shift-invariant spaces

After exploring some topological properties of locally finite-dimensional shift-invariant subspaces of , we show that if provides approximation order , then it provides the corresponding simultaneous approximation order. In the case is generated by a compactly supported function in , it is proved that provides approximation order in the -norm with if and only if the generator is a derivative of a compactly supported function that satisfies the Strang-Fix conditions.

     

主平移不变空间的特征

Shifts-invariant spaces in L1 (R) are investigated. First,based on a study of the system of linearly difference operators, the method of constructing generators with linearly independent shifts is provided. Then the characterizations of the closed shift-invariant subspaces in L1 (R) are given in terms of such generators and the local basis of shift-invariant subspaces.     

Generalized sampling in shift-invariant spaces with multiple stable generators

The aim of this paper is to derive stable generalized sampling in a shift-invariant space with ? stable generators. This is done in the light of the theory of frames in the product Hilbert space (? times). The generalized samples are expressed as the frame coefficients of an appropriate function in with respect to some particular frame in . Since any multiply stable generated shift-invariant space is the image of by means of a bounded invertible operator, the generalized sampling is obtained from some dual frame expansions in . An example in the setting of the Hermite cubic splines is exhibited.     

Dual frames in connected with generalized sampling in shift-invariant spaces

The aim of this article is to derive stable generalized sampling in a shift-invariant space by using some special dual frames in L²(0,1). These sampling formulas involve samples of filtered versions of the functions in the shift-invariant space. The involved samples are expressed as the frame coefficients of an appropriate function in L²(0,1) with respect to some particular frame in L²(0,1). Since any shift-invariant space with stable generator is the image of L²(0,1) by means of a bounded invertible operator, our generalized sampling is derived from some dual frame expansions in L²(0,1).     

平移不变空间的非均匀采样

为了从采样值稳定和惟一地重建信号f(x),函数在采样集{xj,j=∈Z}上的采样值{f(xj),j∈Z}要满足A||f||L^p^p≤∑j∈Z|f(xj)|^p≤B||f||L^p^p.本文研究在平移不变空间V^p[φ]中使上式成立的条件,在几种情况下,得到了使上式成立的采样条件,并建立了由采样值重建f(x)的重建算法,该算法比一般的迭代重建算法收敛更快．     

Multiple sampling and interpolation in Bergman spaces

We study multiple sampling and interpolation problems with unbounded multiplicities in the weighted Bergman space, both in the hilbertian case and the uniform case.     

Multivariate generalized sampling in shift-invariant spaces and its approximation properties

Nowadays the topic of sampling in a shift-invariant space is having a significant impact: it avoids most of the problems associated with classical Shannon's theory. Under appropriate hypotheses, any multivariate function in a shift-invariant space can be recovered from its samples at Zd. However, in many common situations the available data are samples of some convolution operators acting on the function itself: this leads to the problem of multivariate generalized sampling in shift-invariant spaces. This extra information on the functions in the shift-invariant space will allow to sample in an appropriate sub-lattice of Zd. In this paper an L²(Rd) theory involving the frame theory is exhibited. Sampling formulas which are frame expansions for the shift-invariant space are obtained. In the case of overcomplete frame formulas, the search of reconstruction functions with prescribed good properties is allowed. Finally, approximation schemes using these generalized sampling formulas are included.     

Sets of sampling and interpolation in Bergman spaces

Properties of the unions of sampling and interpolation sets for Bergman spaces are discussed in conjunction with the examples given by Seip (1993). Their relationship to the classical interpolation sequences is explored. In addition, the role played by canonical divisors in the study of these sets is examined and an example of a sampling set is constructed in the disk.

     

Using a natural deconvolution for analysis of perturbed integer sampling in shift-invariant spaces

An important cornerstone of both wavelet and sampling theory is shift-invariant spaces, that is, spaces V spanned by a Riesz basis of integer-translates of a single function. Under some mild differentiability and decay assumptions on the Fourier transform of this function, we show that V also is generated by a function with Fourier transform for some g with . We explain why analysis of this particular generating function can be more likely to provide large jitter bounds ε such that any f∈V can be reconstructed from perturbed integer samples f(k+εk) whenever supk∈Z|εk|?ε. We use this natural deconvolution of to further develop analysis techniques from a previous paper. Then we demonstrate the resulting analysis method on the class of spaces for which g has compact support and bounded variation (including all spaces generated by Meyer wavelet scaling functions), on some particular choices of φ for which we know of no previously published bounds and finally, we use it to improve some previously known bounds for B-spline shift-invariant spaces.     

Uncertainty principles in finitely generated shift-invariant spaces with additional invariance

We consider finitely generated shift-invariant spaces (SIS) with additional invariance in L²(R^d)$L^2({\mathbb{R}}^d)$. We prove that if the generators and their translates form a frame, then they must satisfy some stringent restrictions on their behavior at infinity. Part of this work (non-trivially) generalizes recent results obtained in the special case of a principal shift-invariant spaces in L²(R)$L^2(\mathbb{R})$ whose generator and its translates form a Riesz basis.     

Generalized average sampling in shift invariant spaces

Average sampling is motivated by realistic needs, e.g., physical limitation of acquisition devices or characteristics of sampling procedure. As an extension of the average sampling, we study generalized average sampling in shift invariant spaces with frame generators, in which averages are taken from suitable channeled version of signals. Illustrative examples are given.     

Local set theory

In 1945, Eilenberg and MacLane introduced the new mathematical notion of category. Unfortunately, from the very beginning, category theory did not fit into the framework of either Zermelo—Fraenkel set theory or even von Neumann—Bernays—Gödel set-class theory. For this reason, in 1959, MacLane posed the general problem of constructing a new, more flexible, axiomatic set theory which would be an adequate logical basis for the whole of naïve category theory. In this paper, we give axiomatic foundations for local set theory. This theory might be one of the possible solutions of the MacLane problem.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 194–212.Original Russian Text Copyright © 2005 by V. K. Zakharov.This revised version was published online in April 2005 with a corrected issue number.