Shift-invariant Spaces Research Articles

Frames in spaces with finite rate of innovation

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 \(V_q(\Phi, \Lambda)\) modeling signals with finite rate of innovation, such as stream of (different) pulses found in GPS applications, cellular radio and ultra wide-band communication. In particular, the space \(V_q(\Phi, \Lambda)\) is generated by a family of well-localized molecules \(\Phi\) of similar size located on a relatively separated set \(\Lambda\) using \(\ell^q\) coefficients, and hence is locally finitely generated. Moreover that space \(V_q(\Phi, \Lambda)\) 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 \(\Phi\), we show that if the generator \(\Phi\) is a frame for the space \(V_2(\Phi, \Lambda)\) 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 \(\Phi\) for the space \(V_q(\Phi, \Lambda)\) with \(q\ne 2\), and of the polynomial (sub-exponential) decay property of the mask associated with a refinable function that has polynomial (sub-exponential) decay.

Perturbation of Regular Sampling in Shift-Invariant Spaces for Frames

Perturbation theorems for regular sampling in shift-invariant spaces are derived using a generalized perturbation theorem for frames in a Hilbert space, which generalizes the irregular sampling theorem established by Chen Using this generalized irregular sampling theorem, estimates for the maximum perturbation are obtained. Some typical examples illustrate the result

Multiresolution expansion, approximation order and quasiasymptotic behavior of tempered distributions

Multiresolution analysis of tempered distributions is studied through multiresolution analysis on the corresponding test function spaces S r ( R ) , r ∈ N 0 . For a function h, which is smooth enough and of appropriate decay, it is shown that the derivatives of its projections to the corresponding spaces V j , j ∈ Z , in a regular multiresolution analysis of L 2 ( R ) , denoted by h j , multiplied by a polynomial weight converge in sup norm, i.e., h j → h in S r ( R ) as j → ∞ . Analogous result for tempered distributions is obtained by duality arguments. The analysis of the approximation order of the projection operator within the framework of the theory of shift-invariant spaces gives a further refinement of the results. The order of approximation is measured with respect to the corresponding space of test functions. As an application, we give Abelian and Tauberian type theorems concerning the quasiasymptotic behavior of a tempered distribution at infinity.

Functions in Sampling Spaces

Sampling theory in spaces other than the space of band-limited functions has recently received considerable attention. This is in part because the band-limitedness assumption is not very realistic in many applications. In addition, band-limited functions can have very slow decay which translates in poor reconstruction. In this article we study the sampling problem in general shift invariant spaces. We characterize the functions in these spaces and provide necessary and sucient conditions for a function in L 2 (R) to belong to a sampling space. Furthermore we obtain decompositions of a sampling space in sampling subspaces. These decompositions are related with determining sets. Some examples are provided.

Perturbation of frame sequences and its applications to shift-invariant spaces

We generalize the main result in [O. Christensen, H.O. Kim, R.Y. Kim, J.K. Lim, Perturbation of frame sequences in shift-invariant spaces, J. Geom. Anal. 15 (2005) 181–191] in order to make it comparable with existing results. Then we compare the special cases of the three results in the literature in the setting of the perturbation of the generating sets of finitely generated shift-invariant spaces of L 2 ( R d ) .

Nonideal sampling and interpolation from noisy observations in shift-invariant spaces

Digital analysis and processing of signals inherently relies on the existence of methods for reconstructing a continuous-time signal from a sequence of corrupted discrete-time samples. In this paper, a general formulation of this problem is developed that treats the interpolation problem from ideal, noisy samples, and the deconvolution problem in which the signal is filtered prior to sampling, in a unified way. The signal reconstruction is performed in a shift-invariant subspace spanned by the integer shifts of a generating function, where the expansion coefficients are obtained by processing the noisy samples with a digital correction filter. Several alternative approaches to designing the correction filter are suggested, which differ in their assumptions on the signal and noise. The classical deconvolution solutions (least-squares, Tikhonov, and Wiener) are adapted to our particular situation, and new methods that are optimal in a minimax sense are also proposed. The solutions often have a similar structure and can be computed simply and efficiently by digital filtering. Some concrete examples of reconstruction filters are presented, as well as simple guidelines for selecting the free parameters (e.g., regularization) of the various algorithms

Characterization of Oblique Dual Frame Pairs

Given a frame for a subspace W of a Hilbert space H, we consider all possible families of oblique dual frame vectors on an appropriately chosen subspace V. In place of the standard description, which involves computing the pseudoinverse of the frame operator, we develop an alternative characterization which in some cases can be computationally more efficient. We first treat the case of a general frame on an arbitrary Hilbert space, and then specialize the results to shift-invariant frames with multiple generators. In particular, we present explicit versions of our general conditions for the case of shift-invariant spaces with a single generator. The theory is also adapted to the standard frame setting in which the original and dual frames are defined on the same space.

Local bases for refinable spaces

We provide a new representation of a refinable shift invariant space with a compactly supported generator, in terms of functions with a special property of homogeneity. In particular, these functions include all the homogeneous polynomials that are reproducible by the generator, which links this representation to the accuracy of the space. We completely characterize the class of homogeneous functions in the space and show that they can reproduce the generator. As a result we conclude that the homogeneous functions can be constructed from the vectors associated to the spectrum of the scale matrix (a finite square matrix with entries from the mask of the generator). Furthermore, we prove that the kernel of the transition operator has the same dimension as the kernel of this finite matrix. This relation provides an easy test for the linear independence of the integer translates of the generator. This could be potentially useful in applications to approximation theory, wavelet theory and sampling.

Affine frame decompositions and shift-invariant spaces

In this paper, we show that the property of tight affine frame decomposition of functions in L 2 can be extended in a stable way to functions in Sobolev spaces when the generators of the tight affine frames satisfy certain mild regularity and vanishing moment conditions. Applying the affine frame operators Q j on jth levels to any function f in a Sobolev space reveals the detailed information Q j f of f in such tight affine decompositions. We also study certain basic properties of the range of the affine frame operators Q j such as the topological property of closedness and the notion of angles between the ranges for different levels, and thus establishing some interesting connection to (tight) frames of shift-invariant spaces.

REFINABLE SHIFT INVARIANT SPACES IN ℝd

Let φ : ℝd → ℂ be a compactly supported function which satisfies a refinement equation of the form [Formula: see text] where Γ ⊂ ℝd is a lattice, Λ is a finite subset of Γ, and A is a dilation matrix. We prove, under the hypothesis of linear independence of the Γ-translates of φ, that there exists a correspondence between the vectors of the Jordan basis of a finite submatrix of L = [cAi-j]i,j∈Γ and a finite-dimensional subspace [Formula: see text] in the shift-invariant space generated by φ. We provide a basis of [Formula: see text] and show that its elements satisfy a property of homogeneity associated to the eigenvalues of L. If the function φ has accuracy κ, this basis can be chosen to contain a basis for all the multivariate polynomials of degree less than κ. These latter functions are associated to eigenvalues that are powers of the eigenvalues of A-1. Furthermore we show that the dimension of [Formula: see text] coincides with the local dimension of φ, and hence, every function in the shift-invariant space generated by φ can be written locally as a linear combination of translates of the homogeneous functions.

Robustness of sampling and reconstruction and Beurling–Landau-type theorems for shift-invariant spaces

Beurling–Landau-type results are known for a rather small class of functions limited to the Paley–Wiener space and certain spline spaces. Here, we show that the sampling and reconstruction problem in shift-invariant spaces is robust with respect to two classes of probing measures as well as to the underlying shift-invariant space. As an application we enlarge the class of functions for which Beurling–Landau-type results hold.

Characterization of the closedness of the sum of two shift-invariant spaces

We first present a formula for the supremum cosine angle between two closed subspaces of a separable Hilbert space under the assumption that the ‘generators’ form frames for the subspaces. We then characterize the conditions that the sum of two, not necessarily finitely generated, shift-invariant subspaces of L 2 ( R d ) be closed. If the fibers of the generating sets of the shift-invariant subspaces form frames for the fiber spaces a.e., which is satisfied if the shift-invariant subspaces are finitely generated or if the shifts of the generating sets form frames for the respective subspaces, then the characterization is given in terms of the norms of possibly infinite matrices. In particular, if the shift-invariant subspaces are finitely generated, then the characterization is given wholly in terms of the norms of finite matrices.

Some equations relating multiwavelets and multiscaling functions

The local trace function introduced in [Dutkay, The local trace function of shift invariant spaces, J. Operat. Theory 52(2) (2004), 267–291] is used to derive equations that relate multiwavelets and multiscaling functions in the context of a generalized multiresolution analysis, without appealing to filters. A construction of normalized tight frame (NTF) wavelets is given. Particular instances of the construction include NTF and orthonormal wavelet sets.

The infimum cosine angle between two finitely generated shift-invariant spaces and its applications

An expression of the infimum cosine angle between two finite dimensional subspaces is given in terms of Gramians and is applied to find a useful formula for the infimum cosine angle between two finitely generated shift-invariant subspaces of L 2 ( R d ) in terms of the generators. We then present new equivalent conditions for the existence of the oblique projection between two finitely generated shift-invariant subspaces, thereby providing a constructive method to generate oblique dual frames. This result is a generalization of a result of Aldroubi and is closely related with the biorthogonality of two frame multiresolution analyses. Finally, we illustrate our results by examples.

Reconstruction algorithm in lattice-invariant signal spaces

In this paper, we mainly pay attention to the weighted sampling and reconstruction algorithm in lattice-invariant signal spaces. We give the reconstruction formula in lattice-invariant signal spaces, which is a generalization of former results in shift-invariant signal spaces. That is, we generalize and improve Aldroubi, Grochenig and Chen’s results, respectively. So we obtain a general reconstruction algorithm in lattice-invariant signal spaces, which the signal spaces is sufficiently large to accommodate a large number of possible models. They are maybe useful for signal processing and communication theory.

SAMPLING AND OVERSAMPLING IN SHIFT-INVARIANT AND MULTIRESOLUTION SPACES I: VALIDATION OF SAMPLING SCHEMES

We ask what conditions can be placed on generators φ of principal shift invariant spaces to ensure the validity of analogues of the classical sampling theorem for bandlimited signals. Critical rate sampling schemes lead to expansion formulas in terms of samples, while oversampling schemes can lead to expansions in which function values depend only on nearby samples. The basic techniques for validating such schemes are built on the Zak transform and the Poisson summation formula. Validation conditions are phrased in terms of orthogonality, smoothness, and self-similarity, as well as bandlimitedness or compact support of the generator. Effective sampling rates which depend on the length of support of the generator or its Fourier transform are derived.

On the existence of multiresolution analysis for framelets

We show that a compactly supported tight framelet comes from an MRA if the intersection of all dyadic dilations of the space of negative dilates, which is defined as the shift-invariant space generated by the negative scales of a framelet, is trivial. We also construct examples of (non-tight) framelets, which are arbitrarily close to tight frame framelets, such that the corresponding space of negative dilates is equal to the entire space L 2ℝ.

Estimate of aliasing error for non-smooth signals prefiltered by quasi-projections into shift-invariant spaces

An ideal analog-to-digital (A/D) converter prefilters a signal by an ideal lowpass filter. Recent research on A/D conversion based on shift-invariant spaces reveals that prefiltering signals by quasiprojections into shift-invariant spaces provides more flexible choices in designing an A/D conversion system of high accuracy. This paper focuses on the accuracy of such prefiltering, in which the aliasing error e/sub f//sup /spl lambda// is found to behave like e/sub f//sup /spl lambda// /sub 2/ = O(/spl lambda//sup -/spl alpha//) with respect to the dilation /spl lambda/ of the underlying shift-invariant space, provided that the input signal f is Lipschitz-/spl alpha/ continuous. A formula to calculate the coefficient of the decay rate is also figured out in this paper.

The finite section method and problems in frame theory

The finite section method is a convenient tool for approximation of the inverse of certain operators using finite-dimensional matrix techniques. In this paper we demonstrate that the method is very useful in frame theory: it leads to an efficient approximation of the inverse frame operator and also solves related computational problems in frame theory. In the case of a frame which is localized w.r.t. an orthonormal basis we are able to estimate the rate of approximation. The results are applied to the reproducing kernel frame appearing in the theory for shift-invariant spaces generated by a Riesz basis.

Riesz bases in [formula omitted] related to sampling in shift-invariant spaces

The Fourier duality is an elegant technique to obtain sampling formulas in Paley–Wiener spaces. In this paper it is proved that there exists an analogue of the Fourier duality technique in the setting of shift-invariant spaces. In fact, any shift-invariant space V φ with a stable generator φ is the range space of a bounded one-to-one linear operator T between L 2 ( 0 , 1 ) and L 2 ( R ) . Thus, regular and irregular sampling formulas in V φ are obtained by transforming, via T, expansions in L 2 ( 0 , 1 ) with respect to some appropriate Riesz bases.