Jae Kun Lim

Perturbation of frame sequences in shift-invariant spaces

We prove a new perturbation criteria for frame sequences, which generalizes previous results and is easier to apply. In the special case of frames infinitely generated shift-invariant subspaces of L2(ℝd) the condition can be formulated in terms of the norm of a finite Gram matrix and a corresponding rank condition.

Quasi-Biorthogonal Frame Multiresolution Analyses and Wavelets

We introduce the concepts of quasi-biorthogonal frame multiresolution analyses and quasi-biorthogonal frame wavelets which are natural generalizations of biorthogonal multiresolution analyses and biorthogonal wavelets, respectively. Necessary and sufficient conditions for quasi-biorthogonal frame multiresolution analyses to admit quasi-biorthogonal wavelet frames are given, and a non-trivial example of quasi-biorthogonal frame multiresolution analyses admitting quasi-biorthogonal frame wavelets is constructed. Finally, we characterize the pair of quasi-biorthogonal frame wavelets that is associated with quasi-biorthogonal frame multiresolution analyses.

A pair of orthogonal frames

We start with a characterization of a pair of frames to be orthogonal in a shift-invariant space and then give a simple construction of a pair of orthogonal shift-invariant frames. This is applied to obtain a construction of a pair of Gabor orthogonal frames as an example. This is also developed further to obtain constructions of a pair of orthogonal wavelet frames.

Local analysis of frame multiresolution analysis with a general dilation matrix

A multivariate semi-orthogonal frame multiresolution analysis with a general integer dilation matrix and multiple scaling functions is considered. We first derive the formulas of the lengths of the inital (central) shift-invariant space V 0 and the next dilation space V 1 , and, using these formulas, we then address the problem of the number of the elements of a wavelet set, that is, the length of the shift-invariant space W 0 := V 1 ⊖ V 0 . Finally, we show that there does not exist a ‘genuine’ frame multiresolution analysis for which V 0 and V 1 are quasi-stable spaces satisfying the usual length condition.

On Riesz wavelets associated with multiresolution analyses

Abstract We obtain some properties that Riesz wavelets and the corresponding scaling functions should satisfy in order that the Riesz wavelets be associated with multiresolution analyses (MRAs). They are given in terms of the low/high-pass filters and in terms of the Fourier transform by using the newly obtained necessary and sufficient condition for the sum of two principal shift-invariant subspaces to be closed. The properties are used to improve the characterizations of Riesz wavelets associated with MRAs previously obtained by some of the authors.

Applications of Parameterizations of Oblique Duals of a Frame Sequence

Two parameterizations of oblique duals of a given frame sequence are applied to show that the sum of a frame sequence and one of its oblique duals may not be a frame sequence, and that type II dual is locally (and globally) optimal in some sense and that the excesses of the frame sequence and its oblique duals are the same. Other observations on the duals of a frame sequence are included also.

Gramian analysis of multivariate frame multiresolution analyses

We perform a Gramian analysis of a frame multiresolution analysis to give a condition for it to admit a minimal wavelet set and to show that the frame bounds of the natural generator for the wavelet space of a degenerate frame multiresolution analysis shrink.

Sum and direct sum of frame sequences

Casazza, Han and Larson characterized various properties of the direct sum of two frame sequences. We add characterizations of other properties and study the relationship between the direct sum and the sum of frame sequences. In particular, we find a necessary and sufficient condition for the sum of two strongly disjoint (orthogonal) frame sequences (in the same Hilbert space) to be a frame sequence, and thereby show that the sum of two strongly disjoint frame sequences may not be a frame sequence. We also show that the closedness of the sum of the synthesis operators of two frame sequences and that of the sum of the frame operators of the same frame sequences are not related. Other observations are also included.

Schatten-Class operators and frames

By extending the works of Arias et al. and Balazs we present a natural sufficient condition via frames for the membership of the Schatten p-class operators for 1 ≤ p ≤ 2, and show that such a condition is not sufficient to guarantee the membership of the Schatten-class operators for p > 2.By extending the works of Arias et al. and Balazs we present a natural sufficient condition via frames for the membership of the Schatten p-class operators for 1 ≤ p ≤ 2, and show that such a condition is not sufficient to guarantee the membership of the Schatten-class operators for p > 2. Quaestiones Mathematicae 34(2011), 203–211.

Riesz Sequences of Translates and Generalized Duals with Support on (0,1)

If the integer translates of a function ø with compact support generate a frame for a subspace W of L2(ℝ),then it is automatically a Riesz basis for W, and there exists a unique dual Riesz basis belonging to W. Considerable freedom can be obtained by considering oblique duals, i.e., duals not necessarily belonging to W. Extending work by Ben-Artzi and Ron, we characterize the existence of oblique duals generated by a function with support on an interval of length one. If such a generator exists, we show that it can be chosen with desired smoothness. Regardless whether ø is polynomial or not, the same condition implies that a polynomial dual supported on an interval of length one exists.