Processing a radar signal and representations of the discrete Heisenberg group

Baggett, Larry

doi:10.4064/cm-60-61-1-195-203

Cited by 39 publications

(49 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We stress that the results presented here, when applied to Gabor frames, not only produce new results for Gabor frames but also recover, as corollaries, all the existing density results known to hold for Gabor frames. Thus our approach and results both unify and greatly extend the frame density results contained in [20,46,48,35,19], and give new insight into the results in [49,2,42,22,43,51,12], among others.…”

Section: Introductionsupporting

confidence: 68%

Density, Overcompleteness, and Localization of Frames. I. Theory

Bălan

Casazza

Heil

et al. 2006

J Fourier Anal Appl

136

231

View full text Add to dashboard Cite

ABSTRACT. Frames have applications in numerous fields of mathematics and engineering. The fundamental property of frames which makes them so useful is their overcompleteness. In most applications, it is this overcompleteness that is exploited to yield a decomposition that is more stable, more robust, or more compact than is possible using nonredundant systems. This work presents a quantitative framework for describing the overcompleteness of frames. It introduces notions of localization and approximation between two frames F = {f i } i∈I and E = {e j } j∈G (G a discrete abelian group), relating the decay of the expansion of the elements of F in terms of the elements of E via a map a : I → G. A fundamental set of equalities are shown between three seemingly unrelated quantities: The relative measure of F , the relative measure of E -both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements -and the density of the set a(I) in G.Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. In a subsequent article, these results are applied to the case of Gabor frames, producing an array of new results as well as clarifying the meaning of existing results.The notion of localization and related approximation properties introduced in this article are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. A comprehensive examination of the interrelations among these localization and approximation concepts is presented.Math Subject Classifications. 42C15, 46C99.

show abstract

Section: Introductionsupporting

confidence: 68%

Density, Overcompleteness, and Localization of Frames. I. Theory

Bălan

Casazza

Heil

et al. 2006

J Fourier Anal Appl

136

231

View full text Add to dashboard Cite

show abstract

“…Part (a) of Theorem 3 was proved for arbitrary values of αβ by Baggett [6], and for the case that the product αβ is rational by Daubechies [48], with both results appearing in 1990. The operators M βn T αk corresponding to the rectangular lattice αZ × βZ generate a von Neumann algebra, and Baggett's proof made use of this; specifically, he used the representation theory of the discrete Heisenberg group.…”

Section: Incompletenessmentioning

confidence: 98%

History and Evolution of the Density Theorem for Gabor Frames

Heil

2007

J Fourier Anal Appl

174

137

View full text Add to dashboard Cite

ABSTRACT. The Density Theorem for Gabor Frames is one of the fundamental results of time-frequency analysis. This expository survey attempts to reconstruct the long and very involved history of this theorem and to present its context and evolution, from the onedimensional rectangular lattice setting, to arbitrary lattices in higher dimensions, to irregular Gabor frames, and most recently beyond the setting of Gabor frames to abstract localized frames. Related fundamental principles in Gabor analysis are also surveyed, including the Wexler-Raz biorthogonality relations, the Duality Principle, the Balian-Low Theorem, the Walnut and Janssen representations, and the Homogeneous Approximation Property. An extended bibliography is included.

show abstract

“…Now let R 1 denote the regular representation of the discrete group Q A , that is, (1). By the Fourier theory connecting compact and discrete abelian groups, R 1 is equivalent to a representation on the space L 2 ( Q A ) and in fact can be represented as a direct integral of characters as follows.…”

Section: Corollary 24mentioning

confidence: 99%

“…Also, L. Baggett, both on his own and together with K. Merrill and other collaborators [1,2,3], has promoted a representation-theoretic point of view in the study of wavelets and other types of orthonormal bases of L 2 (R n ) associated to discrete groups. In [1], he decomposed the Stone-von Neumann representation of the discrete Heisenberg group on L 2 (R) into a direct integral of representations, and in so doing determined whether or not the translates of Gabor functions parameterized by certain scales spanned a dense subspace of L 2 (R).…”

Section: Introductionmentioning

confidence: 99%

“…In [1], he decomposed the Stone-von Neumann representation of the discrete Heisenberg group on L 2 (R) into a direct integral of representations, and in so doing determined whether or not the translates of Gabor functions parameterized by certain scales spanned a dense subspace of L 2 (R). It is our intention in this paper to use the combined methods of Dai and Larson [6] and Baggett [1] to further study the notion of wavelet sets in R n . Recall from [7] that if A ∈ M (n, Z) ∩ GL(n, Q) satisfies the condition that all of its eigenvalues have modulus strictly greater than one, it is said to be a dilation matrix.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A direct integral decomposition of the wavelet representation

Lim

Packer

Taylor

2001

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. In this paper we use the concept of wavelet sets, as introduced by X. Dai and D. Larson, to decompose the wavelet representation of the discrete group associated to an arbitrary n×n integer dilation matrix as a direct integral of irreducible monomial representations. In so doing we generalize a result of F. Martin and A. Valette in which they show that the wavelet representation is weakly equivalent to the regular representation for the Baumslag-Solitar groups.

show abstract

Processing a radar signal and representations of the discrete Heisenberg group

Cited by 39 publications

References 0 publications

Density, Overcompleteness, and Localization of Frames. I. Theory

Density, Overcompleteness, and Localization of Frames. I. Theory

History and Evolution of the Density Theorem for Gabor Frames

A direct integral decomposition of the wavelet representation

Contact Info

Product

Resources

About