Smoothing minimally supported frequency wavelets: Part II

Hernández, Eugenio; Wang, Xihua; Weiss, Guido

doi:10.1007/bf02647945

Cited by 69 publications

(57 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [17,37,38] e iℓs , ℓ ∈ Z, when restricted to E should constitute an orthonormal basis for L 2 (E) (in other words E is a spectral set for the integer lattice Z), and (ii) The family {2 n E : n ∈ Z} of dilates of E by integral powers of 2 should constitute a measurable partition (i.e. a partition modulo null sets) of R.…”

Section: Definition Of Wavelet Setmentioning

confidence: 99%

Unitary Systems, Wavelet Sets, and Operator-Theoretic Interpolation of Wavelets and Frames

Larson¹

2007

Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore

View full text Add to dashboard Cite

A wavelet is a special case of a vector in a separable Hilbert space that generates a basis under the action of a collection, or system, of unitary operators. We will describe the operator-interpolation approach to wavelet theory using the local commutant of a system. This is really an abstract application of the theory of operator algebras to wavelet theory. The concrete applications of this method include results obtained using specially constructed families of wavelet sets. A frame is a sequence of vectors in a Hilbert space which is a compression of a basis for a larger space. This is not the usual definition in the frame literature, but it is easily equivalent to the usual definition. Because of this compression relationship between frames and bases, the unitary system approach to wavelets (and more generally: wandering vectors) is perfectly adaptable to frame theory. The use of the local commutant is along the same lines as in the wavelet theory. Finally, we discuss constructions of frames with special properties using targeted decompositions of positive operators, and related problems.

show abstract

Section: Definition Of Wavelet Setmentioning

confidence: 99%

Unitary Systems, Wavelet Sets, and Operator-Theoretic Interpolation of Wavelets and Frames

Larson¹

2007

Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore

View full text Add to dashboard Cite

show abstract

“…An orthonormal wavelet whose Fourier transform has the support of smallest possible measure is called a minimally supported frequency (MSF) wavelet. In fact, an MSF wavelet ψ is a wavelet which is associated with a wavelet set W in the sense that the support of ψ is W [1,[4][5][6][7][8][9][10]. One of the earliest wavelets, namely Shannon wavelet for dilation 2, has W = [−2π, −π] ∪ [π, 2π] as its wavelet set, which is a union of two disjoint intervals of R. Wavelet sets in R which are unions of two disjoint intervals and also those which are unions of three disjoint intervals have been characterized by Ha, Kang, Lee and Seo [6].…”

mentioning

confidence: 99%

Construction of Non-MSF Non-MRA Wavelets for L²(R) and H²(R) from MSF Wavelets

Vyas

2009

Bull. Polish Acad. Sci. Math.

View full text Add to dashboard Cite

“…Wavelet sets were also introduced independently and simultaneously as the support sets of MSF (Minimally Supported Frequency) wavelets in the sequence of papers [3], [5], and [6]. (See also the recent excellent book [4].)…”

mentioning

confidence: 99%

On subwavelet sets

Ionascu

Pearcy

1998

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. In this note we give a characterization of subwavelet sets and show that any point x ∈ R\0 has a neighborhood which is contained in a regularized wavelet set. .) The purpose of this note is to provide a characterization of the subwavelet sets and to use this characterization to prove that every point x ∈ R\{0} has a neighborhood contained in a regularized wavelet set. (Regularized wavelet sets are wavelet sets with certain nice properties; see [7].) In particular, this shows that the union of the interiors of all wavelet sets is R\{0}.We begin by introducing some preliminary terminology and notation. The measure space under consideration will always be R together with its σ-ring

show abstract

Smoothing minimally supported frequency wavelets: Part II

Cited by 69 publications

References 4 publications

Unitary Systems, Wavelet Sets, and Operator-Theoretic Interpolation of Wavelets and Frames

Unitary Systems, Wavelet Sets, and Operator-Theoretic Interpolation of Wavelets and Frames

Construction of Non-MSF Non-MRA Wavelets for L²(R) and H²(R) from MSF Wavelets

On subwavelet sets

Contact Info

Product

Resources

About

Smoothing minimally supported frequency wavelets: Part II

Cited by 69 publications

References 4 publications

Unitary Systems, Wavelet Sets, and Operator-Theoretic Interpolation of Wavelets and Frames

Unitary Systems, Wavelet Sets, and Operator-Theoretic Interpolation of Wavelets and Frames

Construction of Non-MSF Non-MRA Wavelets for L2(R) and H2(R) from MSF Wavelets

On subwavelet sets

Contact Info

Product

Resources

About

Construction of Non-MSF Non-MRA Wavelets for L²(R) and H²(R) from MSF Wavelets