Multiresolution analysis on product of zero-dimensional Abelian groups

Lukomskii, S. F.

doi:10.1016/j.jmaa.2011.07.043

Cited by 14 publications

(7 citation statements)

References 9 publications

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“…Note that a general construction of self-similar wavelets generated by partiotoins of the group R n was defined in [27] and is discussed in detail in [39,Sect.2.8]. In [33] this construction was extended to zero-dimensional Abelian groups.…”

Section: Definition 4 (Haar-like Bases) Define a System Of Functionsmentioning

confidence: 99%

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Association schemes on general measure spaces and zero-dimensional Abelian groups

Barg

Skriganov

2015

Advances in Mathematics

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Association schemes form one of the main objects of algebraic combinatorics, classically defined on finite sets. At the same time, direct extensions of this concept to infinite sets encounter some problems even in the case of countable sets, for instance, countable discrete Abelian groups. In an attempt to resolve these difficulties, we define association schemes on arbitrary, possibly uncountable sets with a measure. We study operator realizations of the adjacency algebras of schemes and derive simple properties of these algebras. However, constructing a complete theory in the general case faces a set of obstacles related to the properties of the adjacency algebras and associated projection operators. To develop a theory of association schemes, we focus on schemes on topological Abelian groups where we can employ duality theory and the machinery of harmonic analysis. Using the language of spectrally dual partitions, we prove that such groups support the construction of general Abelian (translation) schemes and establish properties of their spectral parameters (eigenvalues). Addressing the existence question of spectrally dual partitions, we show that they arise naturally on topological zero-* (A. Barg), maksim88138813@mail.ru (M. Skriganov). 143dimensional Abelian groups, for instance, Cantor-type groups or the groups of p-adic numbers. This enables us to construct large classes of examples of dual pairs of association schemes on zero-dimensional groups with respect to their Haar measure, and to compute their eigenvalues and intersection numbers (structural constants). We also derive properties of infinite metric schemes, connecting them with the properties of the non-Archimedean metric on the group. Next we focus on the connection between schemes on zerodimensional groups and harmonic analysis. We show that the eigenvalues have a natural interpretation in terms of Littlewood-Paley wavelet bases, and in the (equivalent) language of martingale theory. For a class of nonmetric schemes constructed in the paper, the eigenvalues coincide with values of orthogonal function systems on zero-dimensional groups. We observe that these functions, which we call Haar-like bases, have the properties of wavelet bases on the group, including in some special cases the self-similarity property. This establishes a seemingly new link between algebraic combinatorics and (non-Archimedean) harmonic analysis. We conclude the paper by studying some analogs of problems of classical coding theory related to the theory of association schemes.

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Section: Definition 4 (Haar-like Bases) Define a System Of Functionsmentioning

confidence: 99%

“…Using the mapping of X → [0, 1] (5.15) (or X → [0, ∞) (5.26)) it is possible to represent the wavelets on X as functions on the real line. For the self-similar case this is done in [33], while the general case has not been studied in detail.…”

Section: Definition 4 (Haar-like Bases) Define a System Of Functionsmentioning

confidence: 99%

Association schemes on general measure spaces and zero-dimensional Abelian groups

Barg

Skriganov

2015

Advances in Mathematics

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“…To mention only a few references on wavelet frames, the reader is referred to [8,10,17] and many references therein. On the other hand, the past decade has also witnessed a tremendous interest in the problem of constructing wavelet bases and frames on various spaces other than R, such as locally compact Abelian groups [15], Vilenkin groups [11], Cantor dyadic groups [20], p-adic fields [1] and zero-dimensional groups [22]. The local field K is a natural model for the structure of wavelet frame systems, as well as a domain upon which one can construct wavelet basis functions.…”

Section: Introductionmentioning

confidence: 99%

a-inner Product on Local Fields of Positive Characteristic

Ahmad¹,

Sheikh²

2018

Journal of Nonlinear Analysis and Application

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“…In recent years, there has been a considerable interest in the problem of constructing wavelet bases on various spaces other than R, such as abstract Hilbert spaces [20], locally compact Abelian groups [8], Cantor dyadic groups [11], p-adic fields [10] and zerodimensional groups [14]. Recently, R.L.…”

Section: Introductionmentioning

confidence: 99%

Construction of biorthogonal wavelet packets on local fields of positive characteristic

Shah¹,

Bhat²

2016

Turk J Math

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Orthogonal wavelet packets lack symmetry which is a much desired property in image and signal processing. The biorthogonal wavelet packets achieve symmetry where the orthogonality is replaced by the biorthogonality. In the present paper, we construct biorthogonal wavelet packets on local fields of positive characteristic and investigate their properties by means of the Fourier transforms. We also show how to obtain several new Riesz bases of the space L 2 (K) by constructing a series of subspaces of these wavelet packets. Finally, we provide the algorithms for the decomposition and reconstruction using these biorthogonal wavelet packets.

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Multiresolution analysis on product of zero-dimensional Abelian groups

Cited by 14 publications

References 9 publications

Association schemes on general measure spaces and zero-dimensional Abelian groups

Association schemes on general measure spaces and zero-dimensional Abelian groups

a-inner Product on Local Fields of Positive Characteristic

Construction of biorthogonal wavelet packets on local fields of positive characteristic

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