Multiresolution analysis on zero-dimensional Abelian groups and wavelets bases

Lukomskii, S. F.

doi:10.1070/sm2010v201n05abeh004088

Cited by 19 publications

(15 citation statements)

References 18 publications

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“…This method was developed for an orthogonal refinable function ϕ with condition supp ϕ ⊂ B 0 (0), where B 0 (0) = {x : |x| p ≤ 1} is the unit ball in the field Q p . Similar results were obtained for arbitrary zero-dimensional group [13]. The condition supp ϕ ⊂ B 0 (0) is very important.…”

Section: Introductionsupporting

confidence: 82%

Trees in Wavelet analysis on Vilenkin groups

Lukomskii

2013

Preprint

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Section: Introductionsupporting

confidence: 82%

Trees in Wavelet analysis on Vilenkin groups

Lukomskii

2013

Preprint

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“…For any zero-dimensional group G the shifts system (ϕ(x− h)) h∈H0 is orthonormal if the condition |φ(χ)| = 1 G ⊥ 0 (χ) is valid. 16 For the Vilenkin group G we can give another condition. 5 3 ].…”

Section: Lemma 33 (Ref 17) the Mask M 0 (χ) Is A Periodic Functionmentioning

confidence: 99%

“…Similar results were obtained for an arbitrary zero-dimensional group. 16 Albeverio, Evdokimov and Skopina 2 proved that if a refinable step function ϕ generates an orthogonal p-adic MRA, then suppφ(χ) ⊂ B 0 (0).…”

Section: Introductionmentioning

confidence: 99%

N-Valid trees in wavelet theory on Vilenkin groups

Lukomskii

Berdnikov

2015

Int. J. Wavelets Multiresolut Inf. Process.

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“…This necessity triggered the appearance of another approach, which employs various graphs as the means to construct orthogonal MRA. In [6,7] another algorithm for construction of ϕ was developed. It doesn't require exhaustive search, but it is valid only for functions |φ(χ)| constant on cosets G ⊥ −1 and taking 2 values only: 0 or 1.…”

Section: Introductionmentioning

confidence: 99%

Necessary and Sufficient Condition for an Orthogonal Scaling Function on Vilenkin Groups

Berdnikov¹

2019

MMI

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There are several approaches to the problem of construction of an orthogonal MRA on Vilenkin groups, but all of them are reduced to the search of the so-called scaling function. In 2005 Yu. Farkov used the so-called "blocked sets" in order to find all possible band-limited scaling functions with compact support for each set of certain parameters and his conditions are necessary and sufficient. S. F. Lukomskii, Iu. S. Kruss and G. S. Berdnikov presented another approach in 2014-2015 which has some advantages over the previous ones and employs the notion from discrete mathematics to achieve the same goals. This approach gives an algorithm for construction of band-limited orthogonal scaling functions with compact support in a concrete fashion using some class of directed graphs, which, in turn, is obtained from the so-called N -valid trees introduced by the same authors in 2012. Up to this point, though, it was not known whether this algorithm is good enough to produce any possible orthogonal scaling function of such a class. This paper describes the aforementioned algorithm and proves that it can be viewed as a necessary and sufficient condition itself, i. e. it produces any possible orthogonal scaling function. Additionally, we get another, more convenient description of the class of directed graphs we are interested in.

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Multiresolution analysis on zero-dimensional Abelian groups and wavelets bases

Cited by 19 publications

References 18 publications

Trees in Wavelet analysis on Vilenkin groups

Trees in Wavelet analysis on Vilenkin groups

N-Valid trees in wavelet theory on Vilenkin groups

Necessary and Sufficient Condition for an Orthogonal Scaling Function on Vilenkin Groups

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