N-Valid trees in wavelet theory on Vilenkin groups

Lukomskii, S. F.; Berdnikov, G. S.

doi:10.1142/s021969131550037x

Cited by 11 publications

(7 citation statements)

References 13 publications

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“…We will prove also that for any n ∈ N there exists a step function ϕ such that 1) ϕ generate an orthogonal MRA, 2) suppφ ⊂ K ⊥ n , 3)φ(K ⊥ n \ K ⊥ n−1 ) ≡ 0. Note that the results of Sections 6, 7 and 8, there are analogues of the corresponding results for Vilenkin groups [12]. Moreover, we use the same methods.…”

Section: Non Haar Waveletsmentioning

confidence: 94%

“…Note that the results of Sections 6, 7 and 8, there are analogues of the corresponding results for Vilenkin groups [12]. Moreover, we use the same methods.…”

Section: Non Haar Waveletsmentioning

confidence: 94%

“…By s = 1 the additive group F (1)+ is a Vilenkin group. Issues of constructing of MRA and wavelets on Vilenkin groups may be found in [7]- [12].…”

Section: Introductionmentioning

confidence: 99%

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Non-Haar MRA on local fields of positive characteristic

Lukomskii

Vodolazov

2016

Journal of Mathematical Analysis and Applications

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We propose a simple method to construct integral periodic mask and corresponding scaling step functions that generate non-Haar orthogonal MRA on the local field F (s) of positive characteristic p. To construct this mask we use two new ideas. First, we consider local field as vector space over the finite field GF (p s ). Second, we construct scaling function by arbitrary tree that has p s vertices. By fixed prime number p there exist p s(p s −2) such trees.

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Section: Non Haar Waveletsmentioning

confidence: 94%

“…Note that the results of Sections 6, 7 and 8, there are analogues of the corresponding results for Vilenkin groups [12]. Moreover, we use the same methods.…”

Section: Non Haar Waveletsmentioning

confidence: 94%

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Non-Haar MRA on local fields of positive characteristic

Lukomskii

Vodolazov

2016

Journal of Mathematical Analysis and Applications

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“…Initially, trees appeared in [8,9], where they were used for construction of Riesz MRA. In [10] authors managed to get rid of restriction supp ϕ(x) ⊂ G −1 . To achieve this, the notion of N -valid tree was introduced.…”

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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“…In the article [13], the concept of N -valid tree was introduced and an algorithm for constructing the mask m (0) and correspondent refinable function ϕ was indicated in the field F (1) . In the articles [14], [5] the mask m (0) and correspondent refinable function ϕ were constructed using graph which is obtained from N -valid tree by adding new arcs.…”

Section: Introductionmentioning

confidence: 99%