Haar Series on Compact Zero-Dimensional Abelian Group

Lukomskii, S. F.

doi:10.18500/1816-9791-2009-9-1-14-19

Cited by 11 publications

(7 citation statements)

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“…Malykhin, Telyakovskii and Kholshchevnikova [82] (see also Astashkin and Semenov [4]) improved the estimation above and proved sharp estimate with factor 1. The first proof of (1.13) with some different constants can be found in Lukomskii [79], a new and shorter proof discribed in this book which improved upper bound and provide similar lower bound can be found in [22]. In particular, for λ := sup n∈N and for any n = ∞ i=1 n i M i and m n we have the following two sided estimate:…”

Section: Partial Sums and Fej éR Means With Respect To The Vilenkin S...mentioning

confidence: 92%

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Bounded operators on Martingale Hardy spaces

Tutberidze¹

2022

Preprint

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The classical Fourier Analysis has been developed in an almost unbelieved way from the first fundamental discoveries by name Fourier. Especially a number of wonderful results have been proved and new directions of such research has been developed e.g. concerning Wavelets Theory, Gabor theory, Time-Frequency Analysis, Fast Fourier Transform, Abstract Harmonic Analysis, etc. One important reason for this is that this development is not only important for improving the "State of the art", but also for its importance in other areas of mathematics and also for several applications (e.g. theory of signal transmission, multiplexing, filtering, image enhancement, coding theory, digital signal processing and pattern recognition.)The classical theory of Fourier series deals with decomposition of a function into sinusoidal waves. Unlike these continuous waves the Vilenkin (Walsh) functions are rectangular waves. The development of the theory of Vilenkin-Fourier series has been strongly influenced by the classical theory of trigonometric series. Because of this it is inevitable to compare results of Vilenkin series to those on trigonometric series. There are many similarities between these theories, but there exist differences also. Much of these can be explained by modern abstract harmonic analysis, which studies orthonormal systems from the point of view of the structure of a topological group.

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Section: Partial Sums and Fej éR Means With Respect To The Vilenkin S...mentioning

confidence: 92%

“…In ( [116], p. 34), the two-sided estimate is proved. In [79], Lukomskii presented the lower estimate with sharp constant 1/4. Malykhin, Telyakovskii and Kholshchevnikova [82] (see also Astashkin and Semenov [4]) improved the estimation above and proved sharp estimate with factor 1.…”

Section: Partial Sums and Fej éR Means With Respect To The Vilenkin S...mentioning

confidence: 97%

Bounded operators on Martingale Hardy spaces

Tutberidze¹

2022

Preprint

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“…where n j ∈ Z m j (j ∈ N + ) and only a finite number of n , March 5, 2018 For the natural number n = ∞ j=1 n j M j , we define functions v and v * by (for details see Lukomskii [38])…”

Section: March 5 2018mentioning

confidence: 99%

Ph.D. Thesis-Martingale Hardy spaces and summability of the one dimensional Vilenkin-Fourier series

Tephnadze

2018

Preprint

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“…It is known that (see Lukomskii [13]) that for every n = ∞ i=1 n i M i , the following two-side estimate is true:…”

Section: Preliminariesmentioning

confidence: 99%

“…Lukomskii [13] proved two-side estimate for Legesgue constants L n with respect to Vilenkin sistem. By using this result we easily obtain that S n k f convergence to f in L 1 -norm, for every integrable function f , if and only if…”

Section: Introductionmentioning

confidence: 99%

On the convergence of partial sums with respect to Vilenkin system on the martingale Hardy spaces

Tephnadze

2018

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In this paper we derive characterizations of boundedness of the subsequences of partial sums with respect to Vilenkin system on the martingale Hardy spaces when 0 < p < 1. Moreover, we find necessary and sufficient conditions for the modulus of continuity of f ∈ Hp martingales, which provide convergence of subsequences of partial sums on the martingale Hardy spaces. It is also proved that these results are the best possible in a special sense. As applications, both some wellknown and new results are pointed out.

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Haar Series on Compact Zero-Dimensional Abelian Group

Cited by 11 publications

References 0 publications

Bounded operators on Martingale Hardy spaces

Bounded operators on Martingale Hardy spaces

Ph.D. Thesis-Martingale Hardy spaces and summability of the one dimensional Vilenkin-Fourier series

On the convergence of partial sums with respect to Vilenkin system on the martingale Hardy spaces

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