Divergence almost everywhere of square partial sums of Fourier-Walsh series of integrable functions

UDC 517.5 We prove that the maximal operator of some means of cubical partial sums of two variable Walsh – Fourier series of integrable functions is of weak type . Moreover, the -means of the function converge a.e. to for , where is the Walsh group for some sequences .

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A criterion for the almost-everywhere convergence of Fourier-Walsh square partial sums of integrable functions

Sbornik Mathematics ◽

10.1070/sm1995v186n07abeh000056 ◽

1995 ◽

Vol 186 (7) ◽

pp. 1057-1070

Author(s):

S V Lukomskii

Keyword(s):

Partial Sums ◽

Almost Everywhere Convergence ◽

Almost Everywhere ◽

Integrable Functions

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Generalized localization for spherical partial sums of multiple Fourier series

Доклады Академии наук ◽

10.31857/s0869-565248917-10 ◽

2019 ◽

Vol 489 (1) ◽

pp. 7-10

Author(s):

R. R. Ashurov

Keyword(s):

Fourier Series ◽

Multiple Fourier Series ◽

Partial Sums ◽

Localization Principle ◽

Open Set ◽

Almost Everywhere

In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the L2-class is proved, that is, if f L2 (ТN) and f = 0 on an open set ТN then it is shown that the spherical partial sums of this function converge to zero almost - everywhere on . It has been previously known that the generalized localization is not valid in Lp (TN) when 1 p 2. Thus the problem of generalized localization for the spherical partial sums is completely solved in Lp (TN), p 1: if p 2 then we have the generalized localization and if p 2, then the generalized localization fails.

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Definition of Set of diagnostic Parameters of System based on the Functional Spaces Theory

SPIIRAS Proceedings ◽

10.15622/sp.2019.18.4.949-975 ◽

2019 ◽

Vol 18 (4) ◽

pp. 949-975 ◽

Cited By ~ 1

Author(s):

Valentin Senchenkov ◽

Damir Absalyamov ◽

Dmitriy Avsyukevich

Keyword(s):

Fourier Series ◽

Experimental Studies ◽

Numerical Procedure ◽

Physical Nature ◽

Convergent Series ◽

Technical Condition ◽

Partial Sums ◽

Diagnostic Tools ◽

Diagnostic Parameters ◽

Integrable Functions

The development of methodical and mathematical apparatus for formation of a set of diagnostic parameters of complex technical systems, the content of which consists of processing the trajectories of the output processes of the system using the theory of functional spaces, is considered in this paper. The trajectories of the output variables are considered as Lebesgue measurable functions. It ensures a unified approach to obtaining diagnostic parameters regardless a physical nature of these variables and a set of their jump-like changes (finite discontinuities of trajectories). It adequately takes into account a complexity of the construction, a variety of physical principles and algorithms of systems operation. A structure of factor-spaces of measurable square Lebesgue integrable functions, ( spaces) is defined on sets of trajectories. The properties of these spaces allow to decompose the trajectories by the countable set of mutually orthogonal directions and represent them in the form of a convergent series. The choice of a set of diagnostic parameters as an ordered sequence of coefficients of decomposition of trajectories into partial sums of Fourier series is substantiated. The procedure of formation of a set of diagnostic parameters of the system, improved in comparison with the initial variants, when the trajectory is decomposed into a partial sum of Fourier series by an orthonormal Legendre basis, is presented. A method for the numerical determination of the power of such a set is proposed. New aspects of obtaining diagnostic information from the vibration processes of the system are revealed. A structure of spaces of continuous square Riemann integrable functions ( spaces) is defined on the sets of vibrotrajectories. Since they are subspaces in the afore mentioned factor-spaces, the general methodological bases for the transformation of vibrotrajectories remain unchanged. However, the algorithmic component of the choice of diagnostic parameters becomes more specific and observable. It is demonstrated by implementing a numerical procedure for decomposing vibrotrajectories by an orthogonal trigonometric basis, which is contained in spaces. The processing of the results of experimental studies of the vibration process and the setting on this basis of a subset of diagnostic parameters in one of the control points of the system is provided. The materials of the article are a contribution to the theory of obtaining information about the technical condition of complex systems. The applied value of the proposed development is a possibility of their use for the synthesis of algorithmic support of automated diagnostic tools.

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Convergence of Marcinkiewicz Means of Integrable Functions with Respect to Two-Dimensional Vilenkin Systems

Georgian Mathematical Journal ◽

10.1515/gmj.2004.467 ◽

2004 ◽

Vol 11 (3) ◽

pp. 467-478

Author(s):

György Gát

Keyword(s):

Maximal Operator ◽

Weak Type ◽

Integrable Function ◽

Two Dimensional ◽

Vilenkin Groups ◽

Almost Everywhere ◽

Integrable Functions ◽

Marcinkiewicz Means

Abstract We prove that the maximal operator of the Marcinkiewicz mean of integrable two-variable functions is of weak type (1, 1) on bounded two-dimensional Vilenkin groups. Moreover, for any integrable function 𝑓 the Marcinkiewicz mean σ 𝑛𝑓 converges to 𝑓 almost everywhere.

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ON A UNIQUENESS THEOREM FOR THE FRANKLIN SYSTEM

Proceedings of the YSU A: Physical and Mathematical Sciences ◽

10.46991/pysu:a/2018.52.2.093 ◽

2018 ◽

Vol 52 (2 (246)) ◽

pp. 93-100

Author(s):

K.A. Navasardyan

Keyword(s):

Uniqueness Theorem ◽

Partial Sums ◽

Uniqueness Theorems ◽

Franklin System ◽

Almost Everywhere ◽

Franklin Series

In this paper we prove that there exist a nontrivial Franklin series and a sequence $ M_n $ such that the partial sums $ S_{M_n} (x) $ of that series converge to 0 almost everywhere and $ \lambda \cdot \text{mes} \{ x : \sup\limits_{n}{\left| S_{M_n} (x) \right|} > \lambda \} \to 0 $ as $ \lambda \to +\infty $. This shows that the boundedness assumption of the ratio $ \dfrac{ M_{n+1}}{M_n} $, used for the proofs of uniqueness theorems in earlier papers, can not be omitted.

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