scholarly journals Integrability of the Majorant of the Fourier Series Partial Sums with Respect to Bases

2005 ◽  
Vol 12 (1) ◽  
pp. 181-188
Author(s):  
George Tkebuchava
Keyword(s):  
Fourier Series ◽  
Orlicz Space ◽  
Baire Category ◽  
Partial Sums

Abstract The majorant of Fourier series partial sums with respect to the system of functions formed by the product of 𝐿([0, 1]) space bases is considered. It is proved that in any Orlicz space wider than 𝐿(log+ 𝐿)𝑑([0, 1]𝑑), 𝑑 ≥ 1, the set of functions with such a majorant is integrable on [0, 1]𝑑 and has the first Baire category.

scholarly journals Convergence in Measure of Logarithmic Means of Quadratical Partial Sums of Double Walsh-Kaczmarz-Fourier Series

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Ushangi Goginava ◽  
Károly Nagy
Keyword(s):  
Fourier Series ◽  
Orlicz Space ◽  
Baire Category ◽  
Partial Sums ◽  
Convergence In Measure ◽  
Logarithmic Means

The main aim of this paper is to prove that the logarithmic means of quadratical partial sums of the double Walsh-Kaczmarz series does not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace ofL log+ L(I2), the set of the functions the logarithmic means of quadratical partial sums of the double Walsh-Kaczmarz series of which converge in measure is of first Baire category.

Convergence in Measure of Partial Sums of Double Vilenkin–Fourier Series

2009 ◽  
Vol 16 (3) ◽  
pp. 507-516
Author(s):  
Ushangi Goginava
Keyword(s):  
Fourier Series ◽  
Orlicz Space ◽  
Baire Category ◽  
Partial Sums ◽  
Convergence In Measure

Abstract The main aim of this paper is to prove that the partial sums of double Vilenkin–Fourier series do not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of 𝐿ln+ 𝐿, the set of functions whose quadratic partial sums of double Vilenkin–Fourier series converge in measure is of first Baire category.

scholarly journals Convergence in Measure of Logarithmic Means of Double Walsh–Fourier Series

2005 ◽  
Vol 12 (4) ◽  
pp. 607-618
Author(s):  
György Gát ◽  
Ushangi Goginava ◽  
George Tkebuchava
Keyword(s):  
Fourier Series ◽  
Orlicz Space ◽  
Baire Category ◽  
Convergence In Measure ◽  
Logarithmic Means

Abstract The main aim of this paper is to prove that the logarithmic means of the double Walsh–Fourier series do not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of 𝐿log 𝐿(𝐼2), the set of functions for which quadratic logarithmic means of the double Walsh–Fourier series converge in measure is of first Baire category.

scholarly journals Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Elena E. Berdysheva ◽  
Nira Dyn ◽  
Elza Farkhi ◽  
Alona Mokhov
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Error Bounds ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Dirichlet Kernel ◽  
Partial Sums ◽  
Functions Of Bounded Variation ◽  
Moduli Of Continuity ◽  
Fourier Approximation

AbstractWe introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

scholarly journals Generalized localization for spherical partial sums of multiple Fourier series

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.

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

2019 ◽  
Vol 18 (4) ◽  
pp. 949-975 ◽  
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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