Divergent Fourier series of integrable functions with respect to orthonormal systems

1998 ◽  
Vol 63 (5) ◽  
pp. 631-637
Author(s):  
A. S. Krantsberg
Keyword(s):  
Fourier Series ◽  
Integrable Functions ◽  
Divergent Fourier Series

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.

Fourier series of integrable functions

2001 ◽  
pp. 159-185
Author(s):  
Casper Goffman ◽  
Togo Nishiura ◽  
Daniel Waterman
Keyword(s):  
Fourier Series ◽  
Integrable Functions

scholarly journals Walsh-Fourier series with coefficients of generalized bounded variation

1989 ◽  
Vol 47 (3) ◽  
pp. 458-465 ◽  
Author(s):  
F. Móricz
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Real Numbers ◽  
Almost Everywhere Convergence ◽  
Almost Everywhere ◽  
Integrable Functions

AbstractWe extend in different ways the class of null sequences of real numbers that are of bounded variation and study the Walsh-Fourier series of integrable functions on the interval [(0, 1) with such coefficients. We prove almost everywhere convergence as well as convergence in the pseu dometric of Lr(0, 1) for 0 < r < 1.

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