New methods for constructing uniformly convergent trigonometric Fourier series

DOI: 10.34759/trd-2022-126-06

Аuthors

Kutysh I. I.

Limited Liability Company «ECOGIBENT», 125080, Moscow, Volokolamskoe shosse, 4

e-mail: ecogibent@mail.ru

Abstract

New results of studies of convergence of trigonometric Fourier series (TFS) with Fourier coefficients constructed by various methods are presented.

Using the concept of the square of the relative norm, the possibility of an analytical representation of a given TFS function is analyzed in detail and it is established that the cause of the divergence of the TFS with a sufficient increase in its degree is the occurrence of the Gibbs effect.

It is shown that when assessing the convergence of the TFS to its function as an independent change of the relative norm, instead of the current value of the degree of the series k, it is reasonable to use the current value of the generalized variable Θ=kπ/n, which allows us to obtain more general results. Moreover, it is sufficient to control only the value of Θ, which determines the amount of calculations.

Recommendations are given for the construction of such Fourier coefficients that ensure uniform convergence of the TFS to its functions f(x) and the correct finding of their first derivatives free of the Gibbs effect.

Uniformly convergent TFSs constructed according to the proposed method are compared with the known Filon and Lanczos series.

In contrast to the Lanczos method, it is proposed to use variables σ-multipliers depending on the new variable ζ, which affects the rate of convergence of the series to its function f(x) and the accuracy of determining the first derivatives of the series without the occurrence of the Gibbs effect.

The results on the construction of uniformly convergent TFSs relate to any maximum-normalized periodic function f(x) in the interval [-π, π] satisfying Dirichlet conditions. Moreover, if the function f(x) has zeros at the ends of a given interval, then it is advisable to build a shortened TFS with decomposition only in terms of sinuses, hence a simpler TFS.

The proposed uniformly converging TFSs can find application in solving various problems of gas dynamics and heat and mass transfer described by partial differential equations.

Keywords:

Fourier coefficients, approximation nodes, Gibbs effect, series step, normalized function, convergence rate of Fourier series, uniformly converging trigonometric Fourier series, square of relative norm, generalized variable, quadrature formula, Fourier series with sine expansion only

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