On summation of Fourier series in finite form

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Abstract

The problem of summation of Fourier series in finite form is formulated in the weak sense, which allows one to consider this problem uniformly both for classically convergent and for divergent series. For series with polynomial Fourier coefficients an,bn∈R[n], it is proved that the sum of a Fourier series can be represented as a linear combination of 1, δ(x), cot⁡x2 and their derivatives. It is shown that this representation can be found in a finite number of steps. For series with rational Fourier coefficients an,bn∈R(n), it is shown that the sum of such a series is always a solution of a linear differential equation with constant coefficients whose right-hand side is a linear combination of 1, δ(x), cot⁡x2 and their derivatives. Thus, the issue of summing a Fourier series with rational coefficients is reduced to the classical problem of the theory of integration in elementary functions.

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1. Introduction The problem of summing a functional series in elementary functions naturally arises when solving problems in mathematical physics [1-6]. If desired, even d’Alembert’s method of solving the wave equation can be considered as a method of summing a Fourier series [7, 8]. Frequently, results on summation in the final form arose as surprising side effects, for example, when accelerating the convergence of series by A.N. Krylov’s method [9-12]. However, the authors of the past avoided considering divergent series, the summation of which, as it seemed then, could yield anything [13, p. 641], [14, Ch. 12, Sect. 4]. With the advent of the theory of generalized functions [15], a reliable basis for considering divergent functional series arose. The surprising fact is that divergent series are usually summed up in a finite form much more easily than convergent ones, and, moreover, the summation of convergent series in a finite form is conveniently reduced to the summation of divergent series. In this paper we illustrate this statement using the example of one-dimensional Fourier series. The possibility of interpreting Krylov’s method in terms of generalized functions was mentioned in [16, p. 32]. 2. Statement of the Problem Definition 1. A periodic function is called piecewise elementary if its period can be divided into a finite number of segments, on each of which an elementary expression in the Liouville sense can be specified for it. We understand the equality between the sum of a Fourier series and a piecewise elementary function in the weak sense [15], which allows a further uniform consideration of the series summation in elementary functions separate from the issue of its pointwise convergence. Definition 2. The Fourier series
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About the authors

Mikhail D. Malykh

RUDN University; Joint Institute for Nuclear Research

Email: malykh_md@pfur.ru
ORCID iD: 0000-0001-6541-6603
Scopus Author ID: 6602318510
ResearcherId: P-8123-2016

Doctor of Physical and Mathematical Sciences, Head of the department of Mathematical Modeling and Artificial Intelligence of RUDN University, research fellow of MLIT JINR

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation; 6 Joliot-Curie St, Dubna, 141980, Russian Federation

Ksaverii Yu. Malyshev

RUDN University; Lomonosov Moscow State University

Author for correspondence.
Email: kmalyshev08102@mail.ru
ORCID iD: 0000-0001-8823-9136
Scopus Author ID: 57221615001

PhD student of the chair of Mathematical Modeling and Artificial Intelligence of RUDN University, engineer of Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation; 1(2) Leninskie Gory, Moscow, 119991, Russian Federation

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Copyright (c) 2024 Malykh M.D., Malyshev K.Y.

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