On summation of Fourier series in finite form - Mikhail D. Malykh; Ksaverii Yu. Malyshev (Discrete and Continuous Models and Applied Computational Science)
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 \(a_n, b_n \in \mathbb{R}[n]\), it is proved that the sum of a Fourier series can be represented as a linear combination of 1, \(δ(x)\), \(\cot \frac{x}{2}\) and their derivatives. It is shown that this representation can be found in a finite number of steps. For series with rational Fourier coefficients \(a_n, b_n \in \mathbb{R}(n)\), 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, \(δ(x)\), \(\cot \frac{x}{2}\) 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.