Some Fast Summations of Fourier Series and Properties of Their Dirichlet-Type Kernels

One of the new approaches to accelerating the Fourier series's convergence is using a specific parametric biorthogonal system. In the simplest case, we discuss the so-called "traditional", simple but quite effective summation algorithms that do not require severe computational costs. This paper constructs, tests, and investigates two algorithms using computational mathematics. It shows that the successful overcoming of the Gibbs phenomenon here is due to new properties of the corresponding Dirichlet-type kernels. In particular, the classical localization principle is violated by this summation. Here, the approximation of the smooth function f(x) defined on segment x ∈ [−1, 1] depends at point x = x0 on its behavior both in the neighborhood of x0 and in the ends of ±1.