Finite Fourier series. The class of trigonometric splines

TL;DR

This paper introduces a class of trigonometric splines derived from finite Fourier series on discrete points, highlighting their properties and potential for further generalizations.

Contribution

It presents a new class of trigonometric splines based on finite Fourier series with orthogonal functions, expanding the scope of spline theory.

Findings

Defines a finite system of orthogonal functions with interpolation properties

Establishes a class of trigonometric splines including polynomial periodic splines

Suggests directions for future research in generalized trigonometric splines

Abstract

Finite trigonometric Fourier series on a set of discrete equidistant points are considered. A finite system of orthogonal functions that have interpolation and certain differential properties on the period is introduced. Finite Fourier series based on this system of functions form a class of trigonometric splines, which includes polynomial periodic simple splines. Trigonometric splines suggest generalizations in several directions and certainly require further research.