Finite Fourier series. The class of 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.