Fundamental trigonometric interpolation and approximating polynomials and splines

TL;DR

This paper explores fundamental trigonometric polynomials and splines on uniform grids, focusing on their use in interpolation and approximation, including least squares methods, with applications in signal processing.

Contribution

It introduces and analyzes fundamental trigonometric polynomials and splines, including least squares variants, for linear and efficient function approximation on uniform grids.

Findings

Fundamental trigonometric polynomials enable linear interpolation.

Trigonometric splines can be used as polynomial analogues.

Applications include signal modeling and process approximation.

Abstract

The paper deals with two fundamental types of trigonometric polynomials and splines on uniform grids, which allow us to construct interpolation approximations that depend linearly on the values of the interpolated function. Fundamental on the same grids are trigonometric LS polynomials and LS splines, which allow us to construct the approximation of functions by the least squares method and also depend linearly on the values of the approximate function. Trigonometric splines were considered as splines with polynomial analogues in some cases. The material is illustrated in graphs. The considered fundamental trigonometric polynomials and splines can be recommended for use in many problems related to approximation of functions, in particular, mathematical modeling of signals and processes.