Polynomial and trigonometric splines

TL;DR

This paper explores the relationship between polynomial and trigonometric splines, demonstrating how they can be unified under certain conditions, thereby combining Fourier series theory with spline theory.

Contribution

It shows that simple polynomial and trigonometric splines are related, allowing transfer of results between the two and unifying their theoretical frameworks.

Findings

Trigonometric splines include polynomial splines as a special case.

Under certain parameters, polynomial and trigonometric splines coincide.

Unified approach enables transfer of results between Fourier series and spline theories.

Abstract

Classes of simple polynomial and simple trigonometric splines given by Fourier series are considered. It is shown that the class of simple trigonometric splines includes the class of simple polynomial splines. For some parameter values, the polynomial splines coincide with the trigonometric ones; this allows to transfer to such trigonometric splines all the results obtained for polynomial splines. Thus, it was possible to combine two powerful theories - the theory of trigonometric Fourier series and the theory of simple polynomial splines. The above material is illustrated by numerous examples.