About norms, semi-norms and variation of trigonometric splines

TL;DR

This paper investigates the properties of norms, semi-norms, and variation in fundamental trigonometric splines of various degrees, highlighting their complete variation and illustrating the least curvature property with graphs.

Contribution

It provides new insights into the variation and semi-norms of trigonometric splines, including cases with parameter-dependent convergence factors and graphical illustrations.

Findings

Complete variation of trigonometric splines of odd and even degrees

Semi-norms of odd-degree trigonometric splines analyzed

Graphical illustration of least curvature property for polynomial splines

Abstract

The work examines norms in of fundamental trigonometric splines of odd and even degrees, which in some cases coincide with polynomial ones. Fundamental trigonometric splines for the case where the con-vergence factors depend on the parameter are also considered. It is shown that an important characteristic of these splines of odd and even degrees is their complete variation. The semi-norms of trigonometric splines of odd degrees are also studied. The given material is illustrated by graphs; so, in particular, an illustration of the property of the least curvature of polynomial splines is given for the first time.