Convolutions power of a characteristic function

TL;DR

This paper explores the properties and computational methods for convolution powers of the characteristic function of [0,1], with applications in spline regularization, spectral analysis, and partitions of unity.

Contribution

It introduces new properties and an algorithm for computing convolution powers of the characteristic function and its derivatives.

Findings

Properties of convolution powers derived

Algorithm for convolution powers developed

Applications in splines and spectral analysis demonstrated

Abstract

This paper deals with the convolution powers of the characteristic function of $[0,1], \chi_{[0,1]}$ and its function-derivatives. The importance that such convolution products have can be seen, for an instance, at \cite{DahmenLatour} where there is the need to find the best differentiable splines as regularization tool to be used to expand functions or in the spectral analysis of signals. Another simple application of these convolution powers is in the construction of a partition of the unity of a very high class of differentiability. Here, some properties of these convolution powers have been obtained which easily led to write the algorithm to produce convolution powers of $\chi_{[0,1]}$ and their function-derivatives. This algorithm was written in {\tt calc}, the program published under GPL that is free to download.