Fractal Convolution: A New Operation between Functions

TL;DR

This paper introduces fractal convolution, a novel binary operation on functions using Iterated Function Systems, with detailed analysis in function spaces and applications to bases and frames in Banach and Hilbert spaces.

Contribution

It presents the first detailed study of fractal convolution in various function spaces, including properties, linear operators, and construction of fractal bases and frames.

Findings

Fractal convolution defines a new operation between functions.

Properties of fractal convolution are characterized in $\, ext{L}^p$ and continuous functions.

Construction of fractal bases and frames in Banach and Hilbert spaces is achieved.

Abstract

In this paper we define an internal binary operation between functions called in the text \emph{fractal convolution}, that applies a pair of mappings into a fractal function. This is done by means of a suitable Iterated Function System. We study in detail the operation in $\mathcal{L}^p$ spaces and in sets of continuous functions, in a different way to previous works of the authors. We develop some properties of the operation and its associated sets. The lateral convolutions with the null function provide linear operators whose characteristics are explored. The last part of the article deals with the construction of convolved fractals bases and frames in Banach and Hilbert spaces of functions.