Weaving frames linked with fractal convolutions

Abstract

Weaving frames have been introduced to deal with some problems in signal processing and wireless sensor networks. More recently, the notion of fractal operator and fractal convolutions have been linked with perturbation theory of Schauder bases and frames. However, the existing literature has established limited connections between the theory of fractals and frame expansions. In this paper, we define weaving frames generated via fractal operators combined with fractal convolutions. The aim is to demonstrate how partial fractal convolutions are associated to Riesz bases, frames, and the concept of weaving frames in a Hilbert space. The context deals with one-sided convolutions, i.e., both left and right partial fractal convolution operators on Lebesgue space \( L^p \) \( (1 \leq p \leq \infty) \). Some applications using partial fractal convolutions with null function have been obtained for the perturbation theory of bases and weaving frames.