Construction of Fractional Framelets in $L^2(\mathbb R)$

Abstract

Framelets generalize orthogonal wavelets by adding the desired properties of redundancy in their systems and flexibility in their construction. These extra features greatly improve their performance over orthogonal wavelets in applications such as image denoising and data processing. The main objective of this paper is to study fractional framelets associted with the fractional refinable functions that are obtained via unitary extension principles. Furthermore all the possible solutions of the matrix equations that arise in the study are obtained. Towards the end it is shown that the problem of extension has always a solution with two fractional framelets.