Fractional biorthogonal wavelets in L²(ℝ)

Abstract

The fractional Fourier transform, which is a generalization of the Fourier transform, has become the focus of many research papers in recent years because of its applications in electrical engineering and optics. In this paper, we introduce the notion of fractional biorthogonal wavelets on $\mathbb{R}$ and obtain the necessary and sufficient conditions for the translates of a single function to form the fractional Riesz bases for their closed linear span. We also provide a complete characterization for the fractional biorthogonality of the translates of fractional scaling functions of two fractional MRA's and the associated fractional biorthogonal wavelet families. Moreover, under mild assumptions on the fractional scaling functions and the corresponding fractional wavelets, we show that the fractional wavelets can generate Reisz bases for $L^2(\mathbb{R}).$

Keywords:

• Frame
• fractional biorthogonal wavelets
• fractional MRA
• fractional Fourier transform

2010 Mathematics Subject Classifications:

• 42C40
• 42C15
• 41A17
• 46F12
• 26A33

Disclosure statement

No potential conflict of interest was reported by the author(s).