  The fractional Fourier transform (FrFT), 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 MRAs 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 $L^2(\mathbb R).$.

Functional Analysis