Abstract
In this article, we introduce the notion of biorthgonoal nonuniform
multiresolution analysis on the spectrum
$\Lambda=\left\{0,
r/N\right\}+2\mathbb Z$,
where $N\ge 1$ is an integer and $r$ is an odd
integer with $1\le r\le 2N-1$ such that
$r$ and $N$ are relatively prime. We first establish the necessary
and sufficient conditions for the translates of a single function to
form the Riesz bases for their closed linear span. We provide the
complete characterization for the biorthogonality of the translates of
scaling functions of two nonuniform multiresolution analysis and the
associated biorthogonal wavelet families. Furthermore, under the mild
assumptions on the scaling functions and the corresponding wavelets
associated with nonuniform multiresolution analysis, we show that the
wavelets can generate Reisz bases.