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\begin{document}

\title{Fractional Nonuniform Multiresolution Analysis in
\$L\^{}2(\textbackslash{}mathbb R)\$}



\author[1]{Hari M. Srivastava}%
\author[2]{Firdous A. Shah}%
\author[2]{Waseem Lone}%
\affil[1]{University of Victoria}%
\affil[2]{University of Kashmir}%


\vspace{-1em}



  
  \date{June 15, 2024}


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\begin{abstract}
In order to provide a significantly richer representation of
non-stationary signals appearing in various disciplines of science and
engineering, we introduce here a novel fractional nonuniform
multiresolution analysis (FrNUMRA) on the spectrum
\$\textbackslash{}Lambda\$ given by \$\textbackslash{}Lambda =
\textbackslash{}left\textbackslash{}\{0,\textbackslash{}frac\{r\}\{N\}\textbackslash{}right\textbackslash{}\}+2\textbackslash{}mathbb\{Z\}\$,
where \$N \textbackslash{}geqq 1\$ is an integer and \$r\$ is an odd
integer with \$ 1 \textbackslash{}leqq r \textbackslash{}leqq 2N-1,\$
such that \$r\$ and \$N\$ are relatively prime. The necessary and
sufficient condition for the existence of nonuniform wavelets of
fractional order is derived and an algorithm is also presented for the
construction of fractional NUMRA starting from a fractional low-pass
filter \$h\_\{0\}\^{}\{\textbackslash{}alpha\}\$ with appropriate
conditions. Moreover, we provide a complete characterization for the
biorthogonality of the translates of the scaling functions of two
fractional nonuniform multiresolution analyses and the associated
fractional biorthogonal wavelet families.%
\end{abstract}%



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