Wavelets with short support

Bin Han and Zuowei Shen


This paper is to construct Riesz wavelets with short support. Wavelets with short support are of interests in both theory and application. In theory, it is known that a B-spline of order $m$ is the shortest supported refinable function among all refinable functions with the same regularity. However, whether the shortest Riesz wavelet with $m$ vanishing moments can be constructed from the multiresolution analysis generated by the B-spline of order $m$ remains open. In applications, wavelets with short support, high order of regularity and high order of vanishing moments are often desirable in signal and image processing, since they provide good time frequency localization and good approximation and lead to fast algorithms. This paper is to provide a theory for the constructions of Riesz wavelets with short support and to give various examples. In particular, we are able to construct the shortest supported Riesz wavelet of $m$ vanishing moments from the multiresolution analysis whose underlying refinable function is a B-spline of order $m$. The support of the wavelets can be made even much shorter by reducing the order of vanishing moments. The study here also provides a new insight of the structures of the spline tight frame systems constructed in \cite{RS:frame} and \cite{DHRS:wf, HM:splinetwf}; and bi-frame systems in \cite{DHRS:wf} and \cite{DH:biframe}.

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