Bandlimited wavelets

Abstract

We provide several general and explicit constructions of bandlimitedwavelets and framelets based on a special class of $2\pi$-periodicfunctions $A_{\delta,\Omega}$. By using bell functions and differentvalues of $\delta$ and $\Omega$, one can obtain orthonormal andRiesz wavelets as well as tight frame wavelets and biframelets. Ineach case, we show that it is possible to produce bandlimitedwavelets lying in the Schwartz class, giving them excellent time andfrequency localization. The results here yield the classicalShannon's orthonormal wavelets and the well-known Meyer'sorthonormal wavelets as examples.The Mixed Unitary Extension Principle is utilized to give asystematic approach in constructing bandlimited framelets. We alsoadapt standard constructions of compactly supported biorthogonalRiesz wavelets to obtain bandlimited biorthogonal Riesz wavelets. Inaddition, it is shown that these bandlimited biorthogonal waveletscan be made interpolatory and lie in the Schwartz class.