<title>Why restrict ourselves to compactly supported basis functions?</title>

Abstract

Compact support is undoubtedly one of the wavelet properties that is given the greatest weight both in theory and applications. It is usually believed to be essential for two main reasons: (1) to have fast numerical algorithms, and (2) to have good time or space localization properties. Here, we argue that this constraint is unnecessarily restrictive and that fast algorithms and good localization can also be achieved with non-compactly supported basis functions. By dropping the compact support requirement, one gains in flexibility. This opens up new perspectives such as fractional wavelets whose key parameters (order, regularity, etc…) are tunable in a continuous fashion. To make our point, we draw an analogy with the closely related task of image interpolation. This is an area where it was believed until very recently that interpolators should be designed to be compactly supported for best results. Today, there is compelling evidence that non-compactly supported interpolators (such as splines, and others) provide the best cost/performance tradeoff.