Wavelets based on proate spheroidal wave functions

The article is concerned with a particular multiresolution analysis (MRA) composed of Paley-Wiener spaces. Their usual wavelet basis consisting of sine functions is replaced by one based on prolate spheroidal wave functions (PSWFs) which have much better time localization than the sinc function. The new wavelets preserve the high energy concentration in both the time and frequency domain inherited from PSWFs. Since the size of the energy concentration interval of PSWFs is one of the most important parameters in some applications, we modify the wavelets at different scales to retain a constant energy concentration interval. This requires a slight modification of the dilation relations, but leads to locally positive kernels. Convergence and other related properties, such as Gibbs phenomenon, of the associated approximations are discussed. A computationally friendly sampling technique is exploited to calculate the expansion coefficients. Several numerical examples are provided to illustrate the theory.