On Sampling Theorem, Wavelets, and Wavelet Transforms

The classical Shannon sampling theorem has resulted in many applications and generalizations. From a multiresolution point of view, it provides the sine scaling function. In this case, for a bandlimited signal, its wavelet series transform (WST) coefficients below a certain resolution level can be exactly obtained from the samples with a sampling rate higher than the Nyquist rate. In this research, we study the properties of cardinal orthogonal scaling functions (COSF), which provide the standard sampling theorem in multiresolution spaces with scaling functions as interpolants. We show that COSF with compact support have and only have one possibility which is the Haar pulse. We present a family of COSF with exponential decay, which are generalizations of the Haar function. With these COSF, an application is the computation of WST coefficients of a signal by the Mallat algorithm. We present some numerical comparisons for different scaling functions to illustrate the advantage of COSF. For signals which are not in multiresolution spaces, we estimate the aliasing error in the sampling theorem by using uniform samples.