FAMILIES OF MULTIRESOLUTION AND WAVELET SPACES WITH OPTIMAL PROPERTIES

Under suitable conditions, if the scaling functions phi1 and phi2 generate the multiresolutions V(j)(phi1) and V(j)(phi2), then their convolution phi1 * phi2 also generates a multiresolution V(j)(phi1 * phi2). Moreover, if p is an appropriate convolution operator from 1, into itself and if phi is a scaling function generating the multiresolution V(j)(phi), then p * phi is a scaling function generating the same multiresolution V(j)(phi) = V(j)(p * phi). Using these two properties, we group the scaling and wavelet functions into equivalent classes and consider various equivalent basis functions of the associated function spaces. We use the n-fold convolution product to construct sequences of multiresolution and wavelet spaces V(j)(phi(n)) and W(j)(phi(n)) with increasing regularity. We discuss the link between multiresolution analysis and Shannon's sampling theory. We then show that the interpolating and orthogonal pre- and post-filters associated with the multiresolution sequence V(0)(phi(n)) asymptotically converge to the ideal lowpass filter of Shannon. We also prove that the filters associated with the sequence of wavelet spaces W(0)(phi(n)) converge to the ideal bandpass filter. Finally, we construct the basic wavelet sequences psi(b)n and show that they tend to Gabor functions. This provides wavelets that are nearly time-frequency optimal. The theory is illustrated with the example of polynomial splines.