Families of multiresolution and wavelet spaces with optimal properties

Abstract

Under suitable conditions, if the scaling functions ϕ1 and ϕ2 generate the multiresolutions V (j)(ϕ1) and V (j)(ϕ2), then their convolution ϕ1*ϕ2also generates a multiresolution V (j)(ϕ1*ϕ2) More over, if p is an appropriate convolution operator from l 2 into itself and if ϕ is a scaling function generating the multiresolution V (j)(ϕ),then p*ϕis a scaling function generating the same multiresolution V (j)(ϕ)=V (j)(p*ϕ). 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)(ϕn) and W (j)(ϕn) 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)(ϕn)asymptotically converge to the ideal lowpass filter of Shannon. We also prove that the filters associated with the sequence of wavelet spaces W (0)(ϕn)convergeto the ideal bandpass filter. Finally, we construct the basic wavelet sequences ψ b nand show that they tend to Gabor functions. Thisprovides wavelets that are nearly time-frequency optimal. The theory is illustrated with the example of polynomial splines.