Spline multiresolutions and wavelet transforms

An extension of the family of Battle/Lemarie spline wavelet transforms is presented. By relaxing the intralevel orthogonality constraint, the authors show how to construct generalized polynomial spline scaling functions and wavelets that span the same multiresolution spaces, but can also exhibit very distinct properties. Particular examples in this family include the B-spline wavelets of compact support that are optimally localized in time-frequency; the cardinal spline wavelets that have the fundamental interpolation property; and the dual spline wavelets (the biorthogonal complement of the B-spline wavelets). A full characterization of the digital filters for the corresponding fast wavelet transform algorithms is provided. The asymptotic properties of these representations are discussed, and the link with Shannon´s sampling theory and the Gabor transform is indicated