Alperts multi-wavelets from spline super-functions

Abstract

For multi-wavelets generalized left eigenvectors of the matrix H, a finite portion of down-sampled convolution matrix H determine the combinations of scaling functions that produce the desired spline or scaling function from which polynomials of desired degree can be reproduced. This condition is used to construct Alpert's multi-wavelets with multiplicity two, three and four and with approximation orders two, three and four respectively. Higher multiplicity Alpert multi-wavelets can also be constructed using this new method.