Oblique and Hierarchical Multiwavelet Bases

Abstract

We develop the theory of oblique multiwavelet bases, which encompasses the orthogonal, semiorthogonal, and biorthogonal cases, and we circumvent the noncommutativity problems that arise in the construction of multiwavelets. Oblique multiwavelets preserve the advantages of orthogonal and biorthogonal wavelets and enhance the flexibility of the theory to accommodate a wider variety of wavelet bases. For example, for a given multiresolution, we can construct supercompact wavelets for which the support is half the size of the shortest orthogonal, semiorthogonal, or biorthogonal wavelet. The theory also produces the h-type, piecewise linear hierarchical bases used in finite element methods, and it allows us to construct new h-type, smooth hierarchical bases, as well as h-type hierarchical bases that use several template functions. For the hierarchical bases, and for all other types of oblique wavelets, the expansion of a function can still be implemented with a perfect reconstruction filter bank. We illustrate the results using the Haar scaling function and the Cohen-Daubechies-Plonka multiscaling function. We also construct a supercompact spline uniwavelet of order 3 and a hierarchical basis that is based on the Hermit cubic spline, and we explicitly give the coefficients of the corresponding filter bank. © 1997 Academic Press.