OBLIQUE AND HIERARCHICAL MULTIWAVELET BASES

We develop the theory of oblique multiwavelet bases, which encompasses the orthogonal, semiorthogonal, and biorthogonal cases, and we circum vent the noncommutativity problems that arise in the construction of m ultiwavelets. Oblique multiwavelets preserve the advantages of orthogo nal and biorthogonal wavelets and enhance the flexibility of the theor y to accommodate a wider variety of wavelet bases. For example, for a given multiresolution, we can construct supercompact wavelets for whic h the support is half the size of the shortest orthogonal, semiorthogo nal, or biorthogonal wavelet. The theory also produces the h-type, pie cewise linear hierarchical bases used in finite element methods, and i t allows us to construct new h-type, smooth hierarchical bases, as wel l as h-type hierarchical bases that use several template functions. Fo r the hierarchical bases, and for all other types of oblique wavelets, the expansion of a function can still be implemented with a perfect r econstruction filter bank. We illustrate the results using the Haar sc aling function and the Cohen-Daubechies-Plonka multiscaling function. We also construct a supercompact spline uniwavelet of order 3 and a hi erarchical basis that is based on the Hermit cubic spline, and we expl icitly give the coefficients of the corresponding filter bank. (C) 199 7 Academic Press.