Multiwavelets on the Interval

Abstract

Smooth orthogonal and biorthogonal multiwavelets on the real line with their scaling function vectors being supported on [−1,1] are of interest in constructing wavelet bases on the interval [0,1] due to their simple structure. In this paper, we shall present a symmetric C2 orthogonal multiwavelet with multiplicity 4 such that its orthogonal scaling function vector is supported on [−1,1], has accuracy order 4, and belongs to the Sobolev space W2.56288. Biorthogonal multiwavelets with multiplicity 4 and vanishing moments of order 4 are also constructed such that the primal scaling function vector is supported on [−1,1], has the Hermite interpolation properties, and belongs to W3.63298, while the dual scaling function vector is supported on [−1,1] and belongs to W1.75833. A continuous dual scaling function vector of the cardinal Hermite interpolant with multiplicity 4 and support [−1,1] is also given. All the wavelet filters constructed in this paper have closed form expressions. Based on the above constructed orthogonal and biorthogonal multiwavelets on the real line, both orthogonal and biorthogonal multiwavelet bases on the interval [0,1] are presented. Such multiwavelet bases on the interval [0,1] have symmetry, small support, high vanishing moments, good smoothness, and simple structures. Furthermore, the sequence norms for the coefficients based on such orthogonal and biorthogonal multiwavelet expansions characterize Sobolev norms ‖·‖Ws([0,1]) for s∈(−2.56288,2.56288) and for s∈(−1.75833,3.63298), respectively.