## 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 $C^2$ 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 $W^{2.56288}$. 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 $W^{3.63298}$ while the dual scaling function vector is supported on $[-1,1]$ and belongs to $W^{1.75833}$. A continuous dual scaling function vector of the cardinal Hermite interpolant with multiplicity $4$ and support $[-1,1]$ is also given. 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 norm $\|.\|_{W^s([0, 1])}$ for $s\in (-2.56288, 2.56288)$ and for $s\in (-1.75833, 3.63298)$, respectively.

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