Biorthogonal Multiwavelets on the Interval: Cubic Hermite Splines

Wolfgang Dahmen, Bin Han, Rong-Qing Jia and Angela Kunoth


Starting with Hermite cubic splines as primal multigenerator, first a dual multigenerator on $\R$ is constructed which consists of continuous functions, has small support and is exact of order two. We then derive multiresolution sequences on the interval while retaining the polynomial exactness on the primal and dual side. This guarantees moment conditions of the corresponding wavelets. The concept of stable completions \cite{CDP} is then used to construct corresponding primal and dual multiwavelets on the interval as follows. An appropriate variation of what is known as hierarchical basis in finite element methods is shown to be an initial completion. This is then in a second step projected into the desired complements spanned by compactly supported biorthogonal multiwavelets. The masks of all multigenerators and multiwavelets are finite so that decomposition and reconstruction algorithms are simple and efficient. Furthermore, in addition to Jackson estimates which follow from the exactness, one can also show Bernstein inequalities for the primal and dual multiresolution. Consequently, sequence norms for the coefficients based on such multiwavelet expansions characterize Sobolev norms $\|\cdot\|_{H^s([0,1])}$ for $s\in (-0.824926,2.5)$. In particular, the multiwavelets form Riesz bases for $L_2([0,1])$.

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