Wavelets of multiplicity 𝑟

Goodman, T. N. T.; Lee, S. L.

doi:10.1090/S0002-9947-1994-1232187-7

Wavelets of multiplicity $r$
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by T. N. T. Goodman and S. L. Lee

Trans. Amer. Math. Soc. 342 (1994), 307-324

DOI: https://doi.org/10.1090/S0002-9947-1994-1232187-7

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Abstract:

A multiresolution approximation ${({V_m})_{m \in {\mathbf {Z}}}}$ of ${L^2}({\mathbf {R}})$ is of multiplicity $r > 0$ if there are r functions ${\phi _1}, \ldots ,{\phi _r}$ whose translates form a Riesz basis for ${V_0}$. In the general theory we derive necessary and sufficient conditions for the translates of ${\phi _1}, \ldots ,{\phi _r},\;{\psi _1}, \ldots ,{\psi _r}$ to form a Riesz basis for ${V_1}$. The resulting reconstruction and decomposition sequences lead to the construction of dual bases for ${V_0}$ and its orthogonal complement ${W_0}$ in ${V_1}$. The general theory is applied in the construction of spline wavelets with multiple knots. Algorithms for the construction of these wavelets for some special cases are given.

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