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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Wavelets of multiplicity $ r$

Authors: T. N. T. Goodman and S. L. Lee
Journal: Trans. Amer. Math. Soc. 342 (1994), 307-324
MSC: Primary 41A15; Secondary 41A30, 42C05, 42C15
MathSciNet review: 1232187
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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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Keywords: Wavelets, multiplicity, decomposition and reconstruction algorithms, duality principle, Riesz basis, cardinal splines with multiple knots
Article copyright: © Copyright 1994 American Mathematical Society

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