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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Wavelets of multiplicity $r$
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by T. N. T. Goodman and S. L. Lee PDF
Trans. Amer. Math. Soc. 342 (1994), 307-324 Request permission

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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Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 342 (1994), 307-324
  • MSC: Primary 41A15; Secondary 41A30, 42C05, 42C15
  • DOI: https://doi.org/10.1090/S0002-9947-1994-1232187-7
  • MathSciNet review: 1232187