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Journal of Fourier Analysis and Applications

, Volume 25, Issue 6, pp 3045–3074 | Cite as

Isomorphism in Wavelets

  • Xingde DaiEmail author
  • Wei Huang
Article
  • 61 Downloads

Abstract

Two scaling functions \(\varphi _A\) and \(\varphi _B\) for Parseval frame wavelets are algebraically isomorphic, \(\varphi _A \simeq \varphi _B\), if they have matching solutions to their (reduced) isomorphic systems of equations. Let A and B be \(d\times d\) and \(s\times s\) expansive dyadic integral matrices with \(d, s\ge 1\) respectively and let \(\varphi _A\) be a scaling function associated with matrix A and generated by a finite solution. There always exists a scaling function \(\varphi _B\) associated with matrix B such that
$$\begin{aligned} \varphi _B \simeq \varphi _A. \end{aligned}$$
An example shows that the assumption on the finiteness of the solutions can not be removed. An algebraic isomorphism with consistency has orthogonality as an invariant.

Keywords

Parseval frame wavelets Isomorphism High dimension Scaling function 

Mathematics Subject Classification

Primary 46N99 47N99 46E99 Secondary 42C40 65T60 

Notes

Acknowledgements

The authors thank Qing Gu and the anonymous referees for their comments that greatly improved the manuscript.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.The University of North Carolina at CharlotteCharlotteUSA
  2. 2.Wells Fargo BankCharlotteUSA

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