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Low-pass filters and representations of the Baumslag Solitar group


Author: Dorin Ervin Dutkay
Journal: Trans. Amer. Math. Soc. 358 (2006), 5271-5291
MSC (2000): Primary 42C40, 28A78, 46L45, 28D05, 22D25
DOI: https://doi.org/10.1090/S0002-9947-06-04230-9
Published electronically: July 21, 2006
MathSciNet review: 2238916
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Abstract | References | Similar Articles | Additional Information

Abstract: We analyze representations of the Baumslag Solitar group

$\displaystyle BS(1,N)=\langle u,t\,\vert\,utu^{-1}=t^N\rangle$

that admit wavelets and show how such representations can be constructed from a given low-pass filter. We describe the direct integral decomposition for some examples and derive from it a general criterion for the existence of solutions for scaling equations. As another application, we construct a Fourier transform for some Hausdorff measures.


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Additional Information

Dorin Ervin Dutkay
Affiliation: Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
Email: ddutkay@math.rutgers.edu

DOI: https://doi.org/10.1090/S0002-9947-06-04230-9
Keywords: Wavelet, representation, low-pass filter, scaling function, Baumslag Solitar group, solenoid, decomposition, ergodic action, fractal
Received by editor(s): July 21, 2004
Published electronically: July 21, 2006
Additional Notes: This work was supported in part by NSF grant DMS0457491
Article copyright: © Copyright 2006 American Mathematical Society

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