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Transactions of the American Mathematical Society

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

 
 

 

Hadamard triples generate self-affine spectral measures


Authors: Dorin Ervin Dutkay, John Haussermann and Chun-Kit Lai
Journal: Trans. Amer. Math. Soc. 371 (2019), 1439-1481
MSC (2010): Primary 42B05, 42A85, 28A25
DOI: https://doi.org/10.1090/tran/7325
Published electronically: October 1, 2018
MathSciNet review: 3885185
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Abstract: Let $ R$ be an expanding matrix with integer entries, and let $ B,L$ be finite integer digit sets so that $ (R,B,L)$ form a Hadamard triple on $ {\mathbb{R}}^d$ in the sense that the matrix

$\displaystyle \frac {1}{\sqrt {\vert\det R\vert}}\left [e^{2\pi i \langle R^{-1}b,\ell \rangle }\right ]_{\ell \in L,b\in B}$    

is unitary. We prove that the associated fractal self-affine measure $ \mu = \mu (R,B)$ obtained by an infinite convolution of atomic measures

$\displaystyle \mu (R,B) = \delta _{R^{-1} B}\ast \delta _{R^{-2}B}\ast \delta _{R^{-3}B}\ast \cdots$    

is a spectral measure, i.e., it admits an orthonormal basis of exponential functions in $ L^2(\mu )$. This settles a long-standing conjecture proposed by Jorgensen and Pedersen and studied by many other authors. Moreover, we also show that if we relax the Hadamard triple condition to an almost-Parseval-frame condition, then we obtain a sufficient condition for a self-affine measure to admit Fourier frames.

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Additional Information

Dorin Ervin Dutkay
Affiliation: Department of Mathematics, University of Central Florida, 4000 Central Florida Blvd., P.O. Box 161364, Orlando, Florida 32816-1364
Email: dorin.dutkay@ucf.edu

John Haussermann
Affiliation: Department of Mathematics, University of Central Florida, 4000 Central Florida Blvd., P.O. Box 161364, Orlando, Florida 32816-1364
Email: jhaussermann@knights.ucf.edu

Chun-Kit Lai
Affiliation: Department of Mathematics, San Francisco State University, 1600 Holloway Avenue, San Francisco, California 94132
Email: cklai@sfsu.edu

DOI: https://doi.org/10.1090/tran/7325
Keywords: Hadamard triples, quasi-product form, self-affine sets, spectral measure
Received by editor(s): October 9, 2016
Received by editor(s) in revised form: June 2, 2017
Published electronically: October 1, 2018
Additional Notes: This work was partially supported by a grant from the Simons Foundation (#228539 to Dorin Dutkay).
The third author was supported by the mini-grant by ORSP of San Francisco State University (Grant No: ST659).
Article copyright: © Copyright 2018 American Mathematical Society