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Transactions of the American Mathematical Society

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



The limits of refinable functions

Authors: Gilbert Strang and Ding-Xuan Zhou
Journal: Trans. Amer. Math. Soc. 353 (2001), 1971-1984
MSC (2000): Primary 42C40, 41A25; Secondary 65F15
Published electronically: January 4, 2001
MathSciNet review: 1813602
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Abstract: A function $\phi$ is refinable ($\phi \in S$) if it is in the closed span of $\{\phi (2x-k)\}$. This set $S$ is not closed in $L_{2}(\mathbb {R})$, and we characterize its closure. A necessary and sufficient condition for a function to be refinable is presented without any information on the refinement mask. The Fourier transform of every $f\in \overline {S} \setminus S$ vanishes on a set of positive measure. As an example, we show that all functions with Fourier transform supported in $[-{\frac {4}{3}}\pi , {\frac {4}{3}}\pi ]$ are the limits of refinable functions. The relation between a refinable function and its mask is studied, and nonuniqueness is proved. For inhomogeneous refinement equations we determine when a solution is refinable. This result is used to investigate refinable components of multiple refinable functions. Finally, we investigate fully refinable functions for which all translates (by any real number) are refinable.

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Additional Information

Gilbert Strang
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Ding-Xuan Zhou
Affiliation: Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kow- loon, Hong Kong, P. R. China

Keywords: Refinable function, Fourier transform, band-limited function, refinement mask, inhomogeneous refinement equation, multiple refinable function, fully refinable function
Received by editor(s): May 15, 1998
Received by editor(s) in revised form: November 3, 1999
Published electronically: January 4, 2001
Additional Notes: Research supported in part by Research Grants Council of Hong Kong.
Article copyright: © Copyright 2001 American Mathematical Society