The limits of refinable functions
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- by Gilbert Strang and Ding-Xuan Zhou PDF
- Trans. Amer. Math. Soc. 353 (2001), 1971-1984 Request permission
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.References
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Additional Information
- Gilbert Strang
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- Email: gs@math.mit.edu
- Ding-Xuan Zhou
- Affiliation: Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kow- loon, Hong Kong, P. R. China
- Email: mazhou@math.cityu.edu.hk
- 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.
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 1971-1984
- MSC (2000): Primary 42C40, 41A25; Secondary 65F15
- DOI: https://doi.org/10.1090/S0002-9947-01-02668-X
- MathSciNet review: 1813602