## Local dual and poly-scale refinability

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## Abstract:

For a compactly supported function $f$, let $S_n(f), n\ge 0$, be the space of all finite linear combinations of $f(M^n\cdot -k), k\in \mathbf Z$. In this paper, we consider the explicit construction of local duals of $f$ and the poly-scale refinability of functions in $S_0(f)$ when $f$ is an $M$-refinable function. We show that for any $M$-refinable function $f$, there exists a local dual of $f$ in $S_N(f)$ for some $N\ge 0$, and that any function in $S_0(f)$ is poly-scale refinable.## References

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## Additional Information

**Qiyu Sun**- Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
- Email: qsun@mail.ucf.edu
- Received by editor(s): December 17, 2002
- Received by editor(s) in revised form: December 8, 2003
- Published electronically: October 14, 2004
- Additional Notes: Partial results of this paper were announced in the 2002 Fall Southeastern Section Meeting of AMS, Orlando, November 9–10, 2002
- Communicated by: David R. Larson
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**133**(2005), 1175-1184 - MSC (2000): Primary 42C40, 41A65
- DOI: https://doi.org/10.1090/S0002-9939-04-07622-1
- MathSciNet review: 2117220