## On a conjecture about MRA Riesz wavelet bases

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- Proc. Amer. Math. Soc.
**134**(2006), 1973-1983 Request permission

## Abstract:

Let $\phi$ be a compactly supported refinable function in $L_2(\mathbb {R})$ such that the shifts of $\phi$ are stable and $\hat \phi (2\xi )=\hat a(\xi )\hat \phi (\xi )$ for a $2\pi$-periodic trigonometric polynomial $\hat a$. A wavelet function $\psi$ can be derived from $\phi$ by $\hat \psi (2\xi ):=e^{-i\xi }\overline {\hat a(\xi +\pi )} \hat \phi (\xi )$. If $\phi$ is an orthogonal refinable function, then it is well known that $\psi$ generates an orthonormal wavelet basis in $L_2(\mathbb {R})$. Recently, it has been shown in the literature that if $\phi$ is a $B$-spline or pseudo-spline refinable function, then $\psi$ always generates a Riesz wavelet basis in $L_2(\mathbb {R})$. It was an open problem whether $\psi$ can always generate a Riesz wavelet basis in $L_2(\mathbb {R})$ for any compactly supported refinable function in $L_2(\mathbb {R})$ with stable shifts. In this paper, we settle this problem by proving that for a family of arbitrarily smooth refinable functions with stable shifts, the derived wavelet function $\psi$ does not generate a Riesz wavelet basis in $L_2(\mathbb {R})$. Our proof is based on some necessary and sufficient conditions on the $2\pi$-periodic functions $\hat a$ and $\hat b$ in $C^{\infty }(\mathbb {R})$ such that the wavelet function $\psi$, defined by $\hat \psi (2\xi ):=\hat b(\xi )\hat \phi (\xi )$, generates a Riesz wavelet basis in $L_2(\mathbb {R})$.## References

- Marcin Bownik,
*Riesz wavelets and generalized multiresolution analyses*, Appl. Comput. Harmon. Anal.**14**(2003), no. 3, 181–194. MR**1984546**, DOI 10.1016/S1063-5203(03)00022-8 - Albert Cohen and Ingrid Daubechies,
*A new technique to estimate the regularity of refinable functions*, Rev. Mat. Iberoamericana**12**(1996), no. 2, 527–591. MR**1402677**, DOI 10.4171/RMI/207 - A. Cohen, Ingrid Daubechies, and J.-C. Feauveau,
*Biorthogonal bases of compactly supported wavelets*, Comm. Pure Appl. Math.**45**(1992), no. 5, 485–560. MR**1162365**, DOI 10.1002/cpa.3160450502 - Ingrid Daubechies,
*Ten lectures on wavelets*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR**1162107**, DOI 10.1137/1.9781611970104 - Ingrid Daubechies, Bin Han, Amos Ron, and Zuowei Shen,
*Framelets: MRA-based constructions of wavelet frames*, Appl. Comput. Harmon. Anal.**14**(2003), no. 1, 1–46. MR**1971300**, DOI 10.1016/S1063-5203(02)00511-0 - B. Dong and Z. W. Shen, Pseudo-splines, wavelets and framelets, preprint, (2004).
- Bin Han,
*Computing the smoothness exponent of a symmetric multivariate refinable function*, SIAM J. Matrix Anal. Appl.**24**(2003), no. 3, 693–714. MR**1972675**, DOI 10.1137/S0895479801390868 - Bin Han,
*Vector cascade algorithms and refinable function vectors in Sobolev spaces*, J. Approx. Theory**124**(2003), no. 1, 44–88. MR**2010780**, DOI 10.1016/S0021-9045(03)00120-5 - Deguang Han and David R. Larson,
*Frames, bases and group representations*, Mem. Amer. Math. Soc.**147**(2000), no. 697, x+94. MR**1686653**, DOI 10.1090/memo/0697 - B. Han and Z. W. Shen, Wavelets with short support, preprint, (2003).
- B. Han and Z. W. Shen, Wavelets from the Loop scheme, J. Fourier Anal. Appl., to appear.
- Rong Qing Jia and Charles A. Micchelli,
*Using the refinement equations for the construction of pre-wavelets. II. Powers of two*, Curves and surfaces (Chamonix-Mont-Blanc, 1990) Academic Press, Boston, MA, 1991, pp. 209–246. MR**1123739** - Rong-Qing Jia, Jianzhong Wang, and Ding-Xuan Zhou,
*Compactly supported wavelet bases for Sobolev spaces*, Appl. Comput. Harmon. Anal.**15**(2003), no. 3, 224–241. MR**2010944**, DOI 10.1016/j.acha.2003.08.003 - Rudolph Lorentz and Peter Oswald,
*Criteria for hierarchical bases in Sobolev spaces*, Appl. Comput. Harmon. Anal.**8**(2000), no. 1, 32–85. MR**1734847**, DOI 10.1006/acha.2000.0275

## Additional Information

**Bin Han**- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
- MR Author ID: 610426
- Email: bhan@math.ualberta.ca
- Received by editor(s): October 1, 2004
- Received by editor(s) in revised form: February 4, 2005
- Published electronically: December 19, 2005
- Additional Notes: This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant G121210654.
- Communicated by: David R. Larson
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**134**(2006), 1973-1983 - MSC (2000): Primary 42C20, 41A15, 41A05
- DOI: https://doi.org/10.1090/S0002-9939-05-08211-0
- MathSciNet review: 2215766