## Illinois Journal of Mathematics

### The path-connectivity of MRA wavelets in $L^2(\mathbb {R}^{d})$

#### Abstract

We show that for any $d\times d$ expansive matrix $A$ with integer entries and $|\det(A)|=2$, the set of all $A$-dilation MRA wavelets is path-connected under the $L^2(\mathbb{R}^d)$ norm topology. We do this through the application of $A$-dilation wavelet multipliers, namely measurable functions $f$ with the property that the inverse Fourier transform of $(f\widehat{\psi})$ is an $A$-dilation wavelet for any $A$-dilation wavelet $\psi$ (where $\widehat{\psi}$ is the Fourier transform of $\psi$). In this process, we have completely characterized all $A$-dilation wavelet multipliers for any integral expansive matrix $A$ with $|\det(A)|=2$.

#### Article information

Source
Illinois J. Math., Volume 54, Number 2 (2010), 601-620.

Dates
First available in Project Euclid: 14 October 2011

https://projecteuclid.org/euclid.ijm/1318598674

Digital Object Identifier
doi:10.1215/ijm/1318598674

Mathematical Reviews number (MathSciNet)
MR2846475

Zentralblatt MATH identifier
1238.42019

#### Citation

Li, Zhongyan; Dai, Xingde; Diao, Yuanan; Huang, Wei. The path-connectivity of MRA wavelets in $L^2(\mathbb {R}^{d})$. Illinois J. Math. 54 (2010), no. 2, 601--620. doi:10.1215/ijm/1318598674. https://projecteuclid.org/euclid.ijm/1318598674

#### References

• X. Dai, D. R. Larson and D. M. Speegle, Wavelet sets in $\mathbb{R}^n$, J. Fourier Anal. Appl. 3(2) (1997), 451–456.
• M. Bownik, A characterization of affine dual frames in $L^2(\mathbb{R}^n)$, Appl. Comput. Harmon. Anal. 8 (2000), 203–221.
• X. Dai and D. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Memoirs Amer. Math. Soc. 134(640) (1998).
• I. Daubechies, Ten lecture on wavelets, CBMS Lecture Notes 61, SIAM, Philadelphia, PA, 1992.
• D. Bakic, I. Krishtal and E. N. Wilson, Parseval frame wavelets with $E_{n}^{(2)}$-dialtions, Appl. Comput. Harmon. Anal. 19 (2005), 386–431.
• Q. Gu and D. Han, On multiresolution analysis (MRA) wavelets in $\mathbb{R}^n$, J. Fourier Anal. Appl. 6(4) (2000), 437–447.
• E. Hernándes and G. Weiss, A first course on wavelets, CRC Press, Boca Raton, FL, 1996.
• J. C. Lagarias and Y. Wang, Haar type orthonormal wavelet basis in $\mathbb{R}^2$, J. Fourier Anal. Appl. 2(1) (1995), 1–14.
• Y. Li, On a class of bidimensional nonseparable wavelet multipliers, J. Math. Anal. Appl. 270 (2002), 543–560.
• Z. Li, X. Dai, Y. Diao and J. Xin, Multipliers, phases and connectivity of MRA wavelets in $L^2(\mathbb{R}^2)$, J. Fourier Anal. Appl. 16(2) (2010), 155–176.
• R. Liang, Some properties of wavelets, Ph.D. Dissertation, University of North Carolina at Charlotte, 1998.
• Y. Meyer, Wavelets and operators, Cambridge Studies in Advanced Mathematics 37, Cambridge Univ. Press, Cambridge, 1992.
• D. Speegle, The s-elementary wavelets are path-connected, Proc. Amer. Math. Soc. 127(1) (1999), 223–233.
• The Wutam Consortium, Basic properties of wavelets, J. Fourier Anal. Appl. 4(4) (1998), 575–594.