On Normalized Tight Frame Wavelet Sets

• Journal title : Kyungpook mathematical journal
• Volume 55, Issue 1,  2015, pp.127-135
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2015.55.1.127
Title & Authors
On Normalized Tight Frame Wavelet Sets
Srivastava, Swati;

Abstract
We determine two-interval normalized tight frame wavelet sets for real dilation $\small{d{\in}(1,{\infty})}$, and characterize all symmetric normalized tight frame wavelet sets. We also construct a normalized tight frame wavelet set which has an infinite number of components accumulating at the origin. These normalized tight frame wavelet sets and their closures possess the same measure. Finally an example of a normalized tight frame wavelet set is provided whose measure is strictly less than the measure of its closure.
Keywords
Wavelet;Frame;Wavelet set;Normalized tight frame wavelet set;Frame polygonal;
Language
English
Cited by
1.
On fixed point sets of wavelet induced isomorphisms and frame induced monomorphisms, International Journal of Wavelets, Multiresolution and Information Processing, 2016, 14, 03, 1650016
References
1.
N. Arcozzi, B. Behera and S. Madan, Large classes of minimally supported frequency wavelets of $L^2({\mathbb{R}})$ and $H^2({\mathbb{R}})$, J. Geom. Anal., 13(2003), 557-579.

2.
M. Bownik and K. R. Hoover, Dimension functions of rationally dilated GMRA's and wavelets, J. Fourier Anal. Appl. 15 (2009), 585-615.

3.
C. K. Chui and X. Shi, Orthonormal wavelets and tight frames with arbitrary real dilations, Appl. Comput. Harmon. Anal., 9(2000), 243-264.

4.
O. Cristensen, An Introduction to Frames and Reisz Bases, Birkhauser, 2003.

5.
X. Dai, Y. Diao and Q. Gu, Frame wavelet sets in $({\mathbb{R}})$, Proc. Amer. Math. Soc., 132(2004), 2567-2575.

6.
Y.-H. Ha, H. Kang, J. Lee and J. K. Seo, Unimodular wavelets for $L^2$ and the Hardy space $H^2$, Michigan Math. J., 41(1994), 345-361.

7.
D. Han and D. R. Larson, Frames, bases and group representation, Mem. Amer. Math. Soc. 697, 2000.

8.
E. Hernandez and G. Weiss, A First Course on Wavelets, CRC Press, 1996.

9.
D. R. Larson, Unitary systems and wavelet sets, Wavelet Analysis and Applications, Birkhauser Basel, 2007, 143-171.

10.
D. Singh, On Multiresolution Analysis, D. Phil. Thesis, University of Allahabad, 2010.

11.
N. K. Shukla and G. C. S. Yadav, A characterization of three-interval scaling sets, Real Anal. Exchange, 35(2009), 121-138.

12.
Z. Zhang, The measure of the closure of a wavelet set may be > $2{\pi}$, J. Cohen and A. I. Zayed (eds.), Wavelets and Multiscale Analysis: Theory and Applications, Applied and Numerical Harmonic Analysis, Springer, 2011.