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The s-elementary frame wavelets are path connected


Authors: X. Dai, Y. Diao, Q. Gu and D. Han
Journal: Proc. Amer. Math. Soc. 132 (2004), 2567-2575
MSC (2000): Primary 46N99
DOI: https://doi.org/10.1090/S0002-9939-04-07271-5
Published electronically: April 8, 2004
MathSciNet review: 2054782
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Abstract: An s-elementary frame wavelet is a function $\psi\in L^2(\mathbb{R})$ which is a frame wavelet and is defined by a Lebesgue measurable set $E\subset\mathbb{R}$ such that $\hat{\psi}= \frac{1}{\sqrt{2\pi}}\chi_E$. In this paper we prove that the family of s-elementary frame wavelets is a path-connected set in the $L^2(\mathbb{R})$-norm. This result also holds for s-elementary $A$-dilation frame wavelets in $L^2(\mathbb{R}^d)$ in general. On the other hand, we prove that the path-connectedness of s-elementary frame wavelets cannot be strengthened to uniform path-connectedness. In fact, the sets of normalized tight frame wavelets and frame wavelets are not uniformly path-connected either.


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Additional Information

X. Dai
Affiliation: Department of Mathematics, University of North Carolina at Charlotte, Charlotte, North Carolina 28223
Email: xdai@uncc.edu

Y. Diao
Affiliation: Department of Mathematics, University of North Carolina at Charlotte, Charlotte, North Carolina 28223

Q. Gu
Affiliation: Department of Mathematics, East China Normal University, Shanghai, People’s Republic of China 200062

D. Han
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816

DOI: https://doi.org/10.1090/S0002-9939-04-07271-5
Keywords: Frames, wavelets, frame wavelets, frame wavelet sets, Fourier transform.
Received by editor(s): March 8, 2002
Received by editor(s) in revised form: February 20, 2003
Published electronically: April 8, 2004
Communicated by: David R. Larson
Article copyright: © Copyright 2004 American Mathematical Society