Available in electronic format
Available in print format
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
     

Frame wavelet sets in $\mathbb{R}$

Author(s): X. Dai; Y. Diao; Q. Gu
Journal: Proc. Amer. Math. Soc. 129 (2001), 2045-2055.
MSC (2000): Primary 46N99, 46B28
Posted: December 28, 2000
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: In this paper, we try to answer an open question raised by Han and Larson, which asks about the characterization of frame wavelet sets. We completely characterize tight frame wavelet sets. We also obtain some necessary conditions and some sufficient conditions for a set $E$ to be a (general) frame wavelet set. Some results are extended to frame wavelet functions that are not defined by frame wavelet set. Several examples are presented and compared with some known results in the literature.


References:

1.
L. Baggett, H. Medina and K. Merrill, Generalized Multiresolution Analyses, and a Construction Procedure for All Wavelet Sets in $\mathbb{R}^n,$ preprint.

2.
X. Dai, Y. Diao and Q. Gu, Normalized Tight Frame Wavelet Sets, preprint.

3.
I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory, 36, (1990), 961-1005. MR 91e:42038

4.
I. Daubechies, Ten Lectures on Wavelets, CBS-NSF Regional Conferences in Applied Mathematics, 61, SIAM, (1992). MR 93e:42045

5.
I. Daubechies, A. Grossman, Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys., 27(5), (1986), 1271-1283. MR 87e:81089

6.
R. J. Duffin, A. C. Shaffer, A class of nonharmonic Fourier Series, Trans. Amer. Math. Soc., 72, (1952), 341-366. MR 13:839a

7.
D. Han and D. Larson, Bases, Frames and Group representations, Memoirs, AMS, to appear.

8.
E. Hernández, G. Weiss, A first course on wavelets, CRC Press, Boca Raton, (1996). MR 97i:42015

9.
S. Mallat, Multiresolution approximations and wavelet orthonormal basis of $ L ^2 (\mathbb{R}), $ Trans. Amer. Math. Soc., 315, (1989) 69-87. MR 90e:42046

10.
Y. Meyer, Ondelettes et operateurs I, Hermann editeurs des sciences et des arts, 1990; Eng. transl, Wavelets and Operators, Camb. Studies in Adv. Math., 37, 1992. MR 93i:42002

11.
Y. Meyer, Wavelets: Algorithms and Applications, transl. from Fr., SIAM, Philadelphia, 1993. MR 95f:94005

12.
B. Sz-Nagy, Expansion theorems of Paley-Wiener type, Duke Math. J., 14, (1947), 975-978.

13.
R. M. Young, An introduction to nonharmonic Fourier series, Academic Press, New York, (1980). MR 81m:42027


Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46N99, 46B28

Retrieve articles in all Journals with MSC (2000): 46N99, 46B28


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
Email: ydiao@uncc.edu

Q. Gu
Affiliation: Department of Mathematics, Beijing University, Beijing, People's Republic of China

DOI: 10.1090/S0002-9939-00-05873-1
PII: S 0002-9939(00)05873-1
Received by editor(s): November 15, 1999
Posted: December 28, 2000
Communicated by: David R. Larson
Copyright of article: Copyright 2000, American Mathematical Society


  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement
Search the AMSPowered by Google