Piecewise linear spectral sequences

Authors:
Youming Liu and Yuesheng Xu

Journal:
Proc. Amer. Math. Soc. **133** (2005), 2297-2308

MSC (2000):
Primary 42C15

Published electronically:
March 21, 2005

MathSciNet review:
2138872

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Abstract | References | Similar Articles | Additional Information

Abstract: We study a class of orthonormal exponential bases for the space and introduce the concept of spectral sequences. We characterize piecewise linear spectral sequences with the knot at and investigate the non-continuity of the piecewise linear spectral sequences. From a special construction of a piecewise constant spectral sequence, the classical Walsh system is recovered.

**1.**Q. Chen, N. Huang, S. Riemenschneider and Y. Xu, A B-spline approach for empirical mode decomposition,*Adv. Comp. Math.*, to appear.**2.**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****3.**Karlheinz Gröchenig,*Foundations of time-frequency analysis*, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001. MR**1843717****4.**Norden E. Huang, Zheng Shen, Steven R. Long, Manli C. Wu, Hsing H. Shih, Quanan Zheng, Nai-Chyuan Yen, Chi Chao Tung, and Henry H. Liu,*The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis*, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.**454**(1998), no. 1971, 903–995. MR**1631591**, 10.1098/rspa.1998.0193**5.**Jeffrey C. Lagarias and Yang Wang,*Spectral sets and factorizations of finite abelian groups*, J. Funct. Anal.**145**(1997), no. 1, 73–98. MR**1442160**, 10.1006/jfan.1996.3008**6.**Stéphane Mallat,*A wavelet tour of signal processing*, Academic Press, Inc., San Diego, CA, 1998. MR**1614527****7.**Charles A. Micchelli and Yuesheng Xu,*Using the matrix refinement equation for the construction of wavelets on invariant sets*, Appl. Comput. Harmon. Anal.**1**(1994), no. 4, 391–401. MR**1310661**, 10.1006/acha.1994.1024**8.**R. E. Paley, A remarkable series of orthogonal functions I,*Proc. Landon Math. Soc.***34**(1932), 241-279.**9.**Yang Wang,*Wavelets, tiling, and spectral sets*, Duke Math. J.**114**(2002), no. 1, 43–57. MR**1915035**, 10.1215/S0012-7094-02-11413-6**10.**Robert M. Young,*An introduction to nonharmonic Fourier series*, Pure and Applied Mathematics, vol. 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR**591684**

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Additional Information

**Youming Liu**

Affiliation:
Department of Applied Mathematics, Beijing Polytechnic University, Pingle Yuan 100, Beijing 100022, People’s Republic of China

**Yuesheng Xu**

Affiliation:
Department of Mathematics, 215 Carnegie Hall, Syracuse University, Syracuse, New York 13244 – and – Institute of Mathematics, Academy of Mathematics and System Sciences, The Chinese Academy of Sciences, Beijing 100080, People’s Republic of China

DOI:
https://doi.org/10.1090/S0002-9939-05-08067-6

Keywords:
Orthonormal exponential bases,
non-constant frequency,
spectral sequences,
Walsh systems

Received by editor(s):
September 11, 2003

Published electronically:
March 21, 2005

Additional Notes:
The first author was supported in part by the Natural Science Foundation of Beijing, No. 1022002

The second author was supported in part by the US National Science Foundation under grants 9973427 and 0312113, by the Natural Science Foundation of China under grant 10371122 and by the Chinese Academy of Sciences under the program “One Hundred Distinguished Chinese Young Scientists”

Communicated by:
David R. Larson

Article copyright:
© Copyright 2005
American Mathematical Society