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
Full-text PDF Free Access
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, https://doi.org/10.1098/rspa.1998.0193
Pedersen, Spectral sets whose spectrum is a lattice with a
base, J. Funct. Anal. 141 (1996), no. 2,
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, https://doi.org/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, https://doi.org/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, https://doi.org/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
- Q. Chen, N. Huang, S. Riemenschneider and Y. Xu, A B-spline approach for empirical mode decomposition, Adv. Comp. Math., to appear.
- I. Daubechies, Ten Lectures on Wavelets, CBMS 61, SIAM, Philadelphia, 1992. MR 1162107 (93e:42045)
- K. Grochenig, Foundations of Time-Frequency Analysis, Birkhauser, Boston, 2001. MR 1843717 (2002h:42001)
- N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Sheng, N. Yen, C. C. Tung and H. H. Liu, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proc. R. Soc. Land. A (1998) 454, 903-995. MR 1631591 (99d:76082)
- J. C. Lagarias and Y. Wang, Spectral sets and factorizations of finite Abelian groups, J. Funct. Anal. 145 (1997), 73-98. MR 1442160 (98b:47011b)
- S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, San Diego, 1998. MR 1614527 (99m:94012)
- C. A. Micchelli and Y. Xu, Using the matrix refinement equation for the construction of wavelets on invariant sets, Appl. Comput. Harmon. Anal. 1 (1994), 391-401. MR 1310661 (96k:42044)
- R. E. Paley, A remarkable series of orthogonal functions I, Proc. Landon Math. Soc. 34 (1932), 241-279.
- Y. Wang, Wavelets, tiling and spectral sets, Duke Math. J., 114 (2002), no. 1, 43-57. MR 1915035 (2003e:42057)
- R. Young, An Introduction to Non-harmonic Fourier Series, Academic Press, London, 1980. MR 0591684 (81m:42027)
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 42C15
Retrieve articles in all journals with MSC (2000): 42C15
Affiliation: Department of Applied Mathematics, Beijing Polytechnic University, Pingle Yuan 100, Beijing 100022, People’s Republic of China
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
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