No Access

REMARKS ON ADAPTIVE FOURIER DECOMPOSITION

Department of Mathematics, Faculty of Science and Technology, University of Macau Taipa, Macau, China

and

Department of Mathematics, Faculty of Mathematics and Computer Science, Wuhan Textile University, Wuhan City, Hubei Province, China

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0219691313500070Cited by:22 (Source: Crossref)

Abstract

This is a continuation of the study of adaptive Fourier decomposition (AFD).¹⁵ Under a mild condition not in terms of smoothness, a convergence rate is provided. We prove that the selection of the parameters corresponding to Fourier series in the average sense is optimal. We also present the transformation matrices between the adaptive rational orthogonal system and the related sequence of the shifted Cauchy kernels and their derivatives.

Supported by University of Macau under research Grant RG056/01-02S/QT/FST.

Keywords:

AMSC: 42A50, 32A30, 32A35, 46J15

References

Q. Chen, L. Li and T. Qian, Int. J. Wavelets, Multiresolut. Inf. Process. 3, 465 (2005), DOI: 10.1142/S0219691305000968. Link, Web of Science, Google Scholar
Q. Chen, L. Li and T. Qian, Phys. D: Nonlinear Phenomena 221, 1 (2006), DOI: 10.1016/j.physd.2006.06.013. Crossref, Web of Science, Google Scholar
G. Davis, S. Mallat and M. Avellaneda, Constructive Approx. 13, 57 (1997), DOI: 10.1007/BF02678430. Crossref, Web of Science, Google Scholar
R. A. Devore and V. N. Temlyakov, Adv. Comput. Math. 5, 173 (1996), DOI: 10.1007/BF02124742. Crossref, Web of Science, Google Scholar
K. J. Lee, The Annals Statist. 20, 608 (1992). Web of Science, Google Scholar
B. Ninness and F. Gustafsson, IEEE Trans. Auto. Control 42, 515 (1997), DOI: 10.1109/9.566661. Crossref, Web of Science, Google Scholar
B. Ninness, H. Hjalmarsson and F. Gustafasson, IEEE Trans. Auto. Control 42, 1384 (1999). Google Scholar
B. Picinbono, IEEE Trans. Signal Process. 45, 552 (1997), DOI: 10.1109/78.558469. Crossref, Web of Science, Google Scholar
T. Qian, J. Integral Equations Approx 17, 159 (2005), DOI: 10.1216/jiea/1181075323. Crossref, Web of Science, Google Scholar
T. Qian, J. Math. Anal. Appl. 314, 526 (2006), DOI: 10.1016/j.jmaa.2005.04.003. Crossref, Web of Science, Google Scholar
T. Qian, Math. Methods Appl. Sci. 29, 1187 (2006), DOI: 10.1002/mma.721. Crossref, Web of Science, Google Scholar
T. Qian, Math. Methods Appl. Sci. 32, 253 (2009), DOI: 10.1002/mma.1032. Crossref, Web of Science, Google Scholar
T. Qian, Q. Chen and L. Li, Phy. D: Nonlinear Phenomena 203, 80 (2005), DOI: 10.1016/j.physd.2005.03.005. Crossref, Web of Science, Google Scholar
T. Qian and E. Wegert, Optimal approximation by Blaschke forms, Complex Variables Elliptic Equations DOI:10.1080/17476933.2011.557152 . Google Scholar
T. Qian and Y. Wang, Adv. Comput. Math. 34, 279 (2011), DOI: 10.1007/s10444-010-9153-4. Crossref, Web of Science, Google Scholar
V. N. Temlyakov, Adv. Comput. Math. 8, 249 (1998), DOI: 10.1023/A:1018900431309. Crossref, Web of Science, Google Scholar
J. L. Walsh , Interpolation and Approximation by Rational Functions in the Complex Plain ( Amer. Math. Soc. Colloq. Publications , 1969 ) . Google Scholar