Characterizations of multi-knot piecewise linear spectral sequences
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# Springer 2006
Characterizations of multi-knot piecewise linear spectral sequences Haitao Li a,b,* and Dunyan Yan c,**,. a
Research and Development Center, 501, China Academy of Railway Sciences, Daliu Shu No. 2, Haidian District, Beijing 100081, P.R. China E-mail: [email protected] b Academy of Mathematics and Systems Science, Chinese Academy of Science, East Road of Zhongguan Chun No. 55, Haidian District, Beijing 100080, P.R. China c School of Information Science and Engineering, Graduate School of the Chinese Academy of Sciences, Beijing 100080, P.R. China E-mail: [email protected]
Accepted 29 September 2005 Communicated by Yuesheng Xu
We study two classes of orthonormal bases for L2 ½0; 1 in this paper. The exponential parts of these bases are multi-knot piecewise linear functions. These bases are called spectral sequences. Characterizations of these multi-knot piecewise linear functions are provided. We also consider an opposite problem for single-knot piecewise linear spectral sequences, where the piecewise linear functions are defined on ½0; Þ and ½; 1. We show that such spectral sequences do not exist except for ¼ 12. Keywords: spectral sequence, orthogonal basis, trigonometric basis AMS 2000 subject classification: 42C15
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Introduction
The classical Fourier functions F :¼ fen : n 2 Zg; where Z denotes the integer set and en ðtÞ :¼ e2int ; t 2 I :¼ ½0; 1; forms an orthonormal basis for the space L2 ðIÞ. This basis plays an important role in many fields such as mathematics, engineering and physics. However, the Fourier basis
* Supported by the Technology and Research project 2002YF015 of the Ministry of Railway of China and by the Natural Science Foundation of China under grant 10371122. ** Supported by the Presidential Foundation of Graduate School of the Chinese Academy of Sciences (yzjj200505). . Corresponding author.
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H. Li, D. Yan / Characterizations of multi-knot piecewise linear spectral sequences
sometimes has serious limitations in many applications [5]. For example, they cannot reflect the instantaneous frequency of nonlinear and non-stationary signals (cf., [2, 3, 5]). In fact, in engineering, if a complex signal s has the form sðtÞ :¼ aðtÞe2iðtÞ ; t 2 I, then is called the phase function and usually is nonlinear. The frequency variation of s is assessed by instantaneous frequency (cf., [2, 3, 5]), which is defined by !ðtÞ ¼ dðtÞ dt ; t 2 I. Huang et al. [5] proposed an algorithm which is called empirical mode decomposition (EMD) to process nonlinear and non-stationary signals. This algorithm can decompose a signal into finite combinations of simple components. These simple components have two characteristics. One is that these components can be derived from signals adaptively; another is that these components can reflect the instantaneous frequency of signals after they are written in the form of aðÞe2iðÞ by using Hilbert transform (cf., [1, 9]). Actually, EMD proposed two questions in the signal analysis and functions decomposition. The first is the adaptiv
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