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Orthonormal bases with nonlinear phases

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Abstract

For adaptive representation of nonlinear signals, the bank \({\cal M}\) of real square integrable functions that have nonlinear phases and nonnegative instantaneous frequencies under the analytic signal method is investigated. A particular class of functions with explicit expressions in \({\cal M}\) is obtained using recent results on the Bedrosian identity. We then construct orthonormal bases for the Hilbert space of real square integrable functions with the basis functions from \({\cal M}\).

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau, People’s Republic of China

    Tao Qian

  2. School of Information Science and Engineering, Graduate University of Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Rui Wang

  3. Department of Mathematics, Syracuse University, Syracuse, NY, 13244, USA

    Yuesheng Xu & Haizhang Zhang

Authors
  1. Tao Qian
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  2. Rui Wang
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  3. Yuesheng Xu
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  4. Haizhang Zhang
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Corresponding author

Correspondence to Yuesheng Xu.

Additional information

Communicated by Qiyu Sun.

Supported in part by Macau Science and Technology Fund 051/2005/A.

Supported in part by the US National Science Foundation under grants CCR-0407476 and DMS-0712827, by National Aeronautics and Space Administration under Cooperative Agreement NNX07AC37A, by the Natural Science Foundation of China under grants 10371122 and 10631080, and by the Education Ministry of the People’s Republic of China under the Changjiang Scholar Chair Professorship Program through Sun Yat-sen University.

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Qian, T., Wang, R., Xu, Y. et al. Orthonormal bases with nonlinear phases. Adv Comput Math 33, 75–95 (2010). https://doi.org/10.1007/s10444-009-9120-0

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  • DOI: https://doi.org/10.1007/s10444-009-9120-0

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