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Inequalities in Mellin-Fourier Signal Analysis

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Wavelet Transforms and Time-Frequency Signal Analysis

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

A specific form of the Mellin transform, referred to as the “scale transform,” is known to be a natural complement to the Fourier transform for wide-band analytic signals. In this chapter, limitations for the simultaneous localization of scale transform pairs are investigated. A number of inequalities are established and discussed, based on various measures of spread (Heisenberg-type inequalities for variance-like measures and Hirschman-type inequalities for entropy). The same issue of maximally concentrating a signal in both scale and frequency domains is also addressed via spread measures, which are applied directly to joint scale-frequency distributions. A simple way of obtaining inequalities for Altes-type distributions is pointed out, new results pertaining to the unitary Bertrand distribution are established, as well as a new form of uncertainty relation for the wavelet transform.

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Authors
  1. Patrick Flandrin
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Editors and Affiliations

  1. Department of Mathematics, University of Central Florida, Orlando, Fl, 32816, USA

    Lokenath Debnath

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© 2001 Springer Science+Business Media New York

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Flandrin, P. (2001). Inequalities in Mellin-Fourier Signal Analysis. In: Debnath, L. (eds) Wavelet Transforms and Time-Frequency Signal Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0137-3_10

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  • DOI: https://doi.org/10.1007/978-1-4612-0137-3_10

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6629-7

  • Online ISBN: 978-1-4612-0137-3

  • eBook Packages: Springer Book Archive

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