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Two-Wavelet Theory

  • M. W. Wong
Part of the Operator Theory: Advances and Applications book series (OT, volume 136)

Abstract

The results hitherto given are for localization operators L F,φ : XX defined in terms of one admissible wavelet φ for the square-integrable representation π: GU(X) of G on X. In this chapter we introduce the notion of a localization operator L F,φψ : XX, which is defined in terms of a symbol F in L l (G) and two admissible wavelets φ and ψ for the square-integrable representation π: GU(X) of G on X. It is proved in this chapter that L F,φ,ψ : XX is in S 1 and a formula for the trace of L F,φ,ψ : XX is given. These results extend, respectively, the corresponding results in Chapter 12 and Chapter 13 from the one-wavelet case to the two-wavelet case. We also give in this chapter the trace class norm inequalities for the localization operator L F,φ,ψ : XX In order to obtain a lower bound for the norm ‖ L F,φ,ψ S 1 of L F,φ,ψ : XX,we need the formula (9.1), which is an analogue of the resolution of the identity formula (6.3) for two admissible wavelets for an irreducible and square-integrable representation π: G → U(X) of G on X.

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Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • M. W. Wong
    • 1
  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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