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Products of Two-Wavelet Multipliers and Their Traces

  • Viorel Catană
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 205)

Abstract

Following Wong’s point of view in his book [12] (see Chapter 21) we give in this paper two formulas for the product of two two-wavelet multipliers \( \psi T_\sigma \bar \phi \) : L 2(ℝ n ) and \( \psi T_\tau \bar \phi \) : L 2(ℝ n )→L 2(ℝ n ), where σ and τ are functions in L 2(ℝ n ) and ϕ and ψ are any functions in L 2(ℝ n )∩L (ℝ n ) such that \( \parallel \phi \parallel _{L^2 (\mathbb{R}^n )} = \parallel \psi \parallel _{L^2 (\mathbb{R}^n )} = 1\). We also give a trace formula and an upper bound estimate on the trace class norm for such a product. Moreover we find sharp estimates on the norm in the trace class of two-wavelet multipliers P σ,ϕ,ψ: L 2(ℝ n )→L 2(ℝ n ) in terms of the symbols σ and the admissible wavelets ϕ and ψ and also we give an inequality about products of positive trace class one-wavelet multipliers. Finally, we give an example of a two-wavelet multiplier which extends Wong’s result concerning the Landau-Pollak-Slepian operator from the one-wavelet case to the two-wavelet case (see Chapter 20 in the book [12] by Wong).

Keywords

Admissible wavelet two-wavelet multiplier trace class operator 

Mathematics Subject Classification (2000)

Primary 42B15 Secondary 47G10 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Viorel Catană
    • 1
  1. 1.Department of Mathematics IUniversity Politehnica of BucharestBucharestRomania

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