Products of Two-Wavelet Multipliers and Their Traces

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(ℝn) and \( \psi T_\tau \bar \phi \) : L 2(ℝn)→L 2(ℝn), where σ and τ are functions in L 2(ℝn) and ϕ and ψ are any functions in L 2(ℝn)∩L ∞(ℝn) 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(ℝn)→L 2(ℝn) 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).Mathematics Subject Classification (2000)Primary 42B15Secondary 47G10KeywordsAdmissible wavelettwo-wavelet multipliertrace class operator