Two-Wavelet Theory

Abstract

The results hitherto given are for localization operators L _F,φ : X → X defined in terms of one admissible wavelet φ for the square-integrable representation π: G → U(X) of G on X. In this chapter we introduce the notion of a localization operator L _F,φψ : X → X, 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: G →U(X) of G on X. It is proved in this chapter that L _F,φ,ψ : X → X is in S ₁ and a formula for the trace of L _F,φ,ψ : X → X 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 : X → X In order to obtain a lower bound for the norm ‖ L _F,φ,ψ ‖ _{S 1} of L _F,φ,ψ : X → X,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.