# 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.*

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