Complex analytic study of Daubechies localization operators

We will consider the inverse problem of eigenvalues of Daubechies localization operator. We will introduce analytic continuation of eigenvalues of Daubechies operator and generating function (Z-transform) of eigenvalues of Daubechies localization operator. Especially, we will clarify the relationship between analytic continuation of eigenvalues of Daubechies localization operator and Z-transform of eigenvalues of Daubechies localization operator.. By making use of inverse Laplace transform and inverse Mellin transform, we will show reconstruction formulas for symbol function of Daubechies localization operator with rotational invariant symbol. In final section we will calculate the trace of Daubechies localization operator.