| Every Weyl transform is in fact a pseudo-differential operator. This well-known fact has been given in the book [12] by Folland. Using various techniques in the book [27] by Wong and introducing amplitude operators, we give another proof of this result. Daubechies operators, first studied by Daubechies as filters in signal analysis, have been shown to be Weyl transforms in the book [26] by Wong. The problem of finding a filter that has the same effect as two filters arranged in series in signal analysis is the same as the computation of the product (or composition) of two Daubechies operators. We find that the symbol of the product of two Daubechies operators is a new twisted convolution of the symbols of the given operators and the product of two Daubechies operators with symbols in L2( Cn ) is, in general, not a Daubechies operator with symbol in L2( Cn ). Because of this fact, we give a necessary and sufficient condition such that the product of two Daubechies operators with symbols in L2( Cn ) is still a Daubechies operator with symbol in L 2( Cn ). We give a subspace M of L 2( Cn ) such that the product of two Daubechies operators with symbols in M is a Daubechies operator with symbol in M. Next, we give trace formulas for Weyl transforms and localization operators, respectively, under suitable assumptions. Finally, we compute the eigenvalues and eigenfunctions of a concentration operator, i.e., a Daubechies operator, with radial symbol for each normalized Hermite function as the admissible wavelet. The method we use in this dissertation differs from that in the paper [2] by Daubechies. Furthermore, we discuss the asymptotic behavior of the eigenvalues. |
