Approximation of Weyl-Heisenberg frame operators
Approximations of the inverse of Weyl-Heisenberg frame (Gabor frame) operators have been studied by \textit{P. G. Casazza} and \textit{O. Christensen} [J. Approx. Theory 103, No. 2, 338--356 (2000; Zbl 0951.46008)] by using finite dimensional truncations. It has been observed that study on approximations of these operators by positive operators for the same kind with nice generators has not done so extensively and the nice generators like compactly supported smooth functions may not always yield frames, however, they generate Bessel sequences (preframes). Thus, approximations using preframe operators rather than frame operators have been investigated in this paper. Also, it has been shown that such approximations are possible in the weak operator topology under mild conditions on the window function \(g\geq 0\) and in the strong operator topology in some special cases. Moreover, the approximations of Weyl-Heisenberg frame operators by Weyl-Heisenberg preframe operators generated by compactly supported smooth functions have been presented.
scientific article; zbMATH DE number 6675403