Asymptotic properties of Gabor frame operators as sampling density tends to infinity (Q1048190)

The convergence of Gabor frame operators which are given as a Riemannian sum of the reconstruction formula of the continuous transform is studied. The author considers first the \(L_2\)-case where necessary and sufficient conditions for existence and convergence are given for the case where frame and dual frame coincides. Afterwards, for the general case a condition is given in terms of analysis and synthesis window function belonging to the Wiener space and their product being locally Riemann-integrable. This condition is shown to be valid in the \(L_p\)-case, \(1\leq p\leq \infty\). To prove it Walnut's representation is used. In the end a counter-example is given which shows that the condition of the product being locally Riemann-integrable is not redundant.