On characterization of the optimal biorthogonal window functions for Gabor transforms

Gabor transforms have been recognized as useful tools in signal analysis. It is known that the solutions for the biorthogonal analysis window function γ(t) given a synthesis window function h(t) in Gabor transforms are not unique in general. Among these solutions, the minimum norm solution has already been given by Wexler and Raz in the discrete-time case and has been studied by Janssen, Ron and Shen, and Daubechies et al. in the continuous-time case. The minimum norm solution in the discrete-time case was also proved to be equal to the most orthogonal-like solution by Qian and Chen. In this note, we consider a general optimal-solution problem, where the minimum norm and the most orthogonal-like solutions are two special cases. We prove that these optimal solutions in many cases are equal. We also prove that it remains true in the continuous-time case.