Grochenig and Balan, Casazza, Heil, and Landau introduced the concepts of localization. The concepts were used to Gabor frames, wavelet frames and sampling theorem in recent years. Here they are applied to the frame of exponential windows with the conclusion that the frame of exponential windows is a Banach frame for a kind of Banach spaces, and the conclusion is also obtained about the relationship between frame bounds, frame density, measure and density of indexing set.
We show that every Bessel sequence （and therefore every frame） in a separable Hilbert space can be expanded to a tight frame by adding some elements. The proof is based on a recent generalization of the frame concept, the g-frame, which illustrates that g-frames could be useful in the study of frame theory. As an application, we prove that any Gabor frame can be expanded to a tight frame by adding one window function.
