Localization of Generalized Frames in Hilbert Spaces: Asymptotic Behavior of Concentration and Toeplitz Operators, Sampling and Interpolation, and Density Results

Suppose H is a separable and complex Hilbert space with a generalized frame (also known as continuous frame) indexed over a metric measure space X. We study the main properties of generalized frames and the operators defined by them, such as concentration operators and Toeplitz operators. Imposing certain localization conditions to the generalized frame, we describe the asymptotic behavior of concentration and Toeplitz operators, and derive important results about the distribution of their eigenvalues. Furthermore, working with multiple generalized frames in H intertwined by a localization conditions, we obtain very general density results. Many examples and applications are shown, among others we obtain necessary density conditions for sampling and interpolation, and these conditions can be applied on classical spaces, such as the Paley-Wiener space, the Bargmann-Fock space, and Gabor systems