Abstract:
This thesis is an exposition of the concept of localization of frames in the problem of irregular sampling in shift-invariant spaces. The given definition of the localization of a frame will appear to be equivalent to an off-diagonal decay of the matrix corre sponding to the frame operator. The proofs of some inverse-closedness theorems of certain classes of matrices having an off-diagonal decay will be given. These theorems imply the localization of the dual frame. Under these localization conditions, the Hilbert space theory can be extended to the family of associated Banach spaces. If the generator of a shift-invariant space satisfies necessary decay conditions, then it will be seen that its reproducing kernel frame will be a localized frame, and the theory will be applicable.
