p-Frames and Shift Invariant Subspaces of L p

Abstract

In this article we investigate the frame properties and closedness for the shift invariant spaceWe derive necessary and sufficient conditions for an indexed family {φ i (· − j) : 1 ≤ i ≤ r, j ∈ Z d } to constitute a p-frame for V p ( ), and to generate a closed shift invariant subspace of L p . A function in the L p -closure of V p ( ) is not necessarily generated by p coefficients. Hence we often hope that V p ( ) itself is closed, i.e., a Banach space. For p = 2, this issue is complicated, but we show that under the appropriate conditions on the frame vectors, there is an equivalence between the concept of p-frames, Banach frames, and the closedness of the space they generate. The relation between a function f ∈ V p ( ) and the coefficients of its representations is neither obvious, nor unique, in general. For the case of p-frames, we are in the context of normed linear spaces, but we are still able to give a characterization of p-frames in terms of the equivalence between the norm of f and an p -norm related to its representations. A Banach frame does not have a dual Banach frame in general, however, for the shift invariant spaces V p ( ), dual Banach frames exist and can be constructed.

Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces

“…The concept of frames generalizes the notion of orthogonal bases and Riesz bases in Hilbert spaces and of unconditional bases in some Banach spaces [2,5,6,12,14,15,20,28,29,46,66,97]. …”

“…They have been extensively studied in recent years (see, for instance, [6,19,52,67,68,69]). Sampling in shift-invariant spaces that are not bandlimited is a suitable and realistic model for many applications, e.g., for taking into account real acquisition and reconstruction devices, for modeling signals with smoother spectrum than is the case with bandlimited functions, or for numerical implementation [9,13,22,26,85,86,103,104,107,110,115,116].…”

Classes of finite equal norm Parseval frames

“…Translates of a single function play a fundamental role in frame theory, timefrequency analysis, sampling theory and more [1,17,18].…”

Some properties of g-p-frames in complex Banach spaces

“…Frames play an important role not only in pure mathematics, but also in signal processing, so it is natural to search for extensions to Banach spaces. p-frame in complex Banach spaces formally defined by Aldroubi et al [3] in 2001. They were reintroduced and developed in 2003 by Chrisrensen and Stoeva [5].…”