p-Frames and Shift Invariant Subspaces of Lp

By Akram Aldroubi, Qiyu Sun, Wai-Shing Tang. 2001

188 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/580aeb66b847ef7e221917c19b04fccfe7b3f9a4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Convolution, Average Sampling, and a Calderon Resolution of the Identity for Shift-Invariant Spaces (2005, background): “Although the assumption that Φ generates a Riesz basis is reasonable, it is not necessarily satisfied in practice and it may not even be true that φ and its shifts generate a frame ([5, 8]).”
Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces (2001, background): “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].”
Stability of Localized Operators (2008, unknown): “operator Ca associated with a summable sequence a= (a(j))j∈Z, it is known that Ca has ℓp-stability for some 1 ≤ p≤ ∞ if and only if ˆa(ξ) 6= 0 for all ξ∈ R (1.3) where ˆa(ξ) := P j∈Z a(j)e −ijξ (c.f. [3,43]). Therefore the ℓp-stability of the convolution operator associated with a summable sequence ”
Frames, Riesz bases, and sampling expansions in Banach spaces via semi-inner products☆ (2011, background): “Before moving on, let us make connections with the existing definitions [1,3,4,16] of frames and Bessel sequences in Banach spaces, and the classical ones [8,10,31,44] for Hilbert spaces.”
SLANTED MATRICES, BANACH FRAMES, AND SAMPLING (2007, background): “Its analogues in Banach spaces, however, are non-trivial (see [6, 8, 21, 32] and references therein).”
Sampling Expansions in Reproducing Kernel Hilbert and Banach Spaces (2009, background): “For instance, it was proved in [3] that under certain natural conditions on the generators, a translation p-frame for a finitely generated shift-invariant subspace is always a Banach frame.”
Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator (2004, background): “2. Some authors use a weaker concept instead of Definition 12. A set E is called a p-frame, if only conditions (a) and (b) hold with Bd = � p [ 7 ].”