[Paper Review] Characterization of atomic decompositions, Banach frames, Xd-frames, duals and synthesis-pseudo-duals, with application to Hilbert frame theory

Diana T. Stoeva|arXiv (Cornell University)|Aug 31, 2011

Mathematical Analysis and Transform Methods35 references4 citations

TL;DR

This paper provides a comprehensive characterization of atomic decompositions, Banach frames, Xd-frames, and their duals in Banach and Hilbert spaces, focusing on the interchangeability of frame and non-frame sequences in series expansions. It introduces synthesis- and analysis-pseudo-duals, establishes conditions under which such non-frame sequences are necessarily frames, and applies the results to Hilbert frame theory, showing that for frames with finite-dimensional kernel of the synthesis operator, all pseudo-duals are dual frames.

ABSTRACT

In this paper we consider series expansions via a frame and a non-frame and the possibilities for interchange of the two sequences, both in the Hilbert and Banach space setting. First we give a characterization of frame-related concepts in Banach spaces (atomic decompositions, Banach frames, $X_d$-Riesz bases, $X_d$-frames, $X_d$-Bessel sequences, and sequences satisfying the lower $X_d$-frame condition). We also determine necessary and sufficient conditions for operators to preserve the type of the concepts listed above. Then we discuss differences and relationships between expansions in a Banach space and its dual space when interchanging the involved sequences. Finally, we apply some of the results to answer problems in Hilbert frame theory. We show that interchanging a frame and a non-Bessel sequence in series expansions is not always possible (leading to differentiation of analysis- and synthesis-pseudo-duals of a frame) and determine an appropriate subspace where interchange can be done. We characterize all the synthesis-pseudo-duals of a frame and determine a class of frames whose synthesis-pseudo-duals (resp. analysis-pseudo-duals) are necessarily frames. We also investigate connections between the lower frame condition and series expansions. Examples are given to illustrate statements in the paper and to show the optimality of some results.

Motivation & Objective

• To characterize atomic decompositions, Banach frames, Xd-frames, and related concepts in Banach spaces.
• To determine necessary and sufficient conditions for operators to preserve frame-type properties.
• To investigate the interchangeability of frame and non-frame sequences in series expansions in Banach and Hilbert spaces.
• To apply the results to Hilbert frame theory, particularly concerning synthesis- and analysis-pseudo-duals.
• To identify classes of frames for which all pseudo-duals are necessarily frames.

Proposed method

• Characterize atomic decompositions, Banach frames, Xd-frames, Xd-Bessel sequences, and lower Xd-frame conditions in Banach spaces.
• Establish operator conditions that preserve frame-type properties using duality and boundedness arguments.
• Analyze series expansions in dual spaces and compare interchangeability of sequences in Banach and Hilbert settings.
• Define synthesis-pseudo-duals (s-pseudo-duals) and analysis-pseudo-duals (a-pseudo-duals) as sequences satisfying only one of the dual frame reconstruction equations.
• Use the kernel of the synthesis operator T_G to determine when pseudo-duals must be frames, leveraging results from [31] and [13].
• Apply functional analytic tools, including ℓ² convergence and orthogonal complement arguments, to prove that (c_i) ∈ ℓ² implies (⟨f_i, g⟩) ∈ ℓ² for g ∈ H.

Experimental results

Research questions

• RQ1 Under what conditions can a non-Bessel sequence be interchanged with a frame in a series expansion?
• RQ2 When are all synthesis- or analysis-pseudo-duals of a frame necessarily frames?
• RQ3 What is the relationship between the lower frame condition and the existence of pseudo-dual sequences?
• RQ4 How do frame properties behave under operator transformations in Banach and Hilbert spaces?
• RQ5 What is the role of the kernel of the synthesis operator in determining the frame nature of pseudo-duals?

Key findings

• For frames with finite-dimensional kernel of the synthesis operator T_G, every s-pseudo-dual and every a-pseudo-dual is necessarily a dual frame.
• The class of frames for which all pseudo-duals are frames includes Riesz bases and certain overcomplete frames with finite-dimensional ker(T_G).
• Interchanging a frame and a non-Bessel sequence in series expansions is not always possible, and such interchange is only valid in a specific subspace.
• All synthesis-pseudo-duals of a frame are characterized as sequences (f_i) satisfying ∑⟨f, g_i⟩f_i = f for all f ∈ H.
• The analysis-pseudo-duals are characterized as sequences (f_i) such that ∑⟨g, f_i⟩g_i = g for all g ∈ H.
• The paper shows that the statement in [34, Prop. 4.10] regarding pseudo-frame representations is incorrect, as demonstrated by Example 5.1.