Frame potential for finite-dimensional Banach spaces

Abstract

We define the frame potential for a Schauder frame on a finite dimensional Banach space as the square of the 2-summing norm of the frame operator. As is the case for frames for Hilbert spaces, we prove that the frame potential can be used to characterize finite unit norm tight frames (FUNTFs) for finite dimensional Banach spaces. We prove the existence of FUNTFs for a variety of spaces, and in particular that every n-dimensional complex Banach space with a 1unconditional basis has a FUNTF of N vectors for every N ≥ n. However, many interesting results on FUNTFs and sums of rank one projections for Hilbert spaces remain unknown for Banach spaces and we conclude the paper with multiple open questions.

Discrete and Continuous Welch Bounds for Banach Spaces with Applications

“…The reason is that frame potential for Banach spaces can not be defined in the way in Definition 3.3. One has to go to the theory of p-summing operators (see [24,66]) to define frame potential in Banach spaces, see [17]. Theorem 2.5 again gives the following result.…”

Paulsen and Projection Problems for Banach Spaces

“…Motivated from the foundational work of Casazza, Han, and Larson [15] and from the fundamental work of Daubechies and DeVore [26], Casazza, Dilworth, Odell, Schlumprecht, and Zsak [8] introduced the notion of approximate Schauder frames for Banach spaces which led to the notion of approximate Schauder frames. Later, in the course of defining frame potential for Banach space and from the notion of Auerbach basis [2,27,32,46], Chavez-Domingues, Freeman, and Kornelson [22] introduced the notion of unit norm tight frames for Banach spaces.…”

“…Definition 2.1. [15,22,31,40,45] Let {τ j } n j=1 be a collection in a Banach space X and {f j } n j=1 be a collection in X * (dual of X ). The pair ({f j } n j=1 , {τ j } n j=1 ) is said to be an approximate Schauder frame (ASF) for X if the frame operator…”

Cross Frame Potential

“…The frame potential of a finite sequence of vectors is a theoretical version of potential energy in physics developed in [2] and later studied in [3,7,15]. In physics, a system of objects acting under a force will move to minimize its potential energy.…”

“…Further results in this direction can be found in [3], where alternative definitions and generalizations of the notion are also pursued. For a study of the frame potential in finitedimensional Banach spaces see [7].…”