[Paper Review] Construction of frame relative to n-Hilbert space

Prasenjit Ghosh, T. K. Samanta|arXiv (Cornell University)|Jan 5, 2021

Mathematical Analysis and Transform Methods10 references4 citations

TL;DR

This paper introduces frames in n-Hilbert spaces, generalizing frame theory from Hilbert spaces to n-normed structures. It establishes that a sequence forms a frame relative to an n-Hilbert space if and only if its pre-frame operator is surjective, and proves that the image of a frame under a bounded linear operator is a frame if and only if the operator is invertible.

ABSTRACT

In this paper, our aim is to introduce the concept of a frame in n-Hilbert space and describe some of their properties. We further discuss tight frame relative to n-Hilbert space. At the end, we study the relationship between frame and bounded linear operator in n-Hilbert space.

Motivation & Objective

• To generalize frame theory from standard Hilbert spaces to n-Hilbert spaces equipped with n-inner products and n-norms.
• To define and study frames and tight frames in the context of n-Hilbert spaces, extending classical frame concepts to higher-order structures.
• To investigate the relationship between frames and bounded linear operators, particularly under what conditions the image of a frame remains a frame.
• To characterize frames in n-Hilbert spaces using the surjectivity of the pre-frame operator and the existence of a pseudo-inverse.
• To establish necessary and sufficient conditions for a sequence to be a frame relative to a fixed (n−1)-tuple of vectors in an n-Hilbert space.

Proposed method

• Define n-inner product and n-norm on a linear space, satisfying axioms including linear dependence characterization and permutation invariance.
• Introduce the concept of a frame in an n-Hilbert space as a sequence {f_i} such that for all f in the space, A‖f|a₂,…,aₙ‖² ≤ ∑|⟨f,f_i|a₂,…,aₙ⟩|² ≤ B‖f|a₂,…,aₙ‖².
• Define the pre-frame operator T_F: l²(ℕ) → X_F by T_F{c_i} = ∑c_i f_i, and its adjoint (analysis operator) T_F^*: H → l²(ℕ).
• Use the frame operator S_F = T_F T_F^* to characterize frame properties, showing it is bounded, positive, self-adjoint, and invertible.
• Apply the pseudo-inverse operator U^† to characterize the surjectivity of the pre-frame operator and derive frame conditions.
• Use the Cauchy-Schwarz inequality and operator norm estimates to derive frame bounds from operator relationships, particularly involving T_F T_F'^* = I_F.

Experimental results

Research questions

• RQ1 What conditions must a sequence satisfy to be a frame in an n-Hilbert space relative to a fixed (n−1)-tuple of vectors?
• RQ2 When is the image of a frame under a bounded linear operator also a frame in the target space?
• RQ3 How can the pre-frame operator be used to characterize frames in n-Hilbert spaces?
• RQ4 What is the role of the pseudo-inverse in establishing frame conditions in n-Hilbert spaces?
• RQ5 Under what conditions does a Bessel sequence become a frame in the n-Hilbert space setting?

Key findings

• A sequence {f_i} is a frame in an n-Hilbert space relative to (a₂,…,aₙ) if and only if its associated pre-frame operator T_F is surjective.
• The frame operator S_F = T_F T_F^* is bounded, positive, self-adjoint, and invertible in the n-Hilbert space setting.
• The image of a frame under a bounded linear operator U is a frame if and only if U is invertible.
• If two Bessel sequences {f_i} and {g_i} have pre-frame operators satisfying T_F (T_F')^* = I_F, then both sequences are frames with lower bounds 1/‖T_F'^*‖² and 1/‖T_F^*‖² respectively.
• The frame condition in n-Hilbert space is characterized by the inequality ‖f|a₂,…,aₙ‖² ≤ ‖T_F'^*‖² ∑|⟨f,f_i|a₂,…,aₙ⟩|², linking operator norms to frame bounds.