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Functional Ghobber-Jaming Uncertainty Principle

K. Mahesh Krishna

TL;DR

This work formulates a Banach space analogue of the Ghobber-Jaming finite dimensional uncertainty principle by introducing $p$-orthonormal bases in a finite dimensional Banach space $\mathcal{X}$ and establishing a quantitative bound under a cross coherence condition. The main result, the Functional Ghobber-Jaming Uncertainty Principle, states that for all $x\in\mathcal{X}$ one has $\|x\| \le \left(1+ \dfrac{1}{1- o(M)^{1/q} o(N)^{1/p} \max_{j,k} |g_k(\tau_j)|}\right) \left[\left(\sum_{j\in M^c} |f_j(x)|^p\right)^{1/p} + \left(\sum_{k\in N^c} |g_k(x)|^p\right)^{1/p}\right]$, and in particular if $x$ is supported on $M$ and $N$ in the respective expansions then $x=0$. The approach leverages an invertible isometry $V$ linking the two $p$-orthonormal bases and analyzes projection operators to connect the tail norms, with a Hilbert space specialization recovering the classical Ghobber-Jaming inequality and discussions of sharpness and extensions to $1$- and $\infty$-orthonormal bases as well as to $p$-Schauder frames. The result provides a concrete finite dimensional Banach space uncertainty bound that generalizes prior linear algebra uncertainty principles to broader normed settings.

Abstract

Let $(\{f_j\}_{j=1}^n, \{τ_j\}_{j=1}^n)$ and $(\{g_k\}_{k=1}^n, \{ω_k\}_{k=1}^n)$ be two p-orthonormal bases for a finite dimensional Banach space $\mathcal{X}$. Let $M,N\subseteq \{1, \dots, n\}$ be such that \begin{align*} o(M)^\frac{1}{q}o(N)^\frac{1}{p}< \frac{1}{\displaystyle \max_{1\leq j,k\leq n}|g_k(τ_j) |}, \end{align*} where $q$ is the conjugate index of $p$. Then for all $x \in \mathcal{X}$, we show that \begin{align}\label{FGJU} (1) \quad \quad \quad \quad \|x\|\leq \left(1+\frac{1}{1-o(M)^\frac{1}{q}o(N)^\frac{1}{p}\displaystyle\max_{1\leq j,k\leq n}|g_k(τ_j)|}\right)\left[\left(\sum_{j\in M^c}|f_j(x)|^p\right)^\frac{1}{p}+\left(\sum_{k\in N^c}|g_k(x) |^p\right)^\frac{1}{p}\right]. \end{align} We call Inequality (1) as \textbf{Functional Ghobber-Jaming Uncertainty Principle}. Inequality (1) improves the uncertainty principle obtained by Ghobber and Jaming \textit{[Linear Algebra Appl., 2011]}.

Functional Ghobber-Jaming Uncertainty Principle

TL;DR

This work formulates a Banach space analogue of the Ghobber-Jaming finite dimensional uncertainty principle by introducing -orthonormal bases in a finite dimensional Banach space and establishing a quantitative bound under a cross coherence condition. The main result, the Functional Ghobber-Jaming Uncertainty Principle, states that for all one has , and in particular if is supported on and in the respective expansions then . The approach leverages an invertible isometry linking the two -orthonormal bases and analyzes projection operators to connect the tail norms, with a Hilbert space specialization recovering the classical Ghobber-Jaming inequality and discussions of sharpness and extensions to - and -orthonormal bases as well as to -Schauder frames. The result provides a concrete finite dimensional Banach space uncertainty bound that generalizes prior linear algebra uncertainty principles to broader normed settings.

Abstract

Let and be two p-orthonormal bases for a finite dimensional Banach space . Let be such that \begin{align*} o(M)^\frac{1}{q}o(N)^\frac{1}{p}< \frac{1}{\displaystyle \max_{1\leq j,k\leq n}|g_k(τ_j) |}, \end{align*} where is the conjugate index of . Then for all , we show that \begin{align}\label{FGJU} (1) \quad \quad \quad \quad \|x\|\leq \left(1+\frac{1}{1-o(M)^\frac{1}{q}o(N)^\frac{1}{p}\displaystyle\max_{1\leq j,k\leq n}|g_k(τ_j)|}\right)\left[\left(\sum_{j\in M^c}|f_j(x)|^p\right)^\frac{1}{p}+\left(\sum_{k\in N^c}|g_k(x) |^p\right)^\frac{1}{p}\right]. \end{align} We call Inequality (1) as \textbf{Functional Ghobber-Jaming Uncertainty Principle}. Inequality (1) improves the uncertainty principle obtained by Ghobber and Jaming \textit{[Linear Algebra Appl., 2011]}.
Paper Structure (2 sections, 9 theorems, 30 equations)

This paper contains 2 sections, 9 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Functional Ghobber-Jaming Uncertainty Principle

Key Result

Theorem 1.1

JAMINGNAZAROV (Nazarov-Jaming Uncertainty Principle) For each $d \in \mathbb{N}$, there exists a universal constant $C_d$ (depends upon $d$) satisfying the following: If $E, F \subseteq \mathbb{R}^d$ are measurable subsets having finite measure, then for all $f \in \mathcal{L}^2 (\mathbb{R}^d)$, In particular, if $f$ is supported on $E$ and $\widehat{f}$ is supported on $F$, then $f=0$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Example 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 5 more