Functional Deutsch Uncertainty Principle

1. Introduction

Let $d\in \mathbb{N}$ and $\widehat{}:{\mathcal{L}}^2\left({\mathbb{R}}^d\right)\to {\mathcal{L}}^2\left({\mathbb{R}}^d\right)$ be the unitary Fourier transform obtained by extending uniquely the bounded linear operator

$$\begin{array}{c}\hfill \widehat{}:{\mathcal{L}}^1\left({\mathbb{R}}^d\right)\cap {\mathcal{L}}^2\left({\mathbb{R}}^d\right)\ni f\mapsto \widehat{f}\in C_0\left({\mathbb{R}}^d\right);\widehat{f}:{\mathbb{R}}^d\ni \xi \mapsto \widehat{f}\left(\xi \right)\underset{{\mathbb{R}}^d}{\int }f\left(x\right)e^{-2\pi i\langle x,\xi \rangle }dx\in \mathbb{C}.\end{array}$$

The Shannon entropy at a function $f\in {\mathcal{L}}^2\left({\mathbb{R}}^d\right)\setminus \left\{0\right\}$ is defined as

$$\begin{array}{c}\hfill S\left(f\right)-\underset{{\mathbb{R}}^d}{\int }{\left|\frac{f\left(x\right)}{\parallel f\parallel }\right|}^2\log {\left|\frac{f\left(x\right)}{\parallel f\parallel }\right|}^{2}dx\end{array}$$

(with the convention $0\log 0=0$) [1]. In 1957, Hirschman proved the following result [2].

Theorem 1.1.

[2] (Hirschman Inequality) For all $f\in {\mathcal{L}}^2\left({\mathbb{R}}^d\right)\setminus \left\{0\right\}$,

$$\begin{array}{c}\hfill S\left(f\right)+S\left(\widehat{f}\right)\ge 0.\end{array}$$

(2)

In the same paper [2] Hirschman conjectured that Inequality (2) can be improved to

$$\begin{array}{c}\hfill S\left(f\right)+S\left(\widehat{f}\right)\ge d(1-\log 2),f\in {\mathcal{L}}^2\left({\mathbb{R}}^d\right)\setminus \left\{0\right\}.\end{array}$$

(3)

Inequality (3) was proved independently in 1975 by Beckner [3] and Bialynicki-Birula and Mycielski [4].

Theorem 1.2.

[3,4] (Hirschman-Beckner-Bialynicki-Birula-Mycielski Uncertainty Principle) For all $f\in {\mathcal{L}}^2\left({\mathbb{R}}^d\right)\setminus \left\{0\right\}$,

$$\begin{array}{c}\hfill S\left(f\right)+S\left(\widehat{f}\right)\ge d(1-\log 2).\end{array}$$

Now one naturally asks whether there is a finite dimensional version of Shannon entropy and uncertainty principle. Let $\mathcal{H}$ be a finite dimensional Hilbert space. Given an orthonormal basis ${\left\{{\tau }_j\right\}}_{j=1}^n$ for $\mathcal{H}$, the (finite) Shannon entropy at a point $h\in {\mathcal{H}}_{\tau }$ is defined as

$$\begin{array}{c}\hfill S_{\tau }\left(h\right)-\sum _{j=1}^n{\left|〈\frac{h}{\parallel h\parallel },{\tau }_j〉\right|}^2\log {\left|〈\frac{h}{\parallel h\parallel },{\tau }_j〉\right|}^2\ge 0,\end{array}$$

where ${\mathcal{H}}_{\tau }\{h\in \mathcal{H}:\langle h,{\tau }_j\rangle \ne 0,1\le j\le n\}$ [5]. In 1983, Deutsch derived following uncertainty principle for Shannon entropy which is fundamental to several developments in Mathematics and Physics [5].

Theorem 1.3.

[5] (Deutsch Uncertainty Principle) Let ${\left\{{\tau }_j\right\}}_{j=1}^n$, ${\left\{{\omega }_j\right\}}_{j=1}^n$ be two orthonormal bases for a finite dimensional Hilbert space $\mathcal{H}$. Then

$$\begin{array}{c}\hfill 2\log n\ge S_{\tau }\left(h\right)+S_{\omega }\left(h\right)\ge -2\log \left(\frac{{\displaystyle 1+\underset{1\le j,k\le n}{max}\left|\langle {\tau }_j,{\omega }_k\rangle \right|}}{2}\right)\ge 0,\forall h\in {\mathcal{H}}_{\tau }.\end{array}$$

(4)

Recently, author derived Banach space versions of Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani uncertainty principle [6], Donoho-Stark approximate support uncertainty principle [7] and Ghobber-Jaming uncertainty principle [8]. We then naturally ask what is the Banach space version of Inequality (4)? In this paper, we are going to answer this question.

2. Functional Deutsch Uncertainty Principle

In the paper, $\mathbb{K}$ denotes $\mathbb{C}$ or $\mathbb{R}$ and $\mathcal{X}$ denotes a finite dimensional Banach space over $\mathbb{K}$. Dual of $\mathcal{X}$ is denoted by ${\mathcal{X}}^{*}$. We need the notion of Parseval p-frames for Banach spaces.

Definition 2.1.

[9,10] Let $\mathcal{X}$ be a finite dimensional Banach space over $\mathbb{K}$. A collection ${\left\{f_j\right\}}_{j=1}^n$ in ${\mathcal{X}}^{*}$ is said to be a Parseval p-frame($1\le p<\infty$) for $\mathcal{X}$ if

$$\begin{array}{c}\hfill {\parallel x\parallel }^p=\sum _{j=1}^n{\left|f_j\left(x\right)\right|}^p,\forall x\in \mathcal{X}.\end{array}$$

(5)

Note that (5) says that $\parallel f_j\parallel \le 1$ for all $1\le j\le n$. Given a Parseval p-frame ${\left\{f_j\right\}}_{j=1}^n$ for $\mathcal{X}$, we define the (finite) p-Shannon entropy at a point $x\in {\mathcal{X}}_f$ as

$$\begin{array}{c}\hfill S_f\left(x\right)-\sum _{j=1}^n{\left|f_j\left(\frac{x}{\parallel x\parallel }\right)\right|}^p\log {\left|f_j\left(\frac{x}{\parallel x\parallel }\right)\right|}^p\ge 0,\end{array}$$

where ${\mathcal{X}}_f\{x\in \mathcal{X}:f_j\left(x\right)\ne 0,1\le j\le n\}$. Following is the fundamental result of this paper.

Theorem 2.2

(Functional Deutsch Uncertainty Principle) Let ${\left\{f_j\right\}}_{j=1}^n$ and ${\left\{g_k\right\}}_{k=1}^m$ be Parseval p-frames for a finite dimensional Banach space $\mathcal{X}$. Then

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{\left(nm\right)}^{\frac{1}{p}}}\le \underset{y\in \mathcal{X},\parallel y\parallel =1}{sup}\left(\underset{1\le j\le n,1\le k\le m}{max}\left|f_j\left(y\right)g_k\left(y\right)\right|\right)}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{c}\log \left(nm\right)\ge S_f\left(x\right)+S_g\left(x\right)\ge -p\log \left({\displaystyle \underset{y\in {\mathcal{X}}_f\cap {\mathcal{X}}_g,\parallel y\parallel =1}{sup}\left(\underset{1\le j\le n,1\le k\le m}{max}\left|f_j\left(y\right)g_k\left(y\right)\right|\right)}\right)>0,\\ \forall x\in {\mathcal{X}}_f\cap {\mathcal{X}}_g.\end{array}\end{array}$$

(6)

Proof. 

Let $z\in \mathcal{X}$ be such that $\parallel z\parallel =1$. Then

$$\begin{array}{cc}\hfill 1& =\left(\sum _{j=1}^n{\left|f_j\left(z\right)\right|}^p\right)\left(\sum _{k=1}^m{\left|g_k\left(z\right)\right|}^p\right)=\sum _{j=1}^n\sum _{k=1}^m{\left|f_j\left(z\right)g_k\left(z\right)\right|}^p\hfill \\ \hfill & \le \sum _{j=1}^n\sum _{k=1}^m{\left({\displaystyle \underset{y\in \mathcal{X},\parallel y\parallel =1}{sup}\left(\underset{1\le j\le n,1\le k\le m}{max}\left|f_j\left(y\right)g_k\left(y\right)\right|\right)}\right)}^p\hfill \\ \hfill & ={\left({\displaystyle \underset{y\in \mathcal{X},\parallel y\parallel =1}{sup}\left(\underset{1\le j\le n,1\le k\le m}{max}\left|f_j\left(y\right)g_k\left(y\right)\right|\right)}\right)}^{p}mn\hfill \end{array}$$

which gives

$$\begin{array}{c}\hfill \frac{1}{mn}\le {\left({\displaystyle \underset{y\in \mathcal{X},\parallel y\parallel =1}{sup}\left(\underset{1\le j\le n,1\le k\le m}{max}\left|f_j\left(y\right)g_k\left(y\right)\right|\right)}\right)}^p.\end{array}$$

Since $1={\sum }_{j=1}^n{\left|f_j\left(\frac{x}{\parallel x\parallel }\right)\right|}^p$ for all $x\in \mathcal{X}\setminus \left\{0\right\}$, $1={\sum }_{k=1}^m{\left|g_k\left(\frac{x}{\parallel x\parallel }\right)\right|}^p$ for all $x\in \mathcal{X}\setminus \left\{0\right\}$ and log function is concave, using Jensen’s inequality (see [11]) we get

$$\begin{array}{cc}\hfill S_f\left(x\right)+S_g\left(x\right)& =\sum _{j=1}^n{\left|f_j\left(\frac{x}{\parallel x\parallel }\right)\right|}^p\log \left(\frac{1}{{\left|f_j\left(\frac{x}{\parallel x\parallel }\right)\right|}^p}\right)+\sum _{k=1}^m{\left|g_k\left(\frac{x}{\parallel x\parallel }\right)\right|}^p\log \left(\frac{1}{{\left|g_k\left(\frac{x}{\parallel x\parallel }\right)\right|}^p}\right)\hfill \\ \hfill & \le \log \left(\sum _{j=1}^n{\left|f_j\left(\frac{x}{\parallel x\parallel }\right)\right|}^p\frac{1}{{\left|f_j\left(\frac{x}{\parallel x\parallel }\right)\right|}^p}\right)+\log \left(\sum _{k=1}^m{\left|g_k\left(\frac{x}{\parallel x\parallel }\right)\right|}^p\frac{1}{{\left|g_k\left(\frac{x}{\parallel x\parallel }\right)\right|}^p}\right)\hfill \\ \hfill & =\log n+\log m=\log \left(nm\right),\forall x\in {\mathcal{X}}_f\cap {\mathcal{X}}_g.\hfill \end{array}$$

Let $x\in {\mathcal{X}}_f\cap {\mathcal{X}}_g$. Then

$$\begin{array}{cc}\hfill S_f\left(x\right)+S_g\left(x\right)& =-\sum _{j=1}^n\sum _{k=1}^m{\left|f_j\left(\frac{x}{\parallel x\parallel }\right)\right|}^p{\left|g_k\left(\frac{x}{\parallel x\parallel }\right)\right|}^p\left[\log {\left|f_j\left(\frac{x}{\parallel x\parallel }\right)\right|}^p+\log {\left|g_k\left(\frac{x}{\parallel x\parallel }\right)\right|}^p\right]\hfill \\ \hfill & =-\sum _{j=1}^n\sum _{k=1}^m{\left|f_j\left(\frac{x}{\parallel x\parallel }\right)\right|}^p{\left|g_k\left(\frac{x}{\parallel x\parallel }\right)\right|}^p\log {\left|f_j\left(\frac{x}{\parallel x\parallel }\right)g_k\left(\frac{x}{\parallel x\parallel }\right)\right|}^p\hfill \\ \hfill & =-p\sum _{j=1}^n\sum _{k=1}^m{\left|f_j\left(\frac{x}{\parallel x\parallel }\right)\right|}^p{\left|g_k\left(\frac{x}{\parallel x\parallel }\right)\right|}^p\log \left|f_j\left(\frac{x}{\parallel x\parallel }\right)g_k\left(\frac{x}{\parallel x\parallel }\right)\right|\hfill \\ \hfill & \ge -p\sum _{j=1}^n\sum _{k=1}^m{\left|f_j\left(\frac{x}{\parallel x\parallel }\right)\right|}^p{\left|g_k\left(\frac{x}{\parallel x\parallel }\right)\right|}^p\log \left({\displaystyle \underset{y\in {\mathcal{X}}_f\cap {\mathcal{X}}_g,\parallel y\parallel =1}{sup}\left(\underset{1\le j\le n,1\le k\le m}{max}\left|f_j\left(y\right)g_k\left(y\right)\right|\right)}\right)\hfill \\ \hfill & =-p\log \left({\displaystyle \underset{y\in {\mathcal{X}}_f\cap {\mathcal{X}}_g,\parallel y\parallel =1}{sup}\left(\underset{1\le j\le n,1\le k\le m}{max}\left|f_j\left(y\right)g_k\left(y\right)\right|\right)}\right)\sum _{j=1}^n\sum _{k=1}^m{\left|f_j\left(\frac{x}{\parallel x\parallel }\right)\right|}^p{\left|g_k\left(\frac{x}{\parallel x\parallel }\right)\right|}^p\hfill \\ \hfill & =-p\log \left({\displaystyle \underset{y\in {\mathcal{X}}_f\cap {\mathcal{X}}_g,\parallel y\parallel =1}{sup}\left(\underset{1\le j\le n,1\le k\le m}{max}\left|f_j\left(y\right)g_k\left(y\right)\right|\right)}\right).\hfill \end{array}$$

□

Corollary 2.3.

Theorem 1.3 follows from Theorem 2.2.

Proof. 

Let ${\left\{{\tau }_j\right\}}_{j=1}^n$, ${\left\{{\omega }_j\right\}}_{j=1}^n$ be two orthonormal bases for a finite dimensional Hilbert space $\mathcal{H}$. Define

$$\begin{array}{c}\hfill f_j:\mathcal{H}\ni h\mapsto \langle h,{\tau }_j\rangle \in \mathbb{K};g_j:\mathcal{H}\ni h\mapsto \langle h,{\omega }_j\rangle \in \mathbb{K},\forall 1\le j\le n.\end{array}$$

Now by using Buzano inequality (see [12,13]) we get

$$\begin{array}{cc}\hfill {\displaystyle \underset{h\in \mathcal{H},\parallel h\parallel =1}{sup}\left(\underset{1\le j,k\le n}{max}\left|f_j\left(h\right)g_k\left(h\right)\right|\right)}& {\displaystyle =\underset{h\in \mathcal{H},\parallel h\parallel =1}{sup}\left(\underset{1\le j,k\le n}{max}|\langle h,{\tau }_j\rangle \left|\right|\langle h,{\omega }_k\rangle |\right)}\hfill \\ \hfill & {\displaystyle \le \underset{h\in \mathcal{H},\parallel h\parallel =1}{sup}\left(\underset{1\le j,k\le n}{max}\left({\parallel h\parallel }^2\frac{\parallel {\tau }_j\parallel \parallel {\omega }_k\parallel +|\langle {\tau }_j,{\omega }_k\rangle |}{2}\right)\right)}\hfill \\ \hfill & =\frac{{\displaystyle 1+\underset{1\le j,k\le n}{max}\left|\langle {\tau }_j,{\omega }_k\rangle \right|}}{2}.\hfill \end{array}$$

□

Theorem 2.2 brings the following question.

Question 2.4.

Given p, m, n and a Banach space $\mathcal{X}$, for which pairs of Parseval p-frames ${\left\{f_j\right\}}_{j=1}^n$ and ${\left\{g_k\right\}}_{k=1}^m$ for $\mathcal{X}$, we have equality in Inequality (6)?

Next we derive a dual inequality of (6). For this we need dual of Definition 2.1.

Definition 2.5.

[14,15,16] Let $\mathcal{X}$ be a finite dimensional Banach space over $\mathbb{K}$. A collection ${\left\{{\tau }_j\right\}}_{j=1}^n$ in $\mathcal{X}$ is said to be a Parseval p-frame ($1\le p<\infty$) for ${\mathcal{X}}^{*}$ if

$$\begin{array}{c}\hfill {\parallel f\parallel }^p=\sum _{j=1}^n{\left|f\left({\tau }_j\right)\right|}^p,\forall f\in {\mathcal{X}}^{*}.\end{array}$$

(7)

Note that (7) says that

$$\begin{array}{c}\hfill {\displaystyle \parallel {\tau }_j\parallel =\underset{f\in {\mathcal{X}}^{*},\parallel f\parallel =1}{sup}|f\left({\tau }_j\right)|\le \underset{f\in {\mathcal{X}}^{*},\parallel f\parallel =1}{sup}{\left(\sum _{j=1}^n{\left|f\left({\tau }_j\right)\right|}^p\right)}^{\frac{1}{p}}=\underset{f\in {\mathcal{X}}^{*},\parallel f\parallel =1}{sup}\parallel f\parallel =1,\forall 1\le j\le n.}\end{array}$$

Given a Parseval p-frame ${\left\{{\tau }_j\right\}}_{j=1}^n$ for ${\mathcal{X}}^{*}$, we define the (finite) p-Shannon entropy at a point $f\in {\mathcal{X}}_{\tau }^{*}$ as

$$\begin{array}{c}\hfill S_{\tau }\left(f\right)-\sum _{j=1}^n{\left|\frac{f\left({\tau }_j\right)}{\parallel f\parallel }\right|}^p\log {\left|\frac{f\left({\tau }_j\right)}{\parallel f\parallel }\right|}^p\ge 0,\end{array}$$

where ${\mathcal{X}}_{\tau }^{*}\{f\in {\mathcal{X}}^{*}:f\left({\tau }_j\right)\ne 0,1\le j\le n\}$. We now have the following dual to Theorem 2.2.

Theorem 2.6.

(Functional Deutsch Uncertainty Principle) Let ${\left\{{\tau }_j\right\}}_{j=1}^n$ and ${\left\{{\omega }_k\right\}}_{k=1}^m$ be two Parseval p-frames for the dual ${\mathcal{X}}^{*}$ of a finite dimensional Banach space $\mathcal{X}$. Then

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{\left(nm\right)}^{\frac{1}{p}}}\le \underset{g\in {\mathcal{X}}^{*},\parallel g\parallel =1}{sup}\left(\underset{1\le j\le n,1\le k\le m}{max}\left|g\left({\tau }_j\right)g\left({\omega }_k\right)\right|\right)}\end{array}$$

and

$$\begin{array}{c}\hfill \begin{array}{c}\log \left(nm\right)\ge S_{\tau }\left(f\right)+S_{\omega }\left(f\right)\ge -p\log \left({\displaystyle \underset{g\in {\mathcal{X}}_{\tau }^{*}\cap {\mathcal{X}}_{\omega }^{*},\parallel g\parallel =1}{sup}\left(\underset{1\le j\le n,1\le k\le m}{max}\left|g\left({\tau }_j\right)g\left({\omega }_k\right)\right|\right)}\right)>0,\\ \forall f\in {\mathcal{X}}_{\tau }^{*}\cap {\mathcal{X}}_{\omega }^{*}.\end{array}\end{array}$$

(8)

Proof. 

Let $h\in {\mathcal{X}}^{*}$ be such that $\parallel h\parallel =1$. Then

$$\begin{array}{cc}\hfill 1& =\left(\sum _{j=1}^n{\left|h\left({\tau }_j\right)\right|}^p\right)\left(\sum _{k=1}^m{\left|h\left({\omega }_k\right)\right|}^p\right)=\sum _{j=1}^n\sum _{k=1}^m{\left|h\left({\tau }_j\right)h\left({\omega }_k\right)\right|}^p\hfill \\ \hfill & \le \sum _{j=1}^n\sum _{k=1}^m{\left({\displaystyle \underset{g\in {\mathcal{X}}^{*},\parallel g\parallel =1}{sup}\left(\underset{1\le j\le n,1\le k\le m}{max}\left|g\left({\tau }_j\right)g\left({\omega }_k\right)\right|\right)}\right)}^p\hfill \\ \hfill & ={\left({\displaystyle \underset{g\in {\mathcal{X}}^{*},\parallel g\parallel =1}{sup}\left(\underset{1\le j\le n,1\le k\le m}{max}\left|g\left({\tau }_j\right)g\left({\omega }_k\right)\right|\right)}\right)}^{p}mn\hfill \end{array}$$

which gives

$$\begin{array}{c}\hfill \frac{1}{mn}\le {\left({\displaystyle \underset{g\in {\mathcal{X}}^{*},\parallel g\parallel =1}{sup}\left(\underset{1\le j\le n,1\le k\le m}{max}\left|g\left({\tau }_j\right)g\left({\omega }_k\right)\right|\right)}\right)}^p.\end{array}$$

Since $1={\sum }_{j=1}^n{\left|\frac{f\left({\tau }_j\right)}{\parallel f\parallel }\right|}^p$ for all $f\in {\mathcal{X}}^{*}\setminus \left\{0\right\}$, $1={\sum }_{k=1}^m{\left|\frac{f\left({\omega }_k\right)}{\parallel f\parallel }\right|}^p$ for all $f\in {\mathcal{X}}^{*}\setminus \left\{0\right\}$ and log function is concave, using Jensen’s inequality we get

$$\begin{array}{cc}\hfill S_{\tau }\left(f\right)+S_{\omega }\left(f\right)& =\sum _{j=1}^n{\left|\frac{f\left({\tau }_j\right)}{\parallel f\parallel }\right|}^p\log \left(\frac{1}{{\left|\frac{f\left({\tau }_j\right)}{\parallel f\parallel }\right|}^p}\right)+\sum _{k=1}^m{\left|\frac{f\left({\omega }_k\right)}{\parallel f\parallel }\right|}^p\log \left(\frac{1}{{\left|\frac{f\left({\omega }_k\right)}{\parallel f\parallel }\right|}^p}\right)\hfill \\ \hfill & \le \log \left(\sum _{j=1}^n{\left|\frac{f\left({\tau }_j\right)}{\parallel f\parallel }\right|}^p\frac{1}{{\left|\frac{f\left({\tau }_j\right)}{\parallel f\parallel }\right|}^p}\right)+\log \left(\sum _{k=1}^m{\left|\frac{f\left({\omega }_k\right)}{\parallel f\parallel }\right|}^p\frac{1}{{\left|\frac{f\left({\omega }_k\right)}{\parallel f\parallel }\right|}^p}\right)\hfill \\ \hfill & =\log n+\log m=\log \left(nm\right),\forall f\in {\mathcal{X}}_{\tau }^{*}\cap {\mathcal{X}}_{\omega }^{*}.\hfill \end{array}$$

Let $f\in {\mathcal{X}}_{\tau }^{*}\cap {\mathcal{X}}_{\omega }^{*}$. Then

$$\begin{array}{cc}\hfill S_{\tau }\left(f\right)+S_{\omega }\left(f\right)& =-\sum _{j=1}^n\sum _{k=1}^m{\left|\frac{f\left({\tau }_j\right)}{\parallel f\parallel }\right|}^p{\left|\frac{f\left({\omega }_k\right)}{\parallel f\parallel }\right|}^p\left[\log {\left|\frac{f\left({\tau }_j\right)}{\parallel f\parallel }\right|}^p+\log {\left|\frac{f\left({\omega }_k\right)}{\parallel f\parallel }\right|}^p\right]\hfill \\ \hfill & =-\sum _{j=1}^n\sum _{k=1}^m{\left|\frac{f\left({\tau }_j\right)}{\parallel f\parallel }\right|}^p{\left|\frac{f\left({\omega }_k\right)}{\parallel f\parallel }\right|}^p\log {\left|\frac{f\left({\tau }_j\right)}{\parallel f\parallel }\frac{f\left({\omega }_k\right)}{\parallel f\parallel }\right|}^p\hfill \\ \hfill & =-p\sum _{j=1}^n\sum _{k=1}^m{\left|\frac{f\left({\tau }_j\right)}{\parallel f\parallel }\right|}^p{\left|\frac{f\left({\omega }_k\right)}{\parallel f\parallel }\right|}^p\log \left|\frac{f\left({\tau }_j\right)}{\parallel f\parallel }\frac{f\left({\omega }_k\right)}{\parallel f\parallel }\right|\hfill \\ \hfill & \ge -p\sum _{j=1}^n\sum _{k=1}^m{\left|\frac{f\left({\tau }_j\right)}{\parallel f\parallel }\right|}^p{\left|\frac{f\left({\omega }_k\right)}{\parallel f\parallel }\right|}^p\log \left({\displaystyle \underset{g\in {\mathcal{X}}^{*},\parallel g\parallel =1}{sup}\left(\underset{1\le j\le n,1\le k\le m}{max}\left|g\left({\tau }_j\right)g\left({\omega }_k\right)\right|\right)}\right)\hfill \\ \hfill & =-p\log \left({\displaystyle \underset{g\in {\mathcal{X}}_{\tau }^{*}\cap {\mathcal{X}}_{\omega }^{*},\parallel g\parallel =1}{sup}\left(\underset{1\le j\le n,1\le k\le m}{max}\left|g\left({\tau }_j\right)g\left({\omega }_k\right)\right|\right)}\right)\sum _{j=1}^n\sum _{k=1}^m{\left|\frac{f\left({\tau }_j\right)}{\parallel f\parallel }\right|}^p{\left|\frac{f\left({\omega }_k\right)}{\parallel f\parallel }\right|}^p\hfill \\ \hfill & =-p\log \left({\displaystyle \underset{g\in {\mathcal{X}}_{\tau }^{*}\cap {\mathcal{X}}_{\omega }^{*},\parallel g\parallel =1}{sup}\left(\underset{1\le j\le n,1\le k\le m}{max}\left|g\left({\tau }_j\right)g\left({\omega }_k\right)\right|\right)}\right).\hfill \end{array}$$

□

Theorem 2.6 again gives the following question.

Question 2.7.

Given p, m, n and a Banach space $\mathcal{X}$, for which pairs of Parseval p-frames ${\left\{{\tau }_j\right\}}_{j=1}^n$ and ${\left\{{\omega }_k\right\}}_{k=1}^m$ for ${\mathcal{X}}^{*}$, we have equality in Inequality (8)?

Author is aware of the improvement of Theorem 1.3 by Maassen and Uffink [17] (cf. [18]) (motivated from a conjecture of Kraus [19]) but unable to derive Maassen-Uffink uncertainty principle from Theorem 2.2.