Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torresani Uncertainty Principle

Preprint

Article

Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torresani Uncertainty Principle

Altmetrics

Downloads

158

Views

Comments

K. Mahesh Krishna^*

This version is not peer-reviewed

Submitted:

05 April 2023

Posted:

05 April 2023

You are already at the latest version

Alerts

Abstract

Let $(\{f_j\}_{j=1}^n, \{\tau_j\}_{j=1}^n)$ and $(\{g_k\}_{k=1}^m, \{\omega_k\}_{k=1}^m)$ be p-Schauder frames for a finite dimensional Banach space $\mathcal{X}$. Then for every $x \in \mathcal{X}\setminus\{0\}$, we show that \begin{align}\label{UE} \|\theta_f x\|_0^\frac{1}{p}\|\theta_g x\|_0^\frac{1}{q} \geq \frac{1}{\displaystyle\max_{1\leq j\leq n, 1\leq k\leq m}|f_j(\omega_k)|}\quad \text{and} \quad \|\theta_g x\|_0^\frac{1}{p}\|\theta_f x\|_0^\frac{1}{q}\geq \frac{1}{\displaystyle\max_{1\leq j\leq n, 1\leq k\leq m}|g_k(\tau_j)|}. \end{align} where \begin{align*} \theta_f: \mathcal{X} \ni x \mapsto (f_j(x) )_{j=1}^n \in \ell^p([n]); \quad \theta_g: \mathcal{X} \ni x \mapsto (g_k(x) )_{k=1}^m \in \ell^p([m]) \end{align*} and $q$ is the conjugate index of $p$. We call Inequality (\ref{UE}) as \textbf{Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torr\'{e}sani Uncertainty Principle}. Inequality (\ref{UE}) improves Ricaud-Torr\'{e}sani uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2013]}. In particular, it improves Elad-Bruckstein uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2002]} and Donoho-Stark uncertainty principle \textit{[SIAM J. Appl. Math., 1989]}.

Keywords:

Subject: Computer Science and Mathematics - Analysis

1. Introduction

While Heisenberg’s uncertainty principle [3] (English translation of original 1927 paper) is one of the greatest inequalities in the first half of the 20 century, Donoho-stark uncertainty principle [1] is one of the greatest inequalities in the second half of the 20 century. For

h \in C^{d}

, let

{∥ h ∥}_{0}

be the number of nonzero entries in h. Let

\hat{} : C^{d} \to C^{d}

be the Fourier transform.

Theorem 1.1. (Donoho-Stark Uncertainty Principle) [1] For every

d \in N

\begin{matrix} {(\frac{{∥ h ∥}_{0} + {∥ \hat{h} ∥}_{0}}{2})}^{2} \geq {∥ h ∥}_{0} {∥ \hat{h} ∥}_{0} \geq d, \forall h \in C^{d} ∖ {0} . \end{matrix}

(2)

In 2002, Elad and Bruckstein generalized Inequality (1) to pairs of orthonormal bases [2]. To state the result we need some notations. Given a collection

{τ_{j}}_{j = 1}^{n}

in a finite dimensional Hilbert space

H

over

K

(

R

C

), we define

\begin{matrix} θ_{τ} : H ∋ h \mapsto θ_{τ} h {(〈 h, τ_{j} 〉)}_{j = 1}^{n} \in K^{n} . \end{matrix}

Theorem 1.2. (Elad-Bruckstein Uncertainty Principle) [2] Let

{τ_{j}}_{j = 1}^{n}

{ω_{j}}_{j = 1}^{n}

be two orthonormal bases for a finite dimensional Hilbert space

H

. Then

\begin{matrix} {(\frac{∥ θ_{τ} {h ∥}_{0} + {∥ θ_{ω} h ∥}_{0}}{2})}^{2} \geq ∥ θ_{τ} {h ∥}_{0} {∥ θ_{ω} h ∥}_{0} \geq \frac{1}{max_{1 \leq j, k \leq n} {| 〈 τ_{j}, ω_{k} 〉 |}^{2}}, \forall h \in H ∖ {0} . \end{matrix}

Note that Theorem 1.1 follows from Theorem 1.2. In 2013, Ricaud and Torrésani showed that orthonormal bases in Theorem 1.2 can be improved to Parseval frames [6].

Theorem 1.3. (Ricaud-Torrésani Uncertainty Principle) [6] Let

{τ_{j}}_{j = 1}^{n}

{ω_{j}}_{j = 1}^{n}

be two Parseval frames for a finite dimensional Hilbert space

H

. Then

\begin{matrix} {(\frac{∥ θ_{τ} {h ∥}_{0} + {∥ θ_{ω} h ∥}_{0}}{2})}^{2} \geq ∥ θ_{τ} {h ∥}_{0} {∥ θ_{ω} h ∥}_{0} \geq \frac{1}{max_{1 \leq j, k \leq n} {| 〈 τ_{j}, ω_{k} 〉 |}^{2}}, \forall h \in H ∖ {0} . \end{matrix}

In this paper, we derive uncertainty principle for finite dimensional Banach spaces which contains Theorem 1.3 as a particular case.

2. Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle

In the paper,

K

denotes

C

R

and

X

denotes a finite dimensional Banach space over

K

. Identity operator on

X

is denoted by

I_{X}

. Dual of

X

is denoted by

X^{*}

. Whenever

1 < p < \infty

, q denotes conjugate index of p. For

d \in N

, standard finite dimensional Banach space

K^{d}

over

K

equipped with standard

{∥ \cdot ∥}_{p}

norm is denoted by

ℓ^{p} ([d])

. Canonical basis for

K^{d}

is denoted by

{δ_{j}}_{j = 1}^{d}

and

{ζ_{j}}_{j = 1}^{d}

be the coordinate functionals associated with

{δ_{j}}_{j = 1}^{d}

. We need the following variant of p-approximate Schauder frames defined by Krishna and Johnson in [5].

Definition 2.1.

Let

X

be a finite dimensional Banach space over

K

. Let

{τ_{j}}_{j = 1}^{n}

be a collection in

X

and

{f_{j}}_{j = 1}^{n}

be a sequence in

X^{*} .

The pair

({f_{j}}_{j = 1}^{n}, {τ_{j}}_{j = 1}^{n})

is said to be ap-Schauder frame(

1 < p < \infty

) for

X

if the following conditions hold.

(i): For every $x \in X$ ,

$\begin{matrix} {∥ x ∥}^{p} = \sum_{j = 1}^{n} {| f_{j} (x) |}^{p} . \end{matrix}$
(ii): For every $x \in X$ ,

$\begin{matrix} x = \sum_{j = 1}^{n} f_{j} (x) τ_{j} . \end{matrix}$

We easily see that condition (i) in Definition 2.1 says that the map

\begin{matrix} θ_{f} : X ∋ x \mapsto {(f_{j} (x))}_{j = 1}^{n} \in ℓ^{p} ([n]) \end{matrix}

is a linear isometry. Like Holub’s characterization of frames for Hilbert spaces [4], following theorem characterizes p-Schauder frames.

Theorem 2.2.

A pair

({f_{j}}_{j = 1}^{n}, {τ_{j}}_{j = 1}^{n})

is a p-Schauder frame for

X

if and only if

\begin{matrix} f_{j} = ζ_{j} U, τ_{j} = V δ_{j}, \forall j = 1, \dots, n, \end{matrix}

where

U : X \to ℓ^{p} ([n])

V : ℓ^{p} ([n]) \to X

are linear operators such that

V U = I_{X}

and U is an isometry.

Following is the crucial result of this paper.

Theorem 2.3. (Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle) Let

({f_{j}}_{j = 1}^{n}, {τ_{j}}_{j = 1}^{n})

and

({g_{k}}_{k = 1}^{m}, {ω_{k}}_{k = 1}^{m})

be p-Schauder frames for a Banach space

X

. Then for every

x \in X ∖ {0}

, we have

\begin{matrix} ∥ θ_{f} {x ∥}_{0}^{\frac{1}{p}} ∥ θ_{g} {x ∥}_{0}^{\frac{1}{q}} \geq \frac{1}{max_{1 \leq j \leq n, 1 \leq k \leq m} | f_{j} (ω_{k}) |} and ∥ θ_{g} {x ∥}_{0}^{\frac{1}{p}} {∥ θ_{f} x ∥}_{0}^{\frac{1}{q}} \geq \frac{1}{max_{1 \leq j \leq n, 1 \leq k \leq m} | g_{k} (τ_{j}) |} . \end{matrix}

(2)

Proof.

Let

x \in X ∖ {0}

and q be the conjugate index of p. First using

θ_{f}

is an isometry and later using

θ_{g}

is an isometry, we get

\begin{matrix} {∥ x ∥}^{p} & = ∥ θ_{f} {x ∥}^{p} = \sum_{j = 1}^{n} | f_{j} {(x) |}^{p} = \sum_{j \in supp (θ_{f} x)} {| f_{j} (x) |}^{p} \\ = \sum_{j \in supp (θ_{f} x)} {|f_{j} (\sum_{k = 1}^{m} g_{k} (x) ω_{k})|}^{p} = \sum_{j \in supp (θ_{f} x)} {|\sum_{k = 1}^{m} g_{k} (x) f_{j} (ω_{k})|}^{p} \\ = \sum_{j \in supp (θ_{f} x)} {|\sum_{k \in supp (θ_{g} x)} g_{k} (x) f_{j} (ω_{k})|}^{p} \leq \sum_{j \in supp (θ_{f} x)} {(\sum_{k \in supp (θ_{g} x)} | g_{k} (x) f_{j} (ω_{k}) |)}^{p} \\ \leq {(max_{1 \leq j \leq n, 1 \leq k \leq m} | f_{j} (ω_{k}) |)}^{p} \sum_{j \in supp (θ_{f} x)} {(\sum_{k \in supp (θ_{g} x)} | g_{k} (x) |)}^{p} \\ = {(max_{1 \leq j \leq n, 1 \leq k \leq m} | f_{j} (ω_{k}) |)}^{p} {∥ θ_{f} x ∥}_{0} {(\sum_{k \in supp (θ_{g} x)} | g_{k} (x) |)}^{p} \\ \leq {(max_{1 \leq j \leq n, 1 \leq k \leq m} | f_{j} (ω_{k}) |)}^{p} {∥ θ_{f} x ∥}_{0} {(\sum_{k \in supp (θ_{g} x)} {| g_{k} (x) |}^{p})}^{\frac{p}{p}} {(\sum_{k \in supp (θ_{g} x)} 1^{q})}^{\frac{p}{q}} \\ = {(max_{1 \leq j \leq n, 1 \leq k \leq m} | f_{j} (ω_{k}) |)}^{p} ∥ θ_{f} {x ∥}_{0} ∥ θ_{g} {x ∥}^{p} {∥ θ_{g} x ∥}_{0}^{\frac{p}{q}} \\ = {(max_{1 \leq j \leq n, 1 \leq k \leq m} | f_{j} (ω_{k}) |)}^{p} ∥ θ_{f} {x ∥}_{0} {∥ x ∥}^{p} {∥ θ_{g} x ∥}_{0}^{\frac{p}{q}} . \end{matrix}

Therefore

\begin{matrix} \frac{1}{max_{1 \leq j \leq n, 1 \leq k \leq m} | f_{j} (ω_{k}) |} \leq ∥ θ_{f} {x ∥}_{0}^{\frac{1}{p}} {∥ θ_{g} x ∥}_{0}^{\frac{1}{q}} . \end{matrix}

On the other way, first using

θ_{g}

is an isometry and

θ_{f}

is an isometry, we get

\begin{matrix} {∥ x ∥}^{p} & = ∥ θ_{g} {x ∥}^{p} = \sum_{k = 1}^{m} | g_{k} {(x) |}^{p} = \sum_{k \in supp (θ_{g} x)} {| g_{k} (x) |}^{p} \\ = \sum_{k \in supp (θ_{g} x)} {|g_{k} (\sum_{j = 1}^{n} f_{j} (x) τ_{j})|}^{p} = \sum_{k \in supp (θ_{g} x)} {|\sum_{j = 1}^{n} f_{j} (x) g_{k} (τ_{j})|}^{p} \\ = \sum_{k \in supp (θ_{g} x)} {|\sum_{j \in supp (θ_{f} x)} f_{j} (x) g_{k} (τ_{j})|}^{p} \leq \sum_{k \in supp (θ_{g} x)} {(\sum_{j \in supp (θ_{f} x)} | f_{j} (x) g_{k} (τ_{j}) |)}^{p} \\ \leq {(max_{1 \leq j \leq n, 1 \leq k \leq m} | g_{k} (τ_{j}) |)}^{p} \sum_{k \in supp (θ_{g} x)} {(\sum_{j \in supp (θ_{f} x)} | f_{j} (x) |)}^{p} \\ = {(max_{1 \leq j \leq n, 1 \leq k \leq m} | g_{k} (τ_{j}) |)}^{p} {∥ θ_{g} x ∥}_{0} {(\sum_{j \in supp (θ_{f} x)} | f_{j} (x) |)}^{p} \\ \leq {(max_{1 \leq j \leq n, 1 \leq k \leq m} | g_{k} (τ_{j}) |)}^{p} {∥ θ_{g} x ∥}_{0} {(\sum_{j \in supp (θ_{f} x)} {| f_{j} (x) |}^{p})}^{\frac{p}{p}} {(\sum_{j \in supp (θ_{f} x)} 1^{q})}^{\frac{p}{q}} \\ = {(max_{1 \leq j \leq n, 1 \leq k \leq m} | g_{k} (τ_{j}) |)}^{p} ∥ θ_{g} {x ∥}_{0} ∥ θ_{f} {x ∥}^{p} {∥ θ_{f} x ∥}_{0}^{\frac{p}{q}} \\ = {(max_{1 \leq j \leq n, 1 \leq k \leq m} | g_{k} (τ_{j}) |)}^{p} ∥ θ_{g} {x ∥}_{0} {∥ x ∥}^{p} {∥ θ_{f} x ∥}_{0}^{\frac{p}{q}} \end{matrix}

Therefore

\begin{matrix} \frac{1}{max_{1 \leq j \leq n, 1 \leq k \leq m} | g_{k} (τ_{j}) |} \leq ∥ θ_{g} {x ∥}_{0}^{\frac{1}{p}} {∥ θ_{f} x ∥}_{0}^{\frac{1}{q}} . \end{matrix}

□

Corollary 2.4.

Theorem 1.3 follows from Theorem 2.3.

Proof.

Let

{τ_{j}}_{j = 1}^{n}

{ω_{j}}_{j = 1}^{n}

be two Parseval frames for a finite dimensional Hilbert space

H

. Define

\begin{matrix} f_{j} : H ∋ h \mapsto 〈 h, τ_{j} 〉 \in K; g_{j} : H ∋ h \mapsto 〈 h, ω_{j} 〉 \in K, \forall 1 \leq j \leq n . \end{matrix}

Then

p = q = 2

θ_{τ} = θ_{f}

θ_{ω} = θ_{g}

and

\begin{matrix} | f_{j} (ω_{k}) | = | 〈 ω_{k}, τ_{j} 〉 |, \forall 1 \leq j, k \leq n . \end{matrix}

□

Theorem 2.3 brings the following question.

Question 2.5.

Given p, m, n and a Banach space

X

, for which pairs of p-Schauder frames

({f_{j}}_{j = 1}^{n}, {τ_{j}}_{j = 1}^{n})

and

({g_{k}}_{k = 1}^{m}, {ω_{k}}_{k = 1}^{m})

, we have equality in Inequality (2)?

References

David L. Donoho and Philip B. Stark. Uncertainty principles and signal recovery. SIAM J. Appl. Math., 49(3):906–931, 1989. [CrossRef]
Michael Elad and Alfred M. Bruckstein. A generalized uncertainty principle and sparse representation in pairs of bases. IEEE Trans. Inform. Theory, 48(9):2558–2567, 2002. [CrossRef]
W. Heisenberg. The physical content of quantum kinematics and mechanics. In John Archibald Wheeler and Wojciech Hubert Zurek, editors, Quantum Theory and Measurement, Princeton Series in Physics, pages 62–84. Princeton University Press, Princeton, NJ, 1983.
James R. Holub. Pre-frame operators, Besselian frames, and near-Riesz bases in Hilbert spaces. Proc. Amer. Math. Soc., 122(3):779–785, 1994. [CrossRef]
K. Mahesh Krishna and P. Sam Johnson. Towards characterizations of approximate Schauder frame and its duals for Banach spaces. J. Pseudo-Differ. Oper. Appl., 12(1):Paper No. 9, 13, 2021. [CrossRef]
Benjamin Ricaud and Bruno Torrésani. Refined support and entropic uncertainty inequalities. IEEE Trans. Inform. Theory, 59(7):4272–4279, 2013. [CrossRef]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

MDPI Initiatives

Important Links

Choose an area of interest and we will send you notifications of new preprints at your preferred frequency.

Disclaimer

Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torresani Uncertainty Principle

Abstract

1. Introduction

2. Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle

References

MDPI Initiatives

Important Links

Subscribe