Unbounded Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle

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Unbounded Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle

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K. MAHESH KRISHNA^*

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18 September 2023

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18 September 2023

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Abstract

Let $(\Omega, \mu)$, $(\Delta, \nu)$ be measure spaces. Let $(\{f_\alpha\}_{\alpha\in \Omega}, \{\tau_\alpha\}_{\alpha\in \Omega})$ and $(\{g_\beta\}_{\beta\in \Delta}, \{\omega_\beta\}_{\beta\in \Delta})$ be unbounded continuous 1-Schauder frames for a Banach space $\mathcal{X}$. Then for every $x \in ( \mathcal{D}(\theta_f) \cap\mathcal{D}(\theta_g))\setminus\{0\}$, we show that \begin{align}\label{UB} \mu(\operatorname{supp}(\theta_f x))\nu(\operatorname{supp}(\theta_g x)) \geq \frac{1}{\left(\displaystyle\sup_{\alpha \in \Omega, \beta \in \Delta}|f_\alpha(\omega_\beta)|\right)\left(\displaystyle\sup_{\alpha \in \Omega , \beta \in \Delta}|g_\beta(\tau_\alpha)|\right)}. \end{align} where \begin{align*} &\theta_f:\mathcal{D}(\theta_f) \ni x \mapsto \theta_fx \in \mathcal{L}^1(\Omega, \mu); \quad \theta_fx: \Omega \ni \alpha \mapsto (\theta_fx) (\alpha):= f_\alpha (x) \in \mathbb{K},\\ &\theta_g: \mathcal{D}(\theta_g) \ni x \mapsto \theta_gx \in \mathcal{L}^1(\Delta, \nu); \quad \theta_gx: \Delta \ni \beta \mapsto (\theta_gx) (\beta):= g_\beta (x) \in \mathbb{K}. \end{align*} We call Inequality (\ref{UB}) as \textbf{Unbounded Donoho-Stark-Elad-Bruckstein-Ricaud-Torr\'{e}sani Uncertainty Principle}. Along with recent \textbf{Functional Continuous Uncertainty Principle} (derived in [arXiv:2308.00312v1 [math.FA], 1 August 2023]), Inequality (\ref{UB}) also 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

MSC: 42C15

1. Introduction

Given a collection

{τ_{j}}_{j = 1}^{n}

in a finite dimensional Hilbert space

H

over

K

(

R

C

), define

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

Most general form of discrete uncertainty principle for finite dimensional Hilbert spaces is the following.

Theorem 1.1. (Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle) [1,2,3] 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}

Recently, Theorem 1.1 has been derived for Banach spaces using the following notion.

Definition 1.2.

[4] Let

(Ω, μ)

be a measure space. Let

{τ_{α}}_{α \in Ω}

be a collection in a Banach space

X

and

{f_{α}}_{α \in Ω}

be a collection in

X^{*}

. The pair

({f_{α}}_{α \in Ω}, {τ_{α}}_{α \in Ω})

is said to be acontinuous p-Schauder framefor

X

(

1 \leq p < \infty

) if the following holds.

(i): For every $x \in X$ , the map $Ω ∋ α \mapsto f_{α} (x) \in K$ is measurable.
(ii): For every $x \in X$ ,

$\begin{matrix} {∥ x ∥}^{p} = \int_{Ω} {| f_{α} (x) |}^{p} d μ (α) . \end{matrix}$
(iii): For every $x \in X$ , the map $Ω ∋ α \mapsto f_{α} (x) τ_{α} \in X$ is weakly measurable.
(iv): For every $x \in X$ ,

$\begin{matrix} x = \int_{Ω} f_{α} (x) τ_{α} d μ (α), \end{matrix}$

where the integral is weak integral.

Given a continuous p-Schauder frame

({f_{α}}_{α \in Ω}, {τ_{α}}_{α \in Ω})

for

X

, define

\begin{matrix} θ_{f} : X ∋ x \mapsto θ_{f} x \in L^{p} (Ω, μ); θ_{f} x : Ω ∋ α \mapsto (θ_{f} x) (α) f_{α} (x) \in K \end{matrix}

Theorem 1.3. (Functional Continuous Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle) [4,5,6] Let

(Ω, μ)

(Δ, ν)

be measure spaces. Let

({f_{α}}_{α \in Ω}

{τ_{α}}_{α \in Ω})

and

({g_{β}}_{β \in Δ}

{ω_{β}}_{β \in Δ})

be continuous p-Schauder frames for a Banach space

X

. Then

(i): for $p > 1$ , we have

$\begin{matrix} μ {(supp (θ_{f} x))}^{\frac{1}{p}} ν {(supp (θ_{g} x))}^{\frac{1}{q}} \geq \frac{1}{sup_{α \in Ω, β \in Δ} | f_{α} (ω_{β}) |}, \forall x \in X ∖ {0}; \\ ν {(supp (θ_{g} x))}^{\frac{1}{p}} μ {(supp (θ_{f} x))}^{\frac{1}{q}} \geq \frac{1}{sup_{α \in Ω, β \in Δ} | g_{β} (τ_{α}) |}, \forall x \in X ∖ {0} . \end{matrix}$

where q is the conjugate index of p.
(ii): for $p = 1$ , we have

$\begin{matrix} μ (supp (θ_{f} x)) \geq \frac{1}{sup_{α \in Ω, β \in Δ} | f_{α} (ω_{β}) |}, ν (supp (θ_{g} x)) \geq \frac{1}{sup_{α \in Ω, β \in Δ} | g_{β} (τ_{α}) |} . \end{matrix}$

(2)

In this paper, we derive an unbounded uncertainty principle which contains Theorem 1.1 as a particular case.

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

We first generalize Definition 1.2 using unbounded linear functionals. Our motivation to do so is the theory of unbounded frames (also known as semi frames or pseudo frames) for Hilbert and Banach spaces, see [7,8,9,10].

Definition 2.1.

Let

(Ω, μ)

be a measure space. Let

{τ_{α}}_{α \in Ω}

be a collection in a Banach space

X

and

{f_{α}}_{α \in Ω}

be a collection of linear functions on

X

(which may not be bounded). The pair

({f_{α}}_{α \in Ω}, {τ_{α}}_{α \in Ω})

is said to be aunbounded continuous p-Schauder frameorcontinuous semi p-Schauder framefor

X

if the following conditions holds.

(i): For every $x \in X$ , the map $Ω ∋ α \mapsto f_{α} (x) \in K$ is measurable.
(ii): The map

$\begin{matrix} θ_{f} : D (θ_{f}) ∋ x \mapsto θ_{f} x \in L^{1} (Ω, μ); θ_{f} x : Ω ∋ α \mapsto (θ_{f} x) (α) f_{α} (x) \in K \end{matrix}$

is well-defined (need not be bounded).
(iii): For every $x \in X$ , the map $Ω ∋ α \mapsto f_{α} (x) τ_{α} \in X$ is weakly measurable.
(iv): For every $x \in D (θ_{f})$ ,

$\begin{matrix} x = \int_{Ω} f_{α} (x) τ_{α} d μ (α), \end{matrix}$

where the integral is weak integral.

Theorem 2.2. (Unbounded Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle) Let

(Ω, μ)

(Δ, ν)

be measure spaces. Let

({f_{α}}_{α \in Ω}, {τ_{α}}_{α \in Ω})

and

({g_{β}}_{β \in Δ}, {ω_{β}}_{β \in Δ})

be unbounded continuous 1-Schauder frames for a Banach space

X

. Then for every

x \in (D (θ_{f}) \cap D (θ_{g})) ∖ {0}

, we have

\begin{matrix} μ (supp (θ_{f} x)) ν (supp (θ_{g} x)) \geq \frac{1}{(sup_{α \in Ω, β \in Δ} | f_{α} (ω_{β}) |) (sup_{α \in Ω, β \in Δ} | g_{β} (τ_{α}) |)} . \end{matrix}

(3)

Proof.

Let

x \in D (θ_{f}) ∖ {0}

. Then

\begin{matrix} ∥ θ_{f} x ∥ & = \int_{Ω} | f_{α} (x) | d μ (α) = \int_{supp (θ_{f} x)} | f_{α} (x) | d μ (α) = \int_{supp (θ_{f} x)} |f_{α} (\int_{Δ} g_{β} (x) ω_{β} d ν (β))| d μ (α) \\ = \int_{supp (θ_{f} x)} |\int_{Δ} g_{β} (x) f_{α} (ω_{β}) d ν (β)| d μ (α) = \int_{supp (θ_{f} x)} |\int_{supp (θ_{g} x)} g_{β} (x) f_{α} (ω_{β}) d ν (β)| d μ (α) \\ \leq \int_{supp (θ_{f} x)} \int_{supp (θ_{g} x)} | g_{β} (x) f_{α} (ω_{β}) | d ν (β) d μ (α) \\ \leq (sup_{α \in Ω, β \in Δ} | f_{α} (ω_{β}) |) \int_{supp (θ_{f} x)} \int_{supp (θ_{g} x)} | g_{β} (x) | d ν (β) d μ (α) \\ = (sup_{α \in Ω, β \in Δ} | f_{α} (ω_{β}) |) μ (supp (θ_{f} x)) \int_{supp (θ_{g} x)} | g_{β} (x) | d ν (β) \\ = (sup_{α \in Ω, β \in Δ} | f_{α} (ω_{β}) |) μ (supp (θ_{f} x)) ∥ θ_{g} x ∥ . \end{matrix}

Therefore

\begin{matrix} \frac{1}{sup_{α \in Ω, β \in Δ} | f_{α} (ω_{β}) |} ∥ θ_{f} x ∥ \leq μ (supp (θ_{f} x)) ∥ θ_{g} x ∥ . \end{matrix}

(4)

On the other way, let

x \in D (θ_{g}) ∖ {0}

. Then

\begin{matrix} ∥ θ_{g} x ∥ & = \int_{Δ} | g_{β} (x) | d ν (β) = \int_{supp (θ_{g} x)} | g_{β} (x) | d ν (β) = \int_{supp (θ_{g} x)} |g_{β} (\int_{Ω} f_{α} (x) τ_{α} d μ (α))| d ν (β) \\ = \int_{supp (θ_{g} x)} |\int_{Ω} f_{α} (x) g_{β} (τ_{α}) d μ (α)| d ν (β) = \int_{supp (θ_{g} x)} |\int_{supp (θ_{f} x)} f_{α} (x) g_{β} (τ_{α}) d μ (α)| d ν (β) \\ \leq \int_{supp (θ_{g} x)} \int_{supp (θ_{f} x)} | f_{α} (x) g_{β} (τ_{α}) | d μ (α) d ν (β) \\ \leq (sup_{α \in Ω, β \in Δ} | g_{β} (τ_{α}) |) \int_{supp (θ_{g} x)} \int_{supp (θ_{f} x)} | f_{α} (x) | d μ (α) d ν (β) \\ = (sup_{α \in Ω, β \in Δ} | g_{β} (τ_{α}) |) ν (supp (θ_{g} x)) \int_{supp (θ_{f} x)} | f_{α} (x) | d μ (α) \\ = (sup_{α \in Ω, β \in Δ} | g_{β} (τ_{α}) |) ν (supp (θ_{g} x)) ∥ θ_{f} x ∥ . \end{matrix}

Therefore

\begin{matrix} \frac{1}{sup_{α \in Ω, β \in Δ} | g_{β} (τ_{α}) |} ∥ θ_{g} x ∥ \leq ν (supp (θ_{g} x)) ∥ θ_{f} x ∥ . \end{matrix}

(5)

Multiplying Inequalities (3) and (4) we get

\begin{matrix} \frac{1}{(sup_{α \in Ω, β \in Δ} | f_{α} (ω_{β}) |) (sup_{α \in Ω, β \in Δ} | g_{β} (τ_{α}) |)} ∥ θ_{f} x ∥ ∥ θ_{g} x ∥ & \leq μ (supp (θ_{f} x)) ν (supp (θ_{g} x)) ∥ θ_{g} x ∥ ∥ θ_{f} x ∥, \\ \forall x \in (D (θ_{f}) \cap D (θ_{g})) ∖ {0} . \end{matrix}

A cancellation of

∥ θ_{f} x ∥ ∥ θ_{g} x ∥

gives the required inequality. □

Corollary . 2.3.

Let

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

and

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

be collections in a Banach space

X

such that

\begin{matrix} x = \sum_{j = 1}^{n} f_{j} (x) τ_{j} = \sum_{k = 1}^{m} g_{k} (x) ω_{k}, \forall x \in X . \end{matrix}

Then for every

x \in X ∖ {0}

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

where

\begin{matrix} θ_{f} : X ∋ x \mapsto {(f_{j} (x))}_{j = 1}^{n} \in ℓ^{1} ([n]); θ_{g} : X ∋ x \mapsto {(g_{k} (x))}_{k = 1}^{m} \in ℓ^{1} ([m]) . \end{matrix}

Even though by multiplying two inequalities in (2) we get Inequality (3) for continuous 1-Schauder frames, observe that the conclusion in (2) is stronger (with stronger assumption) than that of Theorem 2.2.

Note that proof of Theorem 2.2 does not work for unbounded continuous p-Schauder frames for

p > 1

(even by using Holder’s inequality). We are therefore left over with following problem.

Problem . 2.4.

What is the unbounded version of Theorem 2.2 for

p > 1

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Unbounded Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle

Abstract

1. Introduction

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

References

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