An Endpoint Functional Continuous Uncertainty Principle

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An Endpoint Functional Continuous Uncertainty Principle

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K. MAHESH KRISHNA^*

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Submitted:

12 August 2023

Posted:

15 August 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 continuous 1-Schauder frames for a Banach space $\mathcal{X}$. Then for every $x \in \mathcal{X}\setminus\{0\}$, we show that \begin{align*} \mu(\operatorname{supp}(\theta_f x)) \geq \frac{1}{\displaystyle\sup_{\alpha \in \Omega, \beta \in \Delta}|f_\alpha(\omega_\beta)|}, \quad \nu(\operatorname{supp}(\theta_g x))\geq \frac{1}{\displaystyle\sup_{\alpha \in \Omega , \beta \in \Delta}|g_\beta(\tau_\alpha)|}. \end{align*} where \begin{align*} &\theta_f: \mathcal{X} \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{X} \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*} This solves a problem asked by K. M. Krishna in the paper 'Functional Continuous Uncertainty Principle' \textit{[arXiv:2308.00312v1]}.

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

), let

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

Following is the most general form of discrete uncertainty principle for finite dimensional Hilbert spaces.

Theorem 1

(Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle) [2,3,7]). (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 has been greatly improved to continuous families in Banach spaces (even infinite dimensions). To state the result, we need a notion.

Definition 1

([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 a continuous p-Schauder framefor

X

(

1 < 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.

Note that condition (i) in Definition 1 says that the map

\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}

is a linear isometry.

Theorem 2

(Functional Continuous Uncertainty Principle) [4]). (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 for every

x \in X ∖ {0}

, 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}) |}, 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}) |}, \end{matrix}

where q is the conjugate index of p.

Corollary 1

(Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle) [5]). (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 finite dimensional 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}

where q is the conjugate index of p.

In paper [4], it is asked that whether we have version of Theorem 2 for

p = 1

and

p = \infty

. In this paper, we solve the problem for

p = 1

2. Functional Continuous Uncertainty Principle for Continuous 1-Schauder Frames

We clearly have the following definition from Definition 1.

Definition 2.

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 a continuous 1-Schauder framefor

X

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 $x \in X$ ,

$\begin{matrix} ∥ x ∥ = \int_{Ω} | f_{α} (x) | d μ (α) . \end{matrix}$
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.

We note that condition (i) in Definition 2 says that the map

\begin{matrix} θ_{f} : X ∋ 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 a linear isometry.

Theorem 3

(Functional Continuous Uncertainty Principle for Continuous 1-Schauder Frames).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 1-Schauder frames for a Banach space

X

. Then for every

x \in X ∖ {0}

, 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}

In particular,

\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}

and

\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}) |} + \frac{1}{sup_{α \in Ω, β \in Δ} | g_{β} (τ_{α}) |} . \end{matrix}

Proof.

Let

x \in X ∖ {0}

. First using

θ_{f}

is an isometry and later using

θ_{g}

is an isometry, we get

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

Therefore

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

On the other way, first using

θ_{g}

is an isometry and

θ_{f}

is an isometry, we get

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

Therefore

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

□

Corollary 2

(Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle for 1-Schauder Frames). Let

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

and

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

be 1-Schauder frames for a finite dimensional Banach space

X

. Then for every

x \in X ∖ {0}

, we have

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

In particular,

\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}

and

\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}) |} + \frac{1}{max_{1 \leq j \leq n, 1 \leq k \leq m} | g_{k} (τ_{j}) |} . \end{matrix}

Remark 1.

We note that the

L^{1}

-norm uncertainty principles derived in [1,6,8] differ from our result.

References

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K. Mahesh Krishna. Functional continuous uncertainty principle. arXiv:2308.00312v1 [math.FA] 1 August, 2023.
K. Mahesh Krishna. Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani uncertainty principle. arXiv: 2304.03324v1 [math.FA] 5 April, 2023.
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An Endpoint Functional Continuous Uncertainty Principle

Abstract

1. Introduction

2. Functional Continuous Uncertainty Principle for Continuous 1-Schauder Frames

References

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