Functional Continuous Uncertainty Principle

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Functional Continuous Uncertainty Principle

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A peer-reviewed article of this preprint also exists.

K. MAHESH KRISHNA^*

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Submitted:

24 July 2023

Posted:

25 July 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 p-Schauder frames for a Banach space $\mathcal{X}$. Then for every $x \in \mathcal{X}\setminus\{0\}$, we show that \begin{align}\label{CUE} \mu(\operatorname{supp}(\theta_f x))^\frac{1}{p} \nu(\operatorname{supp}(\theta_g x))^\frac{1}{q} \geq \frac{1}{\displaystyle\sup_{\alpha \in \Omega, \beta \in \Delta}|f_\alpha(\omega_\beta)|}, \quad \nu(\operatorname{supp}(\theta_g x))^\frac{1}{p} \mu(\operatorname{supp}(\theta_f x))^\frac{1}{q}\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}^p(\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}^p(\Delta, \nu); \quad \theta_gx: \Delta \ni \beta \mapsto (\theta_gx) (\beta):= g_\beta (x) \in \mathbb{K} \end{align*} and $q$ is the conjugate index of $p$. We call Inequality (\ref{CUE}) as \textbf{Functional Continuous Uncertainty Principle}. It improves the Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torr\'{e}sani Uncertainty Principle obtained by M. Krishna in \textit{[arXiv:2304.03324v1, 2023]}. It also answers a question asked by Prof. Philip B. Stark to the author.

Keywords:

Subject: Computer Science and Mathematics - Analysis

MSC: 42C15 (2020)

1. Introduction

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. Great and surprising inequality of Donoho and Stark which improved our life is the following.

Theorem 1.

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

d \in N

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

(2)

In 2002, Elad and Bruckstein extended Inequality (2) to pairs of orthonormal bases [2]. Given a collection

{τ_{j}}_{j = 1}^{n}

in a finite dimensional Hilbert space

H

over

K

(

R

C

), we define

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

Theorem 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

{(\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} .

In 2013, Ricaud and Torrésani showed that orthonormal bases in Theorem 2 can be improved to Parseval frames [3].

Theorem 3.

(Ricaud-Torrésani Uncertainty Principle) [3] Let

{τ_{j}}_{j = 1}^{n}

{ω_{j}}_{j = 1}^{n}

be two Parseval frames for a finite dimensional Hilbert space

H

. Then

{(\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} .

In 2023, author made a major improvement of Theorem 3.

Theorem 4.

(Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle) [4] 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

∥ 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}) |} .

(3)

By seeing paper [4], Prof. Philip B. Stark, Department of Statistics, University of California, Berkeley, asked the following question to the author.

Question 1.

(Philip B. Stark) What is the infinite dimensional version of Theorem [4]?

In this paper, we derive continuous uncertainty principle for Banach spaces which contains Theorem 4 as a particular case and also answers Question Section 1.

2. Functional Continuous Uncertainty Principle

In the paper,

K

denotes

C

R

and

X

denotes a Banach space (need not be finite dimensional) over

K

. Dual of

X

is denoted by

X^{*}

. Whenever

1 < p < \infty

, q denotes conjugate index of p. We recall the notion of weak integral also known as Pettis integrals [5]. Let

(Ω, μ)

be a measure space and

X

be a Banach space. A function

f : Ω \to X

is said to be weak integrable or Pettis integrable if following conditions hold.

(i): For every $ϕ \in X^{*}$ , the map $ϕ f : Ω \to K$ is measurable and $ϕ f \in L^{1} (Ω, μ)$ .
(ii): For every measurable subset $E \in Ω$ , there exists an (unique) element $x_{E} \in X$ such that

$ϕ (x_{E}) = \int_{E} ϕ (f (α)) d μ (α), \forall ϕ \in X^{*} .$

The element $x_{E} \in X$ is denoted by $\int_{E} f (α) d μ (α)$ . With this notion, we have

$ϕ (\int_{E} f (α) d μ (α)) = \int_{E} ϕ (f (α)) d μ (α), \forall ϕ \in X^{*}, \forall E \in Ω .$

We need the following continuous version of p-Schauder frames [4].

Definition 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 in

X^{*}

. The pair

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

is said to be a continuous p-Schauder frame for

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 $x \in X$ ,

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

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

where the integral is weak integral.

We note that condition (i) in Definition 1 says that the map

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

is a linear isometry. Following is the fundamental result of this paper.

Theorem 5.

(Functional Continuous 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 continuous p-Schauder frames for a Banach space

X

. Then for every

x \in X ∖ {0}

, we have

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}) |} .

(4)

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} = \int_{Ω} | f_{α} {(x) |}^{p} d μ (α) = \int_{supp (θ_{f} x)} {| f_{α} (x) |}^{p} d μ (α) \\ = \int_{supp (θ_{f} x)} {|f_{α} (\int_{Δ} g_{β} (x) ω_{β} d ν (β))|}^{p} d μ (α) = \int_{supp (θ_{f} x)} {|\int_{Δ} g_{β} (x) f_{α} (ω_{β}) d ν (β)|}^{p} d μ (α) \\ = \int_{supp (θ_{f} x)} {|\int_{supp (θ_{g} x)} g_{β} (x) f_{α} (ω_{β}) d ν (β)|}^{p} d μ (α) \leq \int_{supp (θ_{f} x)} {(\int_{supp (θ_{g} x)} | g_{β} (x) f_{α} (ω_{β}) | d ν (β))}^{p} d μ (α) \\ \leq {(sup_{α \in Ω, β \in Δ} | f_{α} (ω_{β}) |)}^{p} \int_{supp (θ_{f} x)} {(\int_{supp (θ_{g} x)} | g_{β} (x) | d ν (β))}^{p} d μ (α) \\ = {(sup_{α \in Ω, β \in Δ} | f_{α} (ω_{β}) |)}^{p} μ (supp (θ_{f} x)) {(\int_{supp (θ_{g} x)} | g_{β} (x) | d ν (β))}^{p} \\ \leq {(sup_{α \in Ω, β \in Δ} | f_{α} (ω_{β}) |)}^{p} μ (supp (θ_{f} x)) {(\int_{supp (θ_{g} x)} {| g_{β} (x) |}^{p} d ν (β))}^{\frac{p}{p}} {(\int_{supp (θ_{g} x)} 1^{q} d ν (β))}^{\frac{p}{q}} \\ = {(sup_{α \in Ω, β \in Δ} | f_{α} (ω_{β}) |)}^{p} μ (supp (θ_{f} x)) {∥ θ_{g} x ∥}^{p} ν {(supp (θ_{g} x))}^{\frac{p}{q}} \\ = {(sup_{α \in Ω, β \in Δ} | f_{α} (ω_{β}) |)}^{p} μ (supp (θ_{f} x)) {∥ x ∥}^{p} ν {(supp (θ_{g} x))}^{\frac{p}{q}} . \end{matrix}

Therefore

\frac{1}{sup_{α \in Ω, β \in Δ} | f_{α} (ω_{β}) |} \leq μ {(supp (θ_{f} x))}^{\frac{1}{p}} ν {(supp (θ_{g} x))}^{\frac{1}{q}} .

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} = \int_{Δ} | g_{β} {(x) |}^{p} d ν (β) = \int_{supp (θ_{g} x)} {| g_{β} (x) |}^{p} d ν (β) \\ = \int_{supp (θ_{g} x)} {|g_{β} (\int_{Ω} f_{α} (x) τ_{α} d μ (α))|}^{p} d ν (β) = \int_{supp (θ_{g} x)} {|\int_{Ω} f_{α} (x) g_{β} (τ_{α}) d μ (α)|}^{p} d ν (β) \\ = \int_{supp (θ_{g} x)} {|\int_{supp (θ_{f} x)} f_{α} (x) g_{β} (τ_{α}) d μ (α)|}^{p} d ν (β) \leq \int_{supp (θ_{g} x)} {(\int_{supp (θ_{f} x)} | f_{α} (x) g_{β} (τ_{α}) | d μ (α))}^{p} d ν (β) \\ \leq {(sup_{α \in Ω, β \in Δ} | g_{β} (τ_{α}) |)}^{p} \int_{supp (θ_{g} x)} {(\int_{supp (θ_{f} x)} | f_{α} (x) | d μ (α))}^{p} d ν (β) \\ = {(sup_{α \in Ω, β \in Δ} | g_{β} (τ_{α}) |)}^{p} ν (supp (θ_{g} x)) {(\int_{supp (θ_{f} x)} | f_{α} (x) | d μ (α))}^{p} \\ \leq {(sup_{α \in Ω, β \in Δ} | g_{β} (τ_{α}) |)}^{p} ν (supp (θ_{g} x)) {(\int_{supp (θ_{f} x)} {| f_{α} (x) |}^{p} d μ (α))}^{\frac{p}{p}} {(\int_{supp (θ_{f} x)} 1^{q} d μ (α))}^{\frac{p}{q}} \\ = {(sup_{α \in Ω, β \in Δ} | g_{β} (τ_{α}) |)}^{p} ν (supp (θ_{g} x)) {∥ θ_{f} x ∥}^{p} μ {(supp (θ_{f} x))}^{\frac{p}{q}} \\ = {(sup_{α \in Ω, β \in Δ} | g_{β} (τ_{α}) |)}^{p} ν (supp (θ_{g} x)) {∥ x ∥}^{p} μ {(supp (θ_{f} x))}^{\frac{p}{q}} . \end{matrix}

Therefore

\frac{1}{sup_{α \in Ω, β \in Δ} | g_{β} (τ_{α}) |} \leq ν {(supp (θ_{g} x))}^{\frac{1}{p}} μ {(supp (θ_{f} x))}^{\frac{1}{q}} .

□

Corollary 1.

Let

(Ω, μ)

(Δ, ν)

be measure spaces. Let

{τ_{α}}_{α \in Ω}

and

{ω_{β}}_{β \in Δ}

be Parseval continuous frames for a Hilbert space

H

. Then for every

h \in H ∖ {0}

, we have

μ (supp (θ_{τ} h)) ν (supp (θ_{ω} h)) \geq \frac{1}{sup_{α \in Ω, β \in Δ} {| 〈 ω_{β}, τ_{α} 〉 |}^{2}}, ν (supp (θ_{ω} h)) μ (supp (θ_{τ} h)) \geq \frac{1}{sup_{α \in Ω, β \in Δ} {| 〈 τ_{α}, ω_{β} 〉 |}^{2}},

where

\begin{matrix} θ_{τ} : H ∋ h \mapsto θ_{τ} h \in L^{2} (Ω, μ); θ_{τ} h : Ω ∋ α \mapsto (θ_{τ} h) (α) 〈 h, τ_{α} 〉 \in K, \\ θ_{ω} : H ∋ h \mapsto θ_{ω} h \in L^{2} (Δ, ν); θ_{ω} h : Δ ∋ β \mapsto (θ_{ω} h) (β) 〈 h, ω_{β} 〉 \in K . \end{matrix}

Corollary 2.

Theorem 4 follows from Theorem 5.

Proof.

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

. Define

Ω {1, \dots, n}

and

Δ {1, \dots, m}

. Take

μ

as counting measure on

Ω

and

ν

as counting measure on

Δ

. □

Theorem 5 brings the following question.

Question 2.

(i) Let

(Ω, μ)

(Δ, ν)

be measure spaces,

X

be a Banach space and

p > 1

. For which pairs of continuous p-Schauder frames

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

and

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

for

X

, we have equality in Inequality (4)?

(ii): What is the version of Theorem 5 for $0 < p \leq 1$ and $p = \infty$ ?

If we can derive the uncertainty principles derived in [6] and [7] for p-Schauder frames, then we hope that we can get continuous versions of them.

Acknowledgments

Author thanks Prof. Philip B. Stark, Department of Statistics, University of California, Berkeley for asking Question 1. Author believes that this paper exists because of Question 1.

References

David, L. Donoho and Philip B. Stark. Uncertainty principles and signal recovery. SIAM J. Appl. Math. 1989, 49, 906–931. [Google Scholar]
Michael Elad and Alfred M. Bruckstein. A generalized uncertainty principle and sparse representation in pairs of bases. IEEE Trans. Inform. Theory 2002, 48, 2558–2567. [CrossRef]
Benjamin Ricaud and Bruno Torrésani. Refined support and entropic uncertainty inequalities. IEEE Trans. Inform. Theory 2013, 59, 4272–4279. [CrossRef]
K. Mahesh Krishna. Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani uncertainty principle. arXiv 2023, arXiv:2304.03324v1.
Michel Talagrand. Pettis integral and measure theory. Mem. Amer. Math. Soc. 1984, 51, ix+224.
K. Mahesh Krishna. Functional Donoho-Stark approximate-support uncertainty principle. arXiv 2023, arXiv:2307.01215v1.
K. Mahesh Krishna. Functional Ghobber-Jaming uncertainty principle. arXiv 2023, arXiv:2306.01014v1.

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