Functional Donoho-Stark Approximate Support Uncertainty Principle

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Functional Donoho-Stark Approximate Support Uncertainty Principle

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K. MAHESH KRISHNA^*

This version is not peer-reviewed

Submitted:

01 July 2023

Posted:

04 July 2023

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Abstract

Let $(\{f_j\}_{j=1}^n, \{\tau_j\}_{j=1}^n)$ and $(\{g_k\}_{k=1}^n, \{\omega_k\}_{k=1}^n)$ be two p-orthonormal bases for a finite dimensional Banach space $\mathcal{X}$. If $ x \in \mathcal{X}\setminus\{0\}$ is such that $\theta_fx$ is $\varepsilon$-supported on $M\subseteq \{1,\dots, n\}$ w.r.t. p-norm and $\theta_gx$ is $\delta$-supported on $N\subseteq \{1,\dots, n\}$ w.r.t. p-norm, then we show that \begin{align}\label{ME} &o(M)^\frac{1}{p}o(N)^\frac{1}{q}\geq \frac{1}{\displaystyle \max_{1\leq j,k\leq n}|f_j(\omega_k) |}\max \{1-\varepsilon-\delta, 0\},\\ &o(M)^\frac{1}{q}o(N)^\frac{1}{p}\geq \frac{1}{\displaystyle \max_{1\leq j,k\leq n}|g_k(\tau_j) |}\max \{1-\varepsilon-\delta, 0\},\label{ME2} \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}^n \in \ell^p([n]) \end{align*} and $q$ is the conjugate index of $p$. We call Inequalities (\ref{ME}) and (\ref{ME2}) as \textbf{Functional Donoho-Stark Approximate Support Uncertainty Principle}. Inequalities (\ref{ME}) and (\ref{ME2}) improve the finite approximate support uncertainty principle obtained by Donoho and Stark \textit{[SIAM J. Appl. Math., 1989]}.

Keywords:

Subject: Computer Science and Mathematics - Analysis

1. Introduction

Let

0 \leq ε < 1

. Recall that a function

f \in L^{2} (R^{d})

is said to be $ε$ -supported on a measurable subset $E \subseteq R^{d}$ (also known as $ε$ -approximately supported as well as $ε$ -essentially supported) [1,9] if

\begin{matrix} {(\int_{E^{c}} {| f (x) |}^{2} d x)}^{\frac{1}{2}} \leq ε {(\int_{R^{d}} {| f (x) |}^{2} d x)}^{\frac{1}{2}} . \end{matrix}

Let

d \in N

and

\hat{} : L^{2} (R^{d}) \to L^{2} (R^{d})

be the unitary Fourier transform obtained by extending uniquely the bounded linear operator

\begin{matrix} \hat{} : L^{1} (R^{d}) \cap L^{2} (R^{d}) ∋ f \mapsto \hat{f} \in C_{0} (R^{d}); \hat{f} : R^{d} ∋ ξ \mapsto \hat{f} (ξ) \int_{R^{d}} f (x) e^{- 2 π i 〈 x, ξ 〉} d x \in C . \end{matrix}

In 1989, Donoho and Stark derived the following uncertainty principle on approximate supports of function and its Fourier transform [1].

Theorem 1.1.

[1] (Donoho-Stark Approximate Support Uncertainty Principle) If

f \in L^{2} (R^{d}) ∖ {0}

is ε-supported on a measurable subset

E \subseteq R^{d}

and

\hat{f}

is δ-supported on a measurable subset

F \subseteq R^{d}

, then

\begin{matrix} m (E) m (F) \geq {(1 - ε - δ)}^{2} . \end{matrix}

Ultimate result in [1] is the finite dimensional Heisenberg uncertainty principle known today as Donoho-Stark uncertainty principle. It is then natural to seek a finite dimensional version of Theorem 1.1. For this, first one needs the notion of approximate support in finite dimensions. Donoho and Stark defined this notion as follows. 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. Given a subset

M \subseteq {1, \dots, n}

, the number of elements in M is denoted by

o (M)

Definition 1.2.

[1] Let

0 \leq ε < 1

. A vector

{(a_{j})}_{j = 1}^{d} \in C^{d}

is said to be

ε

-supported on a subset

M \subseteq {1, \dots, d}

\begin{matrix} {(\sum_{j \in M^{c}} {| a_{j} |}^{2})}^{\frac{1}{2}} \leq ε {(\sum_{j = 1}^{d} {| a_{j} |}^{2})}^{\frac{1}{2}} . \end{matrix}

Finite dimensional version of Theorem 1.1 then reads as follows.

Theorem 1.3.

[1] (Finite Donoho-Stark Approximate Support Uncertainty Principle) If

h \in C^{d} ∖ {0}

is ε-supported on

M \subseteq {1, \dots, d}

and

\hat{h}

is δ-supported on

N \subseteq {1, \dots, d}

, then

\begin{matrix} o (M) o (N) \geq d {(1 - ε - δ)}^{2} . \end{matrix}

In 1990, Smith [8] generalized Theorem 1.3 to Fourier transforms defined on locally compact abelian groups. Recently, Banach space version of finite Donoho-Stark uncertainty principle has been derived in [2]. Therefore we seek a Banach space version of Theorem 1.3. This we obtain in this paper.

2. Functional Donoho-Stark Approximate Support 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 the conjugate index of p. For

d \in N

, the 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

{e_{j}}_{j = 1}^{d}

and

{ζ_{j}}_{j = 1}^{d}

be the coordinate functionals associated with

{e_{j}}_{j = 1}^{d}

Definition 2.1.

[3] Let

X

be a finite dimensional Banach space over

K

. Let

{τ_{j}}_{j = 1}^{n}

be a basis for

X

and let

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

be the coordinate functionals associated with

{τ_{j}}_{j = 1}^{n}

. The pair

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

is said to be a p-orthonormal basis (

1 < p < \infty

) for

X

if the following conditions hold.

(i): $∥ f_{j} ∥ = ∥ τ_{j} ∥ = 1$ for all $1 \leq j \leq n$ .
(ii): For every ${(a_{j})}_{j = 1}^{n} \in K^{n}$ ,

$\begin{matrix} ∥\sum_{j = 1}^{n} a_{j} τ_{j}∥ = {(\sum_{j = 1}^{n} {| a_{j} |}^{p})}^{\frac{1}{p}} . \end{matrix}$

Given a p-orthonormal basis

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

for

X

, we get the following two invertible isometries:

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

Then we have the following proposition.

Proposition 2.2.

Let

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

be a p-orthonormal basis for

X

. Then

(i): $θ_{f}$ is an invertible isometry.
(ii): $θ_{τ}$ is an invertible isometry.
(iii): $θ_{τ} θ_{f} = I_{X}$ .

It is natural to guess the following version of Definition 1.2 for

ℓ^{p} ([n])

Definition 2.3.

Let

0 \leq ε < 1

. A vector

{(a_{j})}_{j = 1}^{n} \in ℓ^{p} ([n])

is said to be ε-supported on a subset

M \subseteq {1 . \dots, n}

w.r.t. p-norm if

\begin{matrix} {(\sum_{j \in M^{c}} {| a_{j} |}^{p})}^{\frac{1}{p}} \leq ε {(\sum_{j = 1}^{n} {| a_{j} |}^{p})}^{\frac{1}{p}} . \end{matrix}

With the above definition we have following theorem.

Theorem 2.4.

(Functional Donoho-Stark Approximate Support Uncertainty Principle) Let

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

and

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

be two p-orthonormal bases for a finite dimensional Banach space

X

. If

x \in X ∖ {0}

is such that

θ_{f} x

is ε-supported on

M \subseteq {1, \dots, n}

w.r.t. p-norm and

θ_{g} x

is δ-supported on

N \subseteq {1, \dots, n}

w.r.t. p-norm, then

\begin{matrix} o {(M)}^{\frac{1}{p}} o {(N)}^{\frac{1}{q}} \geq \frac{1}{max_{1 \leq j, k \leq n} | f_{j} (ω_{k}) |} max {1 - ε - δ, 0}, \end{matrix}

(4)

\begin{matrix} o {(M)}^{\frac{1}{q}} o {(N)}^{\frac{1}{p}} \geq \frac{1}{max_{1 \leq j, k \leq n} | g_{k} (τ_{j}) |} max {1 - ε - δ, 0} . \end{matrix}

(4)

Proof.

For

S \subseteq {1, \dots, n}

, define

P_{S} : ℓ^{p} ([n]) ∋ {(a_{j})}_{j = 1}^{n} \mapsto \sum_{j \in S} a_{j} e_{j} \in ℓ^{p} ([n])

be the canonical projection onto the coordinates indexed by S. Now define

V : = P_{M} θ_{f} θ_{ω} P_{N} : ℓ^{p} ([n]) \to ℓ^{p} ([n])

. Then for

z \in ℓ^{p} ([n])

\begin{matrix} {∥ V z ∥}^{p} = {∥ P_{M} θ_{f} θ_{ω} P_{N} z ∥}^{p} = {∥P_{M} θ_{f} θ_{ω} P_{N} (\sum_{k = 1}^{n} ζ_{k} (z) e_{k})∥}^{p} = {∥P_{M} θ_{f} θ_{ω} (\sum_{k = 1}^{n} ζ_{k} (z) P_{N} e_{k})∥}^{p} \\ = {∥P_{M} θ_{f} θ_{ω} (\sum_{k \in N} ζ_{k} (z) e_{k})∥}^{p} = {∥P_{M} θ_{f} (\sum_{k \in N} ζ_{k} (z) θ_{ω} e_{k})∥}^{p} = {∥P_{M} θ_{f} (\sum_{k \in N} ζ_{k} (z) ω_{k})∥}^{p} \\ = {∥\sum_{k \in N} ζ_{k} (z) P_{M} θ_{f} ω_{k}∥}^{p} = {∥\sum_{k \in N} ζ_{k} (z) P_{M} (\sum_{j = 1}^{n} f_{j} (ω_{k}) e_{j})∥}^{p} = {∥\sum_{k \in N} ζ_{k} (z) \sum_{j = 1}^{n} f_{j} (ω_{k}) P_{M} e_{j}∥}^{p} \\ = {∥\sum_{k \in N} ζ_{k} (z) \sum_{j \in M} f_{j} (ω_{k}) e_{j}∥}^{p} = {∥\sum_{j \in M} (\sum_{k \in N} ζ_{k} (z) f_{j} (ω_{k})) e_{j}∥}^{p} = \sum_{j \in M} {|\sum_{k \in N} ζ_{k} (z) f_{j} (ω_{k})|}^{p} \\ \leq \sum_{j \in M} {(\sum_{k \in N} | ζ_{k} (z) f_{j} (ω_{k}) |)}^{p} \leq {(max_{1 \leq j, k \leq n} | f_{j} (ω_{k}) |)}^{p} \sum_{j \in M} {(\sum_{k \in N} | ζ_{k} (z) |)}^{p} \\ = {(max_{1 \leq j, k \leq n} | f_{j} (ω_{k}) |)}^{p} o (M) {(\sum_{k \in N} | ζ_{k} (z) |)}^{p} \leq {(max_{1 \leq j, k \leq n} | f_{j} (ω_{k}) |)}^{p} o (M) {(\sum_{k \in N} {| ζ_{k} (z) |}^{p})}^{\frac{p}{p}} {(\sum_{k \in N} 1^{q})}^{\frac{p}{q}} \\ = {(max_{1 \leq j, k \leq n} | f_{j} (ω_{k}) |)}^{p} o (M) {(\sum_{k \in N} {| ζ_{k} (z) |}^{p})}^{\frac{p}{p}} o {(N)}^{\frac{p}{q}} \leq {(max_{1 \leq j, k \leq n} | f_{j} (ω_{k}) |)}^{p} o (M) {(\sum_{k = 1}^{n} {| ζ_{k} (z) |}^{p})}^{\frac{p}{p}} o {(N)}^{\frac{p}{q}} \\ = {(max_{1 \leq j, k \leq n} | f_{j} (ω_{k}) |)}^{p} o (M) {∥ z ∥}^{p} o {(N)}^{\frac{p}{q}} . \end{matrix}

Therefore

\begin{matrix} ∥ V ∥ \leq (max_{1 \leq j, k \leq n} | f_{j} (ω_{k}) |) o {(M)}^{\frac{1}{p}} o {(N)}^{\frac{1}{q}} . \end{matrix}

(5)

We now wish to find a lower bound on the operator norm of V. For

x \in X

, we find

\begin{matrix} ∥ θ_{f} x - V θ_{g} x ∥ & \leq ∥ θ_{f} x - P_{M} θ_{f} x ∥ + ∥ P_{M} θ_{f} x - V θ_{g} x ∥ \leq ε ∥ θ_{f} x ∥ + ∥ P_{M} θ_{f} x - V θ_{g} x ∥ \\ = ε ∥ θ_{f} x ∥ + ∥ P_{M} θ_{f} x - P_{M} θ_{f} θ_{ω} P_{N} θ_{g} x ∥ = ε ∥ θ_{f} x ∥ + ∥ P_{M} θ_{f} (x - θ_{ω} P_{N} θ_{g} x) ∥ \\ \leq ε ∥ θ_{f} x ∥ + ∥ x - θ_{ω} P_{N} θ_{g} x ∥ = ε ∥ θ_{f} x ∥ + ∥ θ_{ω} θ_{g} x - θ_{ω} P_{N} θ_{g} x ∥ \\ = ε ∥ θ_{f} x ∥ + ∥ θ_{ω} (θ_{g} x - P_{N} θ_{g} x) ∥ = ε ∥ θ_{f} x ∥ + ∥ θ_{g} x - P_{N} θ_{g} x ∥ \\ \leq ε ∥ θ_{f} x ∥ + δ ∥ θ_{g} x ∥ = ε ∥ x ∥ + δ ∥ x ∥ = (ε + δ) ∥ x ∥ . \end{matrix}

Using triangle inequality, we then get

\begin{matrix} ∥ x ∥ - ∥ V θ_{g} x ∥ = ∥ θ_{f} x ∥ - ∥ V θ_{g} x ∥ \leq ∥ θ_{f} x - V θ_{g} x ∥ \leq (ε + δ) ∥ x ∥, \forall x \in X . \end{matrix}

Since

θ_{g}

is an invertible isometry,

\begin{matrix} (1 - ε - δ) ∥ x ∥ \leq ∥ V θ_{g} x ∥, \forall x \in X \Rightarrow (1 - ε - δ) ∥ y ∥ = (1 - ε - δ) ∥ θ_{g}^{- 1} y ∥ \leq ∥ V y ∥, \forall y \in ℓ^{p} ([n]), \end{matrix}

i.e.,

\begin{matrix} max {1 - ε - δ, 0} \leq ∥ V ∥ . \end{matrix}

(6)

Using Inequalities (5) and (6) we get

\begin{matrix} max {1 - ε - δ, 0} \leq (max_{1 \leq j, k \leq n} | f_{j} (ω_{k}) |) o {(M)}^{\frac{1}{p}} o {(N)}^{\frac{1}{q}} . \end{matrix}

To prove second inequality, define

W : = P_{N} θ_{g} θ_{τ} P_{M} : ℓ^{p} ([n]) \to ℓ^{p} ([n])

. Then for

z \in ℓ^{p} ([n])

\begin{matrix} {∥ W z ∥}^{p} = {∥ P_{N} θ_{g} θ_{τ} P_{M} z ∥}^{p} = {∥P_{N} θ_{g} θ_{τ} P_{M} (\sum_{j = 1}^{n} ζ_{j} (z) e_{j})∥}^{p} = {∥P_{N} θ_{g} θ_{τ} (\sum_{j = 1}^{n} ζ_{j} (z) P_{M} e_{j})∥}^{p} \\ = {∥P_{N} θ_{g} θ_{τ} (\sum_{j \in M} ζ_{j} (z) e_{j})∥}^{p} = {∥P_{N} θ_{g} (\sum_{j \in M} ζ_{j} (z) θ_{τ} e_{j})∥}^{p} = {∥P_{N} θ_{g} (\sum_{j \in M} ζ_{j} (z) τ_{j})∥}^{p} \\ = {∥\sum_{j \in M} ζ_{j} (z) P_{N} θ_{g} τ_{j}∥}^{p} = {∥\sum_{j \in M} ζ_{j} (z) P_{N} (\sum_{k = 1}^{n} g_{k} (τ_{j}) e_{k})∥}^{p} = {∥\sum_{j \in M} ζ_{j} (z) \sum_{k = 1}^{n} g_{k} (τ_{j}) P_{N} e_{k}∥}^{p} \\ = {∥\sum_{j \in M} ζ_{j} (z) \sum_{k \in N} g_{k} (τ_{j}) e_{k}∥}^{p} = {∥\sum_{k \in N} (\sum_{j \in M} ζ_{j} (z) g_{k} (τ_{j})) e_{k}∥}^{p} = \sum_{k \in N} {|\sum_{j \in M} ζ_{j} (z) g_{k} (τ_{j})|}^{p} \\ \leq \sum_{k \in N} {(\sum_{j \in M} | ζ_{j} (z) g_{k} (τ_{j}) |)}^{p} \leq {(max_{1 \leq j, k \leq n} | g_{k} (τ_{j}) |)}^{p} \sum_{k \in N} {(\sum_{j \in M} | ζ_{j} (z) |)}^{p} \\ = {(max_{1 \leq j, k \leq n} | g_{k} (τ_{j}) |)}^{p} o (N) {(\sum_{j \in M} | ζ_{j} (z) |)}^{p} \leq {(max_{1 \leq j, k \leq n} | g_{k} (τ_{j}) |)}^{p} o (N) {(\sum_{j \in M} {| ζ_{j} (z) |}^{p})}^{\frac{p}{p}} {(\sum_{j \in M} 1^{q})}^{\frac{p}{q}} \\ = {(max_{1 \leq j, k \leq n} | g_{k} (τ_{j}) |)}^{p} o (N) {(\sum_{j \in M} {| ζ_{j} (z) |}^{p})}^{\frac{p}{p}} o {(M)}^{\frac{p}{q}} \leq {(max_{1 \leq j, k \leq n} | g_{k} (τ_{j}) |)}^{p} o (N) {(\sum_{j = 1}^{n} {| ζ_{j} (z) |}^{p})}^{\frac{p}{p}} o {(M)}^{\frac{p}{q}} \\ = {(max_{1 \leq j, k \leq n} | g_{k} (τ_{j}) |)}^{p} o (N) {∥ z ∥}^{p} o {(M)}^{\frac{p}{q}} . \end{matrix}

Therefore

\begin{matrix} ∥ W ∥ \leq (max_{1 \leq j, k \leq n} | g_{k} (τ_{j}) |) o {(M)}^{\frac{1}{q}} o {(N)}^{\frac{1}{p}} . \end{matrix}

(7)

Now for

x \in X

\begin{matrix} ∥ θ_{g} x - W θ_{f} x ∥ & \leq ∥ θ_{g} x - P_{N} θ_{g} x ∥ + ∥ P_{N} θ_{g} x - W θ_{f} x ∥ \leq δ ∥ θ_{g} x ∥ + ∥ P_{N} θ_{g} x - W θ_{f} x ∥ \\ = δ ∥ θ_{g} x ∥ + ∥ P_{N} θ_{g} x - P_{N} θ_{g} θ_{τ} P_{M} θ_{f} x ∥ = δ ∥ θ_{g} x ∥ + ∥ P_{N} θ_{g} (x - θ_{τ} P_{M} θ_{f} x) ∥ \\ \leq δ ∥ θ_{g} x ∥ + ∥ x - θ_{τ} P_{M} θ_{f} x ∥ = δ ∥ θ_{g} x ∥ + ∥ θ_{τ} θ_{f} x - θ_{τ} P_{M} θ_{f} x ∥ \\ = δ ∥ θ_{g} x ∥ + ∥ θ_{τ} (θ_{f} x - P_{M} θ_{f} x) ∥ = δ ∥ θ_{g} x ∥ + ∥ θ_{f} x - P_{M} θ_{f} x ∥ \\ \leq δ ∥ θ_{g} x ∥ + ε ∥ θ_{f} x ∥ = δ ∥ x ∥ + ε ∥ x ∥ = (δ + ε) ∥ x ∥ . \end{matrix}

Using triangle inequality and the fact that

θ_{f}

is an invertible isometry we then get

\begin{matrix} max {1 - ε - δ, 0} \leq ∥ W ∥ . \end{matrix}

(8)

Using Inequalities (7) and (8) we get

\begin{matrix} max {1 - ε - δ, 0} \leq (max_{1 \leq j, k \leq n} | g_{k} (τ_{j}) |) o {(M)}^{\frac{1}{q}} o {(N)}^{\frac{1}{p}} . \end{matrix}

□

Corollary 2.5.

Let

{τ_{j}}_{j = 1}^{n}

and

{ω_{j}}_{j = 1}^{n}

be two orthonormal bases for a finite dimensional Hilbert space

H

. Set

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

h \in H ∖ {0}

is such that

θ_{τ} h

is ε-supported on

M \subseteq {1, \dots, n}

and

θ_{ω} h

is δ-supported on

N \subseteq {1, \dots, n}

, then

\begin{matrix} o (M) o (N) \geq \frac{1}{max_{1 \leq j, k \leq n} {| 〈 τ_{j}, ω_{k} 〉 |}^{2}} {(1 - ε - δ)}^{2} . \end{matrix}

In particular, Theorem 1.3 follows from Theorem 2.4.

Proof.

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

and

| f_{j} (ω_{k}) | = | 〈 ω_{k}, τ_{j} 〉 |

for all

1 \leq j, k \leq n .

Theorem 1.3 follows by taking

{τ_{j}}_{j = 1}^{n}

as the standard basis and

{ω_{j}}_{j = 1}^{n}

as the Fourier basis for

C^{n}

. □

Corollary 2.6.

Let

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

and

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

be two p-orthonormal bases for a finite dimensional Banach space

X

. Let

x \in X ∖ {0}

is such that

θ_{f} x

is ε-supported on

M \subseteq {1, \dots, n}

w.r.t. p-norm and

θ_{g} x

is δ-supported on

N \subseteq {1, \dots, n}

w.r.t. p-norm. If

ε + δ \leq 1

, then

\begin{matrix} o {(M)}^{\frac{1}{p}} o {(N)}^{\frac{1}{q}} \geq \frac{1}{max_{1 \leq j, k \leq n} | f_{j} (ω_{k}) |} (1 - ε - δ), \\ o {(M)}^{\frac{1}{q}} o {(N)}^{\frac{1}{p}} \geq \frac{1}{max_{1 \leq j, k \leq n} | g_{k} (τ_{j}) |} (1 - ε - δ) . \end{matrix}

Corollary 2.7.

Let

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

and

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

be two p-orthonormal bases for a finite dimensional Banach space

X

. If

x \in X ∖ {0}

is such that

θ_{f} x

is 0-supported on

M \subseteq {1, \dots, n}

w.r.t. p-norm and

θ_{g} x

is 0-supported on

N \subseteq {1, \dots, n}

w.r.t. p-norm (saying differently,

θ_{f} x

is supported on M and

θ_{g} x

is supported on N), then

\begin{matrix} o {(M)}^{\frac{1}{p}} o {(N)}^{\frac{1}{q}} \geq \frac{1}{max_{1 \leq j, k \leq n} | f_{j} (ω_{k}) |}, o {(M)}^{\frac{1}{q}} o {(N)}^{\frac{1}{p}} \geq \frac{1}{max_{1 \leq j, k \leq n} | g_{k} (τ_{j}) |} . \end{matrix}

Corollary 2.7 is not the Theorem 2.3 in [2] (it is a particular case) because Theorem 2.3 in [2] is derived for p-Schauder frames which is general than p-orthonormal bases. Theorem 2.4 promotes the following question.

Question 2.8.

Given p and a Banach space

X

of dimension n, for which pairs of p-orthonormal bases

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

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

for

X

, subsets

M, N

and

ε, δ

, we have equality in Inequalities (3) and (4)?

Observe that we used

1 < p < \infty

in the proof of Theorem 2.4. Therefore we have the following problem.

Question 2.9.

Whether there are Functional Donoho-Stark Approximate Support Uncertainty Principle (versions of Theorem 2.4) for 1-orthonormal bases and ∞-orthonormal bases?

Keeping

ℓ^{p}

-spaces for

0 < p < 1

as a model space equipped with

\begin{matrix} ∥ {(a_{j})}_{j = 1}^{n} ∥_{p} : = \sum_{j = 1}^{n} {| a_{j} |}^{p}, \forall {(a_{j})}_{j = 1}^{n} \in K^{n}, \end{matrix}

we set following definitions.

Definition 2.10.

Let

X

be a vector space over

K

. We say that

X

is a disc-Banach space if there exists a map called as disc-norm

∥ \cdot ∥ : X \to [0, \infty)

satisfying the following conditions.

(i): If $x \in X$ is such that $∥ x ∥ = 0$ , then $x = 0$ .
(ii): $∥ x + y ∥ \leq ∥ x ∥ + ∥ y ∥$ for all $x, y \in X$ .
(iii): $∥ λ x ∥ \leq | λ | ∥ x ∥$ for all $x \in X$ and for all $λ \in K$ with $| λ | \geq 1$ .
(iv): $∥ λ x ∥ \geq | λ | ∥ x ∥$ for all $x \in X$ and for all $λ \in K$ with $| λ | \leq 1$ .
(v): $X$ is complete w.r.t. the metric $d (x, y) : = ∥ x - y ∥$ for all $x, y \in X$ .

Definition 2.11.

Let

X

be a finite dimensional disc-Banach space over

K

. Let

{τ_{j}}_{j = 1}^{n}

be a basis for

X

and let

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

be the coordinate functionals associated with

{τ_{j}}_{j = 1}^{n}

. The pair

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

is said to be a p-orthonormal basis (

1 < p < \infty

) for

X

if the following conditions hold.

(i): $∥ f_{j} ∥ = ∥ τ_{j} ∥ = 1$ for all $1 \leq j \leq n$ .
(ii): For every ${(a_{j})}_{j = 1}^{n} \in K^{n}$ ,

$\begin{matrix} ∥\sum_{j = 1}^{n} a_{j} τ_{j}∥ = \sum_{j = 1}^{n} {| a_{j} |}^{p} . \end{matrix}$

Then we also have the following question.

Question 2.12.

Whether there are versions of Theorem 2.4 for p-orthonormal bases

0 < p < 1

We wish to mention that in [2] the functional uncertainty principle was derived for p-Schauder frames which is general than p-orthonormal bases. Thus it is desirable to derive Theorem 2.4 or a variation of it for p-Schauder frames, which we can’t.

We end by asking the following curious question whose motivation is the recently proved Balian-Low theorem (which is also an uncertainty principle) for Gabor systems in finite dimensional Hilbert spaces [5,6,7].

Question 2.13.

Whether there is a Functional Balian-Low Theorem (which we like to call Functional Balian-Low-Lammers-Stampe-Nitzan-Olsen Theorem) for Gabor-Schauder systems in finite dimensional Banach spaces (Gabor-Schauder system is as defined in [4])?

References

David L. Donoho and Philip B. Stark. Uncertainty principles and signal recovery. SIAM J. Appl. Math., 49(3):906–931, 1989. [CrossRef]
K. Mahesh Krishna. Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani uncertainty principle. arXiv: 2304.03324v1 [math.FA] 5 April, 2023. [CrossRef]
K. Mahesh Krishna. Functional Ghobber-Jaming uncertainty principle. arXiv: 2306.01014v1 [math.FA] 1 June, 2023. [CrossRef]
K. Mahesh Krishna. Group-frames for Banach space. arXiv: 2305.01499v1 [math.FA] 1 May, 2023. [CrossRef]
M. Lammers and S Stampe. The finite Balian-Low conjecture. 2015 International Conference on Sampling Theory and Applications (SampTA), Washington, DC, USA, pages 139–143, 2015.
Shahaf Nitzan and Jan-Fredrik Olsen. A quantitative Balian-Low theorem. J. Fourier Anal. Appl., 19(5):1078–1092, 2013. [CrossRef]
Shahaf Nitzan and Jan-Fredrik Olsen. Balian-Low type theorems in finite dimensions. Math. Ann., 373(1-2):643–677, 2019. [CrossRef]
Kennan T. Smith. The uncertainty principle on groups. SIAM J. Appl. Math., 50(3):876–882, 1990. [CrossRef]
David N. Williams. New mathematical proof of the uncertainty relation. Amer. J. Phys., 47(7):606–607, 1979. [CrossRef]

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Functional Donoho-Stark Approximate Support Uncertainty Principle

Abstract

1. Introduction

2. Functional Donoho-Stark Approximate Support Uncertainty Principle

References

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