Functional Ghobber-Jaming Uncertainty Principle

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Functional Ghobber-Jaming Uncertainty Principle

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K. MAHESH KRISHNA^*

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Submitted:

01 June 2023

Posted:

02 June 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 $\mathcal{X}$. Let $M,N\subseteq \{1, \dots, n\}$ be such that \begin{align*} o(M)^\frac{1}{q}o(N)^\frac{1}{p}< \frac{1}{\displaystyle \max_{1\leq j,k\leq n}|g_k(\tau_j) |}, \end{align*} where $q$ is the conjugate index of $p$. Then for all $x \in \mathcal{X}$, we show that \begin{align}\label{FGJU} (1) \quad \quad \quad \quad \|x\|\leq \left(1+\frac{1}{1-o(M)^\frac{1}{q}o(N)^\frac{1}{p}\displaystyle\max_{1\leq j,k\leq n}|g_k(\tau_j)|}\right)\left[\left(\sum_{j\in M^c}|f_j(x)|^p\right)^\frac{1}{p}+\left(\sum_{k\in N^c}|g_k(x) |^p\right)^\frac{1}{p}\right]. \end{align} We call Inequality (1) as \textbf{Functional Ghobber-Jaming Uncertainty Principle}. Inequality (1) improves the uncertainty principle obtained by Ghobber and Jaming \textit{[Linear Algebra Appl., 2011]}.

Keywords:

Subject: Computer Science and Mathematics - Analysis

MSC: 42C15; 46B03; 46B04

1. Introduction

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 2007, Jaming [1] extended the uncertainty principle obtained by Nazarov for

R

in 1993 [2] (cf. [3]). In the following theorem, Lebesgue measure on

R^{d}

is denoted by m. Mean width of a measurable subset E of

R^{d}

having finite measure is denoted by

w (E)

Theorem 1

([1,2]). (Nazarov-Jaming Uncertainty Principle) For each

d \in N

, there exists a universal constant

C_{d}

(depends upon d) satisfying the following: If

E, F \subseteq R^{d}

are measurable subsets having finite measure, then for all

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

\begin{matrix} \int_{R^{d}} {| f (x) |}^{2} d x & \leq C_{d} e^{C_{d} min {m (E) m (F), m {(E)}^{\frac{1}{d}} w (F), m {(F)}^{\frac{1}{d}} w (E)}} & [\int_{E^{c}} {| f (x) |}^{2} d x + \int_{F^{c}} {| \hat{f} (ξ) |}^{2} d ξ] . \end{matrix}

(1)

In particular, if f is supported on E and

\hat{f}

is supported on F, then

f = 0

Theorem 1 and the milestone paper [4] of Donoho and Stark which derived finite dimensional uncertainty principles, motivated Ghobber and Jaming [5] to ask what is the exact finite dimensional analogue of Theorem 1? Ghobber and Jaming were able to derive the following beautiful theorem. Given a subset

M \subseteq {1, \dots, n}

, the number of elements in M is denoted by

o (M)

Theorem 2

([5]). (Ghobber-Jaming Uncertainty Principle) Let

{τ_{j}}_{j = 1}^{n}

and

{ω_{j}}_{j = 1}^{n}

be orthonormal bases for the Hilbert space

C^{n}

. If

M, N \subseteq {1, \dots, n}

are such that

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

(2)

then for all

h \in C^{n}

\begin{matrix} ∥ h ∥ \leq (1 + \frac{1}{1 - \sqrt{o (M) o (N)} max_{1 \leq j, k \leq n} | 〈 τ_{j}, ω_{k} 〉 |}) [{(\sum_{j \in M^{c}} {| 〈 h, τ_{j} 〉 |}^{2})}^{\frac{1}{2}} + {(\sum_{k \in N^{c}} {| 〈 h, ω_{k} 〉 |}^{2})}^{\frac{1}{2}}] . \end{matrix}

In particular, if h is supported on M in the expansion using basis

{τ_{j}}_{j = 1}^{n}

and h is supported on N in the expansion using basis

{ω_{j}}_{j = 1}^{n}

, then

h = 0

It is reasonable to ask whether there is a Banach space version of Ghobber-Jaming Uncertainty Principle, which when restricted to Hilbert space, reduces to Theorem 2? We are going to answer this question in the paper.

2. Functional Ghobber-Jaming 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

, 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

{δ_{j}}_{j = 1}^{d}

and

{ζ_{j}}_{j = 1}^{d}

be the coordinate functionals associated with

{δ_{j}}_{j = 1}^{d}

. Motivated from the properties of orthonormal bases for Hilbert spaces, we set the following notion of p-orthonormal bases which is also motivated from the notion of p-approximate Schauder frames [6] and p-unconditional Schauder frames [7].

Definition 1.

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 ap-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})

, we easily see from Definition 1 that

\begin{matrix} ∥ x ∥ = ∥\sum_{j = 1}^{n} f_{j} (x) τ_{j}∥ = {(\sum_{j = 1}^{n} {| f_{j} (x) |}^{p})}^{\frac{1}{p}}, \forall x \in X . \end{matrix}

Example 1.

The pair

({ζ_{j}}_{j = 1}^{d}, {δ_{j}}_{j = 1}^{d})

is a p-orthonormal basis for

ℓ^{p} ([d])

Like orthonormal bases for Hilbert spaces, the following theorem characterizes all p-orthonormal bases.

Theorem 3.

Let

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

be a p-orthonormal basis for

X

. Then a pair

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

is a p-orthonormal basis for

X

if and only if there is an invertible linear isometry

V : X \to X

such that

\begin{matrix} g_{j} = f_{j} V^{- 1}, ω_{j} = V τ_{j}, \forall 1 \leq j \leq n . \end{matrix}

Proof.

(\Rightarrow)

Define

V : X ∋ x \mapsto \sum_{j = 1}^{n} f_{j} (x) ω_{j} \in X

. Since

{ω_{j}}_{j = 1}^{n}

is a basis for

X

, V is invertible with inverse

V^{- 1} : X ∋ x \mapsto \sum_{j = 1}^{n} g_{j} (x) τ_{j} \in X

. For

x \in X

\begin{matrix} ∥ V x ∥ = ∥\sum_{j = 1}^{n} f_{j} (x) ω_{j}∥ = {(\sum_{j = 1}^{n} {| f_{j} (x) |}^{p})}^{\frac{1}{p}} = ∥\sum_{j = 1}^{n} f_{j} (x) τ_{j}∥ = ∥ x ∥ . \end{matrix}

Therefore V is isometry. Note that we clearly have

ω_{j} = V τ_{j}, \forall 1 \leq j \leq n .

Now let

1 \leq j \leq n

. Then

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

(\Leftarrow)

Since V is invertible,

{ω_{j}}_{j = 1}^{n}

is a basis for

X

. Now we see that

g_{j} (ω_{k}) = f_{j} (V^{- 1} V τ_{k}) = f_{j} (τ_{k}) = δ_{j, k}

for all

1 \leq j, k \leq n

. Therefore

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

is the coordinate functionals associated with

{ω_{j}}_{j = 1}^{n}

. Since V is an isometry, we have

∥ ω_{j} ∥ = 1

for all

1 \leq j \leq n

. Since V is also invertible, we have

\begin{matrix} ∥ g_{j} ∥ & = sup_{x \in X, ∥ x ∥ \leq 1} | g_{j} (x) | = sup_{x \in X, ∥ x ∥ \leq 1} | f_{j} (V^{- 1} x) | = sup_{V y \in X, ∥ V y ∥ \leq 1} | f_{j} (y) | \\ = sup_{V y \in X, ∥ y ∥ \leq 1} | f_{j} (y) | = ∥ f_{j} ∥ = 1, \forall 1 \leq j \leq n . \end{matrix}

Finally, for every

{(a_{j})}_{j = 1}^{n} \in K^{n}

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

□

In the next result we show that Example 1 is prototypical as long as we consider p-orthonormal bases.

Theorem 4.

X

has a p-orthonormal basis

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

, then

X

is isometrically isomorphic to

ℓ^{p} ([n])

Proof.

Define

V : X ∋ x \mapsto \sum_{j = 1}^{n} f_{j} (x) δ_{j} \in ℓ^{p} ([n])

. By doing a similar calculation as in the direct part in the proof of Theorem 3, we see that V is an invertible isometry. □

Now we derive main result of this paper.

Theorem 5.(Functional Ghobber-Jaming 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 p-orthonormal bases for

X

. If

M, N \subseteq {1, \dots, n}

are such that

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

then for all

x \in X

\begin{matrix} ∥ x ∥ \leq (1 + \frac{1}{1 - o {(M)}^{\frac{1}{q}} o {(N)}^{\frac{1}{p}} max_{1 \leq j, k \leq n} | g_{k} (τ_{j}) |}) [{(\sum_{j \in M^{c}} {| f_{j} (x) |}^{p})}^{\frac{1}{p}} + {(\sum_{k \in N^{c}} {| g_{k} (x) |}^{p})}^{\frac{1}{p}}] . \end{matrix}

(3)

In particular, if x is supported on M in the expansion using basis

{τ_{j}}_{j = 1}^{n}

and x is supported on N in the expansion using basis

{ω_{k}}_{k = 1}^{n}

, then

x = 0

Proof.

Given

S \subseteq {1, \dots, n}

, define

\begin{matrix} P_{S} x \sum_{j \in S} f_{j} (x) τ_{j}, \forall x \in {X, ∥ x ∥}_{S, f} {(\sum_{j \in S} {| f_{j} (x) |}^{p})}^{\frac{1}{p}}, {∥ x ∥}_{S, g} {(\sum_{j \in S} {| g_{j} (x) |}^{p})}^{\frac{1}{p}} . \end{matrix}

Also define

V : X ∋ x \mapsto \sum_{k = 1}^{n} g_{k} (x) τ_{k} \in X

. Then V is an invertible isometry. Using V we make following important calculations:

\begin{matrix} ∥ P_{S} x ∥ = ∥\sum_{j \in S} f_{j} (x) τ_{j}∥ = {(\sum_{j \in S} {| f_{j} (x) |}^{p})}^{\frac{1}{p}} = {∥ x ∥}_{S, f}, \forall x \in X \end{matrix}

and

\begin{matrix} ∥ P_{S} V x ∥ & = ∥\sum_{j \in S} f_{j} (V x) τ_{j}∥ = ∥\sum_{j \in S} f_{j} (\sum_{k = 1}^{n} g_{k} (x) τ_{k}) τ_{j}∥ = ∥\sum_{j \in S} \sum_{k = 1}^{n} g_{k} (x) f_{j} (τ_{k}) τ_{j}∥ \\ = ∥\sum_{j \in S} g_{j} (x) τ_{j}∥ = {(\sum_{j \in S} {| g_{j} (x) |}^{p})}^{\frac{1}{p}} = {∥ x ∥}_{S, g}, \forall x \in X . \end{matrix}

Now let

y \in X

be such that

{j \in {1, \dots, n} : f_{j} (y) \neq 0} \subseteq M .

Then

∥ P_{N} V y ∥ = ∥ P_{N} V P_{M} y ∥ \leq ∥ P_{N} V P_{M} ∥ ∥ y ∥

and

\begin{matrix} {∥ y ∥}_{N^{c}, g} = ∥ P_{N^{c}} V y ∥ = ∥ V y - P_{N} V y ∥ \geq ∥ V y ∥ - ∥ P_{N} V y ∥ = ∥ y ∥ - ∥ P_{N} V y ∥ \geq ∥ y ∥ - ∥ P_{N} V P_{M} ∥ ∥ y ∥ . \end{matrix}

Therefore

\begin{matrix} {∥ y ∥}_{N^{c}, g} \geq (1 - ∥ P_{N} V P_{M} ∥) ∥ y ∥ . \end{matrix}

(4)

Let

x \in X

. Note that

P_{M} x

satisfies

{j \in {1, \dots, n} : f_{j} (P_{M} x) \neq 0} \subseteq M .

Now using (4) we get

\begin{matrix} ∥ x ∥ & = ∥ P_{M} x + P_{M^{c}} x ∥ \leq ∥ P_{M} x ∥ + ∥ P_{M^{c}} x ∥ \leq \frac{1}{1 - ∥ P_{N} V P_{M} ∥} ∥ P_{M} {x ∥}_{N^{c}, g} + ∥ P_{M^{c}} x ∥ \\ = \frac{1}{1 - ∥ P_{N} V P_{M} ∥} ∥ P_{N^{c}} V P_{M} x ∥ + ∥ P_{M^{c}} x ∥ = \frac{1}{1 - ∥ P_{N} V P_{M} ∥} ∥ P_{N^{c}} V (x - P_{M^{c}} x) ∥ + ∥ P_{M^{c}} x ∥ \\ \leq \frac{1}{1 - ∥ P_{N} V P_{M} ∥} ∥ P_{N^{c}} V x ∥ + \frac{1}{1 - ∥ P_{N} V P_{M} ∥} ∥ P_{N^{c}} V P_{M^{c}} x ∥ + ∥ P_{M^{c}} x ∥ \\ \leq \frac{1}{1 - ∥ P_{N} V P_{M} ∥} ∥ P_{N^{c}} V x ∥ + \frac{1}{1 - ∥ P_{N} V P_{M} ∥} ∥ P_{M^{c}} x ∥ + ∥ P_{M^{c}} x ∥ \\ = \frac{1}{1 - ∥ P_{N} V P_{M} ∥} ∥ P_{N^{c}} V x ∥ + (1 + \frac{1}{1 - ∥ P_{N} V P_{M} ∥}) ∥ P_{M^{c}} x ∥ \\ \leq ∥ P_{N^{c}} V x ∥ + \frac{1}{1 - ∥ P_{N} V P_{M} ∥} ∥ P_{N^{c}} V x ∥ + (1 + \frac{1}{1 - ∥ P_{N} V P_{M} ∥}) ∥ P_{M^{c}} x ∥ \\ = (1 + \frac{1}{1 - ∥ P_{N} V P_{M} ∥}) [∥ P_{N^{c}} V x ∥ + ∥ P_{M^{c}} x ∥] = (1 + \frac{1}{1 - ∥ P_{N} V P_{M} ∥}) {[∥ x ∥}_{N^{c}, g} + ∥ P_{M^{c}} x ∥] \\ = (1 + \frac{1}{1 - ∥ P_{N} V P_{M} ∥}) [{(\sum_{j \in M^{c}} {| f_{j} (x) |}^{p})}^{\frac{1}{p}} + {(\sum_{k \in N^{c}} {| g_{k} (x) |}^{p})}^{\frac{1}{p}}] . \end{matrix}

For

x \in X

, we now find

\begin{matrix} ∥ P_{N} V P_{M} {x ∥}^{p} = {∥\sum_{k \in N} f_{k} (V P_{M} x) τ_{k}∥}^{p} = {(\sum_{k \in N} {| f_{k} (V P_{M} x) |}^{p})}^{\frac{1}{p}} = \sum_{k \in N} {|(f_{k} V) (\sum_{j \in M} f_{j} (x) τ_{j})|}^{p} \\ = \sum_{k \in N} {|\sum_{j \in M} f_{j} (x) f_{k} (V τ_{j})|}^{p} = \sum_{k \in N} {|\sum_{j \in M} f_{j} (x) f_{k} (\sum_{r = 1}^{n} g_{r} (τ_{j}) τ_{r})|}^{p} = \sum_{k \in N} {|\sum_{j \in M} f_{j} (x) \sum_{r = 1}^{n} g_{r} (τ_{j}) f_{k} (τ_{r})|}^{p} \\ = \sum_{k \in N} {|\sum_{j \in M} f_{j} (x) g_{k} (τ_{j})|}^{p} \leq \sum_{k \in N} {(\sum_{j \in M} | f_{j} (x) 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} | f_{j} (x) |)}^{p} \\ = {(max_{1 \leq j, k \leq n} | g_{k} (τ_{j}) |)}^{p} o (N) {(\sum_{j \in M} | f_{j} (x) |)}^{p} \leq {(max_{1 \leq j, k \leq n} | g_{k} (τ_{j}) |)}^{p} o (N) {(\sum_{j \in M} {| f_{j} (x) |}^{p})}^{\frac{p}{p}} {(\sum_{j \in M} 1^{q})}^{\frac{p}{q}} \\ \leq {(max_{1 \leq j, k \leq n} | g_{k} (τ_{j}) |)}^{p} o (N) {(\sum_{j = 1}^{n} {| f_{j} (x) |}^{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) {∥ x ∥}^{p} o {(M)}^{\frac{p}{q}} . \end{matrix}

Therefore

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

which gives the theorem. □

Corollary 1.

Theorem 2 follows from Theorem 5.

Proof.

Let

{τ_{j}}_{j = 1}^{n}

{ω_{j}}_{j = 1}^{n}

be two orthonormal bases 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

and

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

for all

1 \leq j, k \leq n .

□

By interchanging p-orthonormal bases in Theorem 5 we get the following theorem.

Theorem 6.(Functional Ghobber-Jaming 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 p-orthonormal bases for

X

. If

M, N \subseteq {1, \dots, n}

are such that

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

then for all

x \in X

\begin{matrix} ∥ x ∥ \leq (1 + \frac{1}{1 - o {(M)}^{\frac{1}{q}} o {(N)}^{\frac{1}{p}} max_{1 \leq j, k \leq n} | f_{j} (ω_{k}) |}) [{(\sum_{k \in M^{c}} {| g_{k} (x) |}^{p})}^{\frac{1}{p}} + {(\sum_{j \in N^{c}} {| f_{j} (x) |}^{p})}^{\frac{1}{p}}] . \end{matrix}

In particular, if x is supported on M in the expansion using basis

{ω_{k}}_{k = 1}^{n}

and x is supported on N in the expansion using basis

{τ_{j}}_{j = 1}^{n}

, then

x = 0

Observe that the constant

\begin{matrix} C_{d} e^{C_{d} min {m (E) m (F), m {(E)}^{\frac{1}{d}} w (F), m {(F)}^{\frac{1}{d}} w (E)}} \end{matrix}

in Inequality (1) is depending upon subsets E, F and not on the entire domain

R

of functions f,

\hat{f}

. Thus it is natural to ask whether there is a constant sharper in Inequality (3) depending upon subsets M, N and not on

{1, \dots, n}

. A careful observation in the proof of Theorem 5 gives following result.

Theorem 7.

Let

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

and

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

be p-orthonormal bases for

X

. If

M, N \subseteq {1, \dots, n}

are such that

\begin{matrix} o {(M)}^{\frac{1}{q}} o {(N)}^{\frac{1}{p}} < \frac{1}{max_{j \in M, k \in N} | g_{k} (τ_{j}) |}, \end{matrix}

then for all

x \in X

\begin{matrix} ∥ x ∥ \leq (1 + \frac{1}{1 - o {(M)}^{\frac{1}{q}} o {(N)}^{\frac{1}{p}} max_{j \in M, k \in N} | g_{k} (τ_{j}) |}) [{(\sum_{j \in M^{c}} {| f_{j} (x) |}^{p})}^{\frac{1}{p}} + {(\sum_{k \in N^{c}} {| g_{k} (x) |}^{p})}^{\frac{1}{p}}] . \end{matrix}

Similarly we have the following result from Theorem 6.

Theorem 8.

Let

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

and

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

be p-orthonormal bases for

X

. If

M, N \subseteq {1, \dots, n}

are such that

\begin{matrix} o {(M)}^{\frac{1}{q}} o {(N)}^{\frac{1}{p}} < \frac{1}{max_{j \in N, k \in M} | f_{j} (ω_{k}) |}, \end{matrix}

then for all

x \in X

\begin{matrix} ∥ x ∥ \leq (1 + \frac{1}{1 - o {(M)}^{\frac{1}{q}} o {(N)}^{\frac{1}{p}} max_{j \in N, k \in M} | f_{j} (ω_{k}) |}) [{(\sum_{k \in M^{c}} {| g_{k} (x) |}^{p})}^{\frac{1}{p}} + {(\sum_{j \in N^{c}} {| f_{j} (x) |}^{p})}^{\frac{1}{p}}] . \end{matrix}

Theorem 5 brings the following question.

Question 9.

Given p and a Banach space

X

of dimension n, for which subsets

M, N \subseteq {1, \dots, n}

and 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

, we have equality in Inequality (3)?

It is clear that we used

1 < p < \infty

in the proof of Theorem 5. However, Definition 1 can easily be extended to include cases

p = 1

and

p = \infty

. This therefore leads to the following question.

Question 10.

Whether there are Functional Ghobber-Jaming Uncertainty Principle (versions of Theorem 5) for 1-orthonormal bases and ∞-orthonormal bases?

We end by mentioning that Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle for finite dimensional Banach spaces is derived in [8] (actually, in [8] the functional uncertainty principle was derived for p-Schauder frames which is general than p-orthonormal bases. Thus it is worth to derive Theorem 5 or a variation of it for p-Schauder frames, which we are unable).

References

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Functional Ghobber-Jaming Uncertainty Principle

Abstract

1. Introduction

2. Functional Ghobber-Jaming Uncertainty Principle

References

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