Preprint
Article

Functional Ghobber-Jaming Uncertainty Principle

Altmetrics

Downloads

125

Views

94

Comments

0

This version is not peer-reviewed

Submitted:

01 June 2023

Posted:

02 June 2023

You are already at the latest version

Alerts
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

1. Introduction

Let d N and ^ : L 2 ( R d ) L 2 ( R d ) be the unitary Fourier transform obtained by extending uniquely the bounded linear operator
^ : L 1 ( R d ) L 2 ( R d ) f f ^ C 0 ( R d ) ; f ^ : R d ξ f ^ ( ξ ) R d f ( x ) e 2 π i x , ξ d x C .
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 N , there exists a universal constant C d (depends upon d) satisfying the following: If E , F R d are measurable subsets having finite measure, then for all f L 2 ( R d ) ,
R d | f ( x ) | 2 d x C d e C d min { m ( E ) m ( F ) , m ( E ) 1 d w ( F ) , m ( F ) 1 d w ( E ) } E c | f ( x ) | 2 d x + F c | f ^ ( ξ ) | 2 d ξ .
In particular, if f is supported on E and 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 { 1 , , 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 { 1 , , n } are such that
o ( M ) o ( N ) < 1 max 1 j , k n | τ j , ω k | 2 ,
then for all h C n ,
h 1 + 1 1 o ( M ) o ( N ) max 1 j , k n | τ j , ω k | j M c | h , τ j | 2 1 2 + k N c | h , ω k | 2 1 2 .
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 or 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 < , q denotes conjugate index of p. For d N , the standard finite dimensional Banach space K d over K equipped with standard · p norm is denoted by 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 < ) for X if the following conditions hold.
(i) 
f j = τ j = 1 for all 1 j n .
(ii) 
For every ( a j ) j = 1 n K n ,
j = 1 n a j τ j = j = 1 n | a j | p 1 p .
Given a p-orthonormal basis ( { f j } j = 1 n , { τ j } j = 1 n ) , we easily see from Definition 1 that
x = j = 1 n f j ( x ) τ j = j = 1 n | f j ( x ) | p 1 p , x X .
Example 1. 
The pair ( { ζ j } j = 1 d , { δ j } j = 1 d ) is a p-orthonormal basis for 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 X such that
g j = f j V 1 , ω j = V τ j , 1 j n .
Proof. ( ) Define V : X x j = 1 n f j ( x ) ω j X . Since { ω j } j = 1 n is a basis for X , V is invertible with inverse V 1 : X x j = 1 n g j ( x ) τ j X . For x X ,
V x = j = 1 n f j ( x ) ω j = j = 1 n | f j ( x ) | p 1 p = j = 1 n f j ( x ) τ j = x .
Therefore V is isometry. Note that we clearly have ω j = V τ j , 1 j n . Now let 1 j n . Then
f j ( V 1 x ) = f j k = 1 n g k ( x ) τ k = k = 1 n g k ( x ) f j ( τ k ) = g j ( x ) , x X .
( ) 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 j , k 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 j n . Since V is also invertible, we have
g j = sup x X , x 1 | g j ( x ) | = sup x X , x 1 | f j ( V 1 x ) | = sup V y X , V y 1 | f j ( y ) | = sup V y X , y 1 | f j ( y ) | = f j = 1 , 1 j n .
Finally, for every ( a j ) j = 1 n K n ,
j = 1 n a j ω j = j = 1 n a j V τ j = V j = 1 n a j τ j = j = 1 n a j τ j = j = 1 n | a j | p 1 p .
In the next result we show that Example 1 is prototypical as long as we consider p-orthonormal bases.
Theorem 4. 
If X has a p-orthonormal basis ( { f j } j = 1 n , { τ j } j = 1 n ) , then X is isometrically isomorphic to p ( [ n ] ) .
Proof. 
Define V : X x j = 1 n f j ( x ) δ j 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 { 1 , , n } are such that
o ( M ) 1 q o ( N ) 1 p < 1 max 1 j , k n | g k ( τ j ) | ,
then for all x X ,
x 1 + 1 1 o ( M ) 1 q o ( N ) 1 p max 1 j , k n | g k ( τ j ) | j M c | f j ( x ) | p 1 p + k N c | g k ( x ) | p 1 p .
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 { 1 , , n } , define
P S x j S f j ( x ) τ j , x X , x S , f j S | f j ( x ) | p 1 p , x S , g j S | g j ( x ) | p 1 p .
Also define V : X x k = 1 n g k ( x ) τ k X . Then V is an invertible isometry. Using V we make following important calculations:
P S x = j S f j ( x ) τ j = j S | f j ( x ) | p 1 p = x S , f , x X
and
P S V x = j S f j ( V x ) τ j = j S f j k = 1 n g k ( x ) τ k τ j = j S k = 1 n g k ( x ) f j ( τ k ) τ j = j S g j ( x ) τ j = j S | g j ( x ) | p 1 p = x S , g , x X .
Now let y X be such that { j { 1 , , n } : f j ( y ) 0 } M . Then P N V y = P N V P M y P N V P M y and
y N c , g = P N c V y = V y P N V y V y P N V y = y P N V y y P N V P M y .
Therefore
y N c , g ( 1 P N V P M ) y .
Let x X . Note that P M x satisfies { j { 1 , , n } : f j ( P M x ) 0 } M . Now using (4) we get
x = P M x + P M c x P M x + P M c x 1 1 P N V P M P M x N c , g + P M c x = 1 1 P N V P M P N c V P M x + P M c x = 1 1 P N V P M P N c V ( x P M c x ) + P M c x 1 1 P N V P M P N c V x + 1 1 P N V P M P N c V P M c x + P M c x 1 1 P N V P M P N c V x + 1 1 P N V P M P M c x + P M c x = 1 1 P N V P M P N c V x + 1 + 1 1 P N V P M P M c x P N c V x + 1 1 P N V P M P N c V x + 1 + 1 1 P N V P M P M c x = 1 + 1 1 P N V P M [ P N c V x + P M c x ] = 1 + 1 1 P N V P M [ x N c , g + P M c x ] = 1 + 1 1 P N V P M j M c | f j ( x ) | p 1 p + k N c | g k ( x ) | p 1 p .
For x X , we now find
P N V P M x p = k N f k ( V P M x ) τ k p = k N | f k ( V P M x ) | p 1 p = k N ( f k V ) j M f j ( x ) τ j p = k N j M f j ( x ) f k ( V τ j ) p = k N j M f j ( x ) f k r = 1 n g r ( τ j ) τ r p = k N j M f j ( x ) r = 1 n g r ( τ j ) f k ( τ r ) p = k N j M f j ( x ) g k ( τ j ) p k N j M | f j ( x ) g k ( τ j ) | p max 1 j , k n | g k ( τ j ) | p k N j M | f j ( x ) | p = max 1 j , k n | g k ( τ j ) | p o ( N ) j M | f j ( x ) | p max 1 j , k n | g k ( τ j ) | p o ( N ) j M | f j ( x ) | p p p j M 1 q p q max 1 j , k n | g k ( τ j ) | p o ( N ) j = 1 n | f j ( x ) | p p p j M 1 q p q = max 1 j , k n | g k ( τ j ) | p o ( N ) x p o ( M ) p q .
Therefore
P N V P M max 1 j , k n | g k ( τ j ) | o ( N ) 1 p o ( M ) 1 q
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
f j : H h h , τ j K ; g j : H h h , ω j K , 1 j n .
Then p = q = 2 and | f j ( ω k ) | = | ω k , τ j | for all 1 j , k 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 { 1 , , n } are such that
o ( M ) 1 q o ( N ) 1 p < 1 max 1 j , k n | f j ( ω k ) | ,
then for all x X ,
x 1 + 1 1 o ( M ) 1 q o ( N ) 1 p max 1 j , k n | f j ( ω k ) | k M c | g k ( x ) | p 1 p + j N c | f j ( x ) | p 1 p .
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
C d e C d min { m ( E ) m ( F ) , m ( E ) 1 d w ( F ) , m ( F ) 1 d w ( E ) }
in Inequality (1) is depending upon subsets E, F and not on the entire domain R of functions f, 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 , , 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 { 1 , , n } are such that
o ( M ) 1 q o ( N ) 1 p < 1 max j M , k N | g k ( τ j ) | ,
then for all x X ,
x 1 + 1 1 o ( M ) 1 q o ( N ) 1 p max j M , k N | g k ( τ j ) | j M c | f j ( x ) | p 1 p + k N c | g k ( x ) | p 1 p .
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 { 1 , , n } are such that
o ( M ) 1 q o ( N ) 1 p < 1 max j N , k M | f j ( ω k ) | ,
then for all x X ,
x 1 + 1 1 o ( M ) 1 q o ( N ) 1 p max j N , k M | f j ( ω k ) | k M c | g k ( x ) | p 1 p + j N c | f j ( x ) | p 1 p .
Theorem 5 brings the following question.
Question 9. 
Given p and a Banach space X of dimension n, for which subsets M , N { 1 , , 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 < in the proof of Theorem 5. However, Definition 1 can easily be extended to include cases p = 1 and p = . 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

  1. Jaming, P. Nazarov’s uncertainty principles in higher dimension. J. Approx. Theory 2007, 149, 30–41. [Google Scholar] [CrossRef]
  2. Nazarov, F.L. Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type. Algebra Analiz 1993, 5, 3–66. [Google Scholar]
  3. Havin, V.; Jöricke, B. The uncertainty principle in harmonic analysis; A Series of Modern Surveys in Mathematics, Springer-Verlag, Berlin, 1994; Volume 28, pp. xii+543. [CrossRef]
  4. Donoho, D.L.; Stark, P.B. Uncertainty principles and signal recovery. SIAM J. Appl. Math. 1989, 49, 906–931. [Google Scholar] [CrossRef]
  5. Ghobber, S.; Jaming, P. On uncertainty principles in the finite dimensional setting. Linear Algebra Appl. 2011, 435, 751–768. [Google Scholar] [CrossRef]
  6. Krishna, K.M.; Johnson, P.S. Towards characterizations of approximate Schauder frame and its duals for Banach spaces. J. Pseudo-Differ. Oper. Appl. 2021, 12, 13. [Google Scholar] [CrossRef]
  7. Krishna, K.M. Group-Frames for Banach Spaces. arXiv, 2023; arXiv: 2305.01499v1. [Google Scholar]
  8. Krishna, K.M. Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle. arXiv 2023 arXiv: 2304.03324v1.
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.
Prerpints.org logo

Preprints.org is a free preprint server supported by MDPI in Basel, Switzerland.

Subscribe

© 2024 MDPI (Basel, Switzerland) unless otherwise stated