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Noncommutative Donoho-Stark-Elad-Bruckstein-Ricaud-Torresani Uncertainty Principle

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01 June 2024

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05 June 2024

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Abstract
Let $\{\tau_n\}_{n=1}^\infty$ and $\{\omega_m\}_{m=1}^\infty$ be two modular Parseval frames for a Hilbert C*-module $\mathcal{E}$. Then for every $x \in \mathcal{E}\setminus\{0\}$, we show that \begin{align}\label{UE} \|\theta_\tau x \|_0 \|\theta_\omega x \|_0 \geq \frac{1}{\sup_{n, m \in \mathbb{N}} \|\langle \tau_n, \omega_m\rangle \|^2}. \end{align} We call Inequality (\ref{UE}) as \textbf{Noncommutative Donoho-Stark-Elad-Bruckstein-Ricaud-Torr\'{e}sani Uncertainty Principle}. Inequality (\ref{UE}) is the noncommutative analogue of breakthrough Ricaud-Torr\'{e}sani uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2013]}. In particular, Inequality (\ref{UE}) extends Elad-Bruckstein uncertainty principle \textit{[IEEE Trans. Inform. Theory, 2002]} and Donoho-Stark uncertainty principle \textit{[SIAM J. Appl. Math., 1989]}.
Keywords: 
Subject: Computer Science and Mathematics  -   Analysis

MSC:  42C15; 46L08

1. Introduction

In 1989, Donoho and Stark derived following uncertainty principle which is one of the greatest inequality of all time in both pure and applied Mathematics [1]. For h ∈ C d , let ∥ h ∥ 0 be the number of nonzero entries in h. Let   ^ : C d → C d be the Fourier transform defined by
( a j ) j = 0 d − 1 ^ 1 d ∑ j = 0 d − 1 a j e − 2 π i j k d k = 0 d − 1 , ∀ ( a j ) j = 0 d − 1 ∈ C d .
Theorem 1.
(Donoho-Stark Uncertainty Principle) [1,2] For every d ∈ N ,
∥ h ∥ 0 + ∥ h ^ ∥ 0 2 2 ≥ ∥ h ∥ 0 ∥ h ^ ∥ 0 ≥ d , ∀ h ∈ C d ∖ { 0 } .
By noting that Fourier transform is unitary and unitary operators are in one to one correspondence with orthonormal bases, in 2002, Elad and Bruckstein generalized Inequality (1) to arbitrary orthonormal bases [3]. To state the result we need some notations. Given a collection { τ j } j = 1 n in a finite dimensional Hilbert space H over K ( R or C ), we define
θ τ : H ∋ h ↦ θ τ h : = ( 〈 h , τ j 〉 ) j = 1 n ∈ K n .
Theorem 2.
(Elad-Bruckstein Uncertainty Principle) [3,4] Let { τ j } j = 1 n , { ω j } j = 1 n be two orthonormal bases for a finite dimensional Hilbert space H . Then
∥ θ τ h ∥ 0 + ∥ θ ω h ∥ 0 2 2 ≥ ∥ θ τ h ∥ 0 ∥ θ ω h ∥ 0 ≥ 1 max 1 ≤ j , k ≤ n | 〈 τ j , ω k 〉 | 2 , ∀ h ∈ H ∖ { 0 } .
In 2013, Ricaud and Torrésani showed that orthonormal bases in Theorem 2 can be improved to Parseval frames [5]. Recall that a collection { τ j } j = 1 n in a finite dimensional Hilbert space H is said to be a Parseval frame for H [6] if
∥ h ∥ 2 = ∑ j = 1 n | 〈 h , τ j 〉 | 2 , ∀ h ∈ H .
Theorem 3.
(Ricaud-Torrésani Uncertainty Principle) [5] Let { τ j } j = 1 n , { ω j } j = 1 n be two Parseval frames for a finite dimensional Hilbert space H . Then
∥ θ τ h ∥ 0 + ∥ θ ω h ∥ 0 2 2 ≥ ∥ θ τ h ∥ 0 ∥ θ ω h ∥ 0 ≥ 1 max 1 ≤ j , k ≤ n | 〈 τ j , ω k 〉 | 2 , ∀ h ∈ H ∖ { 0 } .
The main purpose of this paper is to generalize and derive a noncommutative version of Theorem 3. For this we want generalization of Hilbert spaces known as Hilbert C*-modules. Hilbert C*-modules are first introduced by Kaplansky [7] for modules over commutative C*-algebras and later developed for modules over arbitrary C*-algebras by Paschke [8] and Rieffel [9].
Definition 1.
[7,8,9] Let A be a unital C*-algebra. A left module E over A is said to be a (left) Hilbert C*-module if there exists a map 〈 · , · 〉 : E × E → A such that the following hold.
(i) 〈 x , x 〉 ≥ 0 , ∀ x ∈ E . If x ∈ E satisfies 〈 x , x 〉 = 0 , then x = 0 .
(ii) 〈 x + y , z 〉 = 〈 x , z 〉 + 〈 y , z 〉 , ∀ x , y , z ∈ E .
(iii) 〈 a x , y 〉 = a 〈 x , y 〉 , ∀ x , y ∈ E . ∀ a ∈ A .
(iv) 〈 x , y 〉 = 〈 y , x 〉 * , ∀ x , y ∈ E .
(v) E is complete w.r.t. the norm ∥ x ∥ : = ∥ 〈 x , x 〉 ∥ , ∀ x ∈ E .
We are going to use the following inequality.
Lemma 1.
[8] (Noncommutative Cauchy-Schwarz inequality) If E is a Hilbert C*-module over A , then
〈 x , y 〉 〈 y , x 〉 ≤ ∥ 〈 y , y 〉 ∥ 〈 x , x 〉 , ∀ x , y ∈ E .
Given a unital C*-algebra A , define
ℓ 2 ( N , A ) { a n } n = 1 ∞ : a n ∈ A , ∀ n ∈ N , ∑ n = 1 ∞ a n a n * converges in A .
Modular A -inner product on ℓ 2 ( N , A ) is defined as
〈 { a n } n = 1 ∞ , { b n } n = 1 ∞ 〉 : = ∑ n = 1 ∞ a n b n * , ∀ { a n } n = 1 ∞ , { b n } n = 1 ∞ ∈ ℓ 2 ( N , A ) .
Hence the norm on ℓ 2 ( N , A ) becomes
∥ { a n } n = 1 ∞ ∥ : = ∑ n = 1 ∞ a n a n * 1 2 , ∀ { a n } n = 1 ∞ ∈ ℓ 2 ( N , A ) .

2. Noncommutative Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle

We start by recalling the definition of Parseval frames for Hilbert C*-modules by Frank and Larson [10].
Definition 2.
[10] Let E be a Hilbert C*-module over a unital C*-algebra A . A collection { τ n } n = 1 ∞ in E is said to be a modular Parseval frame for E if
〈 x , x 〉 = ∑ n = 1 ∞ 〈 x , τ n 〉 〈 τ n , x 〉 , ∀ x ∈ E .
As shown in [10] a modular Parseval frame { τ n } n = 1 ∞ for E gives an adjointable isometry
θ τ : E ∋ x ↦ θ τ x : = { 〈 x , τ n 〉 } n = 1 ∞ ∈ ℓ 2 ( N , A )
with adjoint
θ τ * : ℓ 2 ( N , A ) ∋ { a n } n = 1 ∞ ↦ θ τ * { a n } n = 1 ∞ : = ∑ n = 1 ∞ a n τ n ∈ E .
We this preliminaries we can derive noncommutative analogue of Theorem 3. In the following theorem, given a subset Λ ⊆ N , we set the notation
o ( Λ ) : = Number of elements in Λ .
Theorem 4.
(Noncommutative Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle) For any two modular Parseval frames { τ n } n = 1 ∞ and { ω m } m = 1 ∞ for a Hilbert C*-module E , we have
∥ θ τ x ∥ 0 + ∥ θ ω x ∥ 0 2 2 ≥ ∥ θ τ x ∥ 0 ∥ θ ω x ∥ 0 ≥ 1 sup n , m ∈ N ∥ 〈 τ n , ω m 〉 ∥ 2 , ∀ x ∈ E , x ≠ 0 .
Proof. 
Let x ∈ E be nonzero. Using Lemma 1 and the well-known fact in C*-algebra that `norm respects ordering of positive elements’, we get
∥ x ∥ 2 = ∥ 〈 x , x 〉 ∥ = ∑ n = 1 ∞ 〈 x , τ n 〉 〈 τ n , x 〉 = ∑ n ∈ supp ( θ τ x ) 〈 x , τ n 〉 〈 τ n , x 〉 = ∑ n ∈ supp ( θ τ x ) ∑ m = 1 ∞ 〈 x , ω m 〉 ω m , τ n τ n , ∑ k = 1 ∞ 〈 x , ω k 〉 ω k = ∑ n ∈ supp ( θ τ x ) ∑ m ∈ supp ( θ ω x ) 〈 x , ω m 〉 ω m , τ n τ n , ∑ k ∈ supp ( θ ω x ) 〈 x , ω k 〉 ω k = ∑ n ∈ supp ( θ τ x ) ∑ m ∈ supp ( θ ω x ) 〈 x , ω m 〉 〈 τ n , ω m 〉 * ∑ k ∈ supp ( θ ω x ) 〈 x , ω k 〉 〈 τ n , ω k 〉 * ≤ ∑ n ∈ supp ( θ τ x ) ∑ m ∈ supp ( θ ω x ) 〈 τ n , ω m 〉 〈 τ n , ω m 〉 * ∑ k ∈ supp ( θ ω x ) 〈 x , ω k 〉 〈 ω k , x 〉 ≤ ∑ n ∈ supp ( θ τ x ) ∑ m ∈ supp ( θ ω x ) ∥ 〈 τ n , ω m 〉 〈 τ n , ω m 〉 * ∥ ∑ k ∈ supp ( θ ω x ) 〈 x , ω k 〉 〈 ω k , x 〉 ≤ sup n , m ∈ N ∥ 〈 τ n , ω m 〉 ∥ 2 ∑ n ∈ supp ( θ τ x ) ∑ m ∈ supp ( θ ω x ) 1 · ∑ k ∈ supp ( θ ω x ) 〈 x , ω k 〉 〈 ω k , x 〉 ≤ sup n , m ∈ N ∥ 〈 τ n , ω m 〉 ∥ 2 o ( supp ( θ τ x ) o ( supp ( θ ω x ) ∑ k ∈ supp ( θ ω x ) 〈 x , ω k 〉 〈 ω k , x 〉 = sup n , m ∈ N ∥ 〈 τ n , ω m 〉 ∥ 2 ∥ θ τ x ∥ 0 ∥ θ ω x ∥ 0 ∑ k ∈ supp ( θ ω x ) 〈 x , ω k 〉 〈 ω k , x 〉 = sup n , m ∈ N ∥ 〈 τ n , ω m 〉 ∥ 2 ∥ θ τ x ∥ 0 ∥ θ ω x ∥ 0 ∥ 〈 x , x 〉 ∥ = sup n , m ∈ N ∥ 〈 τ n , ω m 〉 ∥ 2 ∥ θ τ x ∥ 0 ∥ θ ω x ∥ 0 ∥ x ∥ 2 .
By canceling ∥ x ∥ we get the stated inequality. □
Using Chebotarev theorem, in 2005, Tao [11,12] improved Theorem 1 for prime dimensions d.
Theorem 5.
(Tao Uncertainty Principle ) [11] For every prime p,
∥ h ∥ 0 + ∥ h ^ ∥ 0 ≥ p + 1 , ∀ h ∈ C p ∖ { 0 } .
In view of Theorem 5 we make the following conjecture.
Conjecture 6.
Let p be a prime and A be a unital C*-algebra with invariant basis number property. Let   ^ : A p → A p be the noncommutative Fourier transform defined by
( a j ) j = 0 p − 1 ^ 1 p ∑ j = 0 p − 1 a j e − 2 π i j k p k = 0 p − 1 , ∀ ( a j ) j = 0 p − 1 ∈ A p .
Then
∥ x ∥ 0 + ∥ x ^ ∥ 0 ≥ p + 1 , ∀ x ∈ A p ∖ { 0 } .

References

  1. Donoho, D.L.; Stark, P.B. Uncertainty principles and signal recovery. SIAM J. Appl. Math. 1989, 49, 906–931. [Google Scholar] [CrossRef]
  2. Terras, A. Fourier analysis on finite groups and applications; Vol. 43, London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 1999; pp. x+442. [CrossRef]
  3. Elad, M.; Bruckstein, A.M. A generalized uncertainty principle and sparse representation in pairs of bases. IEEE Trans. Inform. Theory 2002, 48, 2558–2567. [Google Scholar] [CrossRef]
  4. Elad, M. Sparse and redundant representations : From theory to applications in signal and image processing; Springer, New York, 2010; pp. xx+376. [CrossRef]
  5. Ricaud, B.; Torrésani, B. Refined support and entropic uncertainty inequalities. IEEE Trans. Inform. Theory 2013, 59, 4272–4279. [Google Scholar] [CrossRef]
  6. Benedetto, J.J.; Fickus, M. Finite normalized tight frames. Adv. Comput. Math. 2003, 18, 357–385. [Google Scholar] [CrossRef]
  7. Kaplansky, I. Modules over operator algebras. Amer. J. Math. 1953, 75, 839–858. [Google Scholar] [CrossRef]
  8. Paschke, W.L. Inner product modules over B*-algebras. Trans. Amer. Math. Soc. 1973, 182, 443–468. [Google Scholar] [CrossRef]
  9. Rieffel, M.A. Induced representations of C*-algebras. Advances in Math. 1974, 13, 176–257. [Google Scholar] [CrossRef]
  10. Frank, M.; Larson, D.R. Frames in Hilbert C*-modules and C*-algebras. J. Operator Theory 2002, 48, 273–314. [Google Scholar]
  11. Tao, T. An uncertainty principle for cyclic groups of prime order. Math. Res. Lett. 2005, 12, 121–127. [Google Scholar] [CrossRef]
  12. Ceccherini-Silberstein, T.; Scarabotti, F.; Tolli, F. Discrete harmonic analysis : Representations, number theory, expanders, and the Fourier transform; Vol. 172, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2018; pp. xiii+573. [CrossRef]
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