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An Endpoint Functional Continuous Uncertainty Principle

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12 August 2023

Posted:

15 August 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 1-Schauder frames for a Banach space $\mathcal{X}$. Then for every $x \in \mathcal{X}\setminus\{0\}$, we show that \begin{align*} \mu(\operatorname{supp}(\theta_f x)) \geq \frac{1}{\displaystyle\sup_{\alpha \in \Omega, \beta \in \Delta}|f_\alpha(\omega_\beta)|}, \quad \nu(\operatorname{supp}(\theta_g x))\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}^1(\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}^1(\Delta, \nu); \quad \theta_gx: \Delta \ni \beta \mapsto (\theta_gx) (\beta):= g_\beta (x) \in \mathbb{K}. \end{align*} This solves a problem asked by K. M. Krishna in the paper 'Functional Continuous Uncertainty Principle' \textit{[arXiv:2308.00312v1]}.
Keywords: 
Uncertainty Principle, Parseval Frame, Banach space.
Subject: 
Computer Science and Mathematics  -   Analysis

MSC:  42C15

1. Introduction

Given a collection { τ j } j = 1 n in a finite dimensional Hilbert space H over K ( R or C ), let
θ τ : H h θ τ h ( h , τ j ) j = 1 n K n .
Following is the most general form of discrete uncertainty principle for finite dimensional Hilbert spaces.
Theorem 1 
(Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle) [2,3,7]). (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 } .
Recently, Theorem 1 has been greatly improved to continuous families in Banach spaces (even infinite dimensions). To state the result, we need a notion.
Definition 1
([4]). Let ( Ω , μ ) be a measure space. Let { τ α } α Ω be a collection in a Banach space X and { f α } α Ω be a collection in X * . The pair ( { f α } α Ω , { τ α } α Ω ) is said to be a continuous p-Schauder framefor X ( 1 < p < ) if the following holds.
(i)
For every x X , the map Ω α f α ( x ) K is measurable.
(ii)
For every x X ,
x p = Ω | f α ( x ) | p d μ ( α ) .
(iii)
For every x X , the map Ω α f α ( x ) τ α X is weakly measurable.
(iv)
For every x X ,
x = Ω f α ( x ) τ α d μ ( α ) ,
where the integral is weak integral.
Note that condition (i) in Definition 1 says that the map
θ f : X x θ f x L p ( Ω , μ ) ; θ f x : Ω α ( θ f x ) ( α ) f α ( x ) K
is a linear isometry.
Theorem 2 
(Functional Continuous Uncertainty Principle) [4]). (Let ( Ω , μ ) , ( Δ , ν ) be measure spaces. Let ( { f α } α Ω , { τ α } α Ω ) and ( { g β } β Δ , { ω β } β Δ ) be continuous p-Schauder frames for a Banach space X . Then for every x X { 0 } , we have
μ ( supp ( θ f x ) ) 1 p ν ( supp ( θ g x ) ) 1 q 1 sup α Ω , β Δ | f α ( ω β ) | , ν ( supp ( θ g x ) ) 1 p μ ( supp ( θ f x ) ) 1 q 1 sup α Ω , β Δ | g β ( τ α ) | ,
where q is the conjugate index of p.
Corollary 1 
(Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle) [5]). (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 X { 0 } , we have
θ f x 0 1 p θ g x 0 1 q 1 max 1 j n , 1 k m | f j ( ω k ) | and θ g x 0 1 p θ f x 0 1 q 1 max 1 j n , 1 k m | g k ( τ j ) | ,
where q is the conjugate index of p.
In paper [4], it is asked that whether we have version of Theorem 2 for p = 1 and p = . In this paper, we solve the problem for p = 1 .

2. Functional Continuous Uncertainty Principle for Continuous 1-Schauder Frames

We clearly have the following definition from Definition 1.
Definition 2.
Let ( Ω , μ ) be a measure space. Let { τ α } α Ω be a collection in a Banach space X and { f α } α Ω be a collection in X * . The pair ( { f α } α Ω , { τ α } α Ω ) is said to be a continuous 1-Schauder framefor X if the following holds.
i
For every x X , the map Ω α f α ( x ) K is measurable.
ii
For every x X ,
x = Ω | f α ( x ) | d μ ( α ) .
iii
For every x X , the map Ω α f α ( x ) τ α X is weakly measurable.
iv
For every x X ,
x = Ω f α ( x ) τ α d μ ( α ) ,
where the integral is weak integral.
We note that condition (i) in Definition 2 says that the map
θ f : X x θ f x L 1 ( Ω , μ ) ; θ f x : Ω α ( θ f x ) ( α ) f α ( x ) K
is a linear isometry.
Theorem 3
(Functional Continuous Uncertainty Principle for Continuous 1-Schauder Frames).Let ( Ω , μ ) , ( Δ , ν ) be measure spaces. Let ( { f α } α Ω , { τ α } α Ω ) and ( { g β } β Δ , { ω β } β Δ ) be continuous 1-Schauder frames for a Banach space X . Then for every x X { 0 } , we have
μ ( supp ( θ f x ) ) 1 sup α Ω , β Δ | f α ( ω β ) | , ν ( supp ( θ g x ) ) 1 sup α Ω , β Δ | g β ( τ α ) | .
In particular,
μ ( supp ( θ f x ) ) ν ( supp ( θ g x ) ) 1 sup α Ω , β Δ | f α ( ω β ) | sup α Ω , β Δ | g β ( τ α ) |
and
μ ( supp ( θ f x ) ) + ν ( supp ( θ g x ) ) 1 sup α Ω , β Δ | f α ( ω β ) | + 1 sup α Ω , β Δ | g β ( τ α ) | .
Proof. 
Let x X { 0 } . First using θ f is an isometry and later using θ g is an isometry, we get
x = θ f x = Ω | f α ( x ) | d μ ( α ) = supp ( θ f x ) | f α ( x ) | d μ ( α ) = supp ( θ f x ) f α Δ g β ( x ) ω β d ν ( β ) d μ ( α ) = supp ( θ f x ) Δ g β ( x ) f α ( ω β ) d ν ( β ) d μ ( α ) = supp ( θ f x ) supp ( θ g x ) g β ( x ) f α ( ω β ) d ν ( β ) d μ ( α ) supp ( θ f x ) supp ( θ g x ) | g β ( x ) f α ( ω β ) | d ν ( β ) d μ ( α ) sup α Ω , β Δ | f α ( ω β ) | supp ( θ f x ) supp ( θ g x ) | g β ( x ) | d ν ( β ) d μ ( α ) = sup α Ω , β Δ | f α ( ω β ) | μ ( supp ( θ f x ) ) supp ( θ g x ) | g β ( x ) | d ν ( β ) = sup α Ω , β Δ | f α ( ω β ) | μ ( supp ( θ f x ) ) θ g x = sup α Ω , β Δ | f α ( ω β ) | μ ( supp ( θ f x ) ) x .
Therefore
1 sup α Ω , β Δ | f α ( ω β ) | μ ( supp ( θ f x ) ) .
On the other way, first using θ g is an isometry and θ f is an isometry, we get
x = θ g x = Δ | g β ( x ) | d ν ( β ) = supp ( θ g x ) | g β ( x ) | d ν ( β ) = supp ( θ g x ) g β Ω f α ( x ) τ α d μ ( α ) d ν ( β ) = supp ( θ g x ) Ω f α ( x ) g β ( τ α ) d μ ( α ) d ν ( β ) = supp ( θ g x ) supp ( θ f x ) f α ( x ) g β ( τ α ) d μ ( α ) d ν ( β ) supp ( θ g x ) supp ( θ f x ) | f α ( x ) g β ( τ α ) | d μ ( α ) d ν ( β ) sup α Ω , β Δ | g β ( τ α ) | supp ( θ g x ) supp ( θ f x ) | f α ( x ) | d μ ( α ) d ν ( β ) = sup α Ω , β Δ | g β ( τ α ) | ν ( supp ( θ g x ) ) supp ( θ f x ) | f α ( x ) | d μ ( α ) = sup α Ω , β Δ | g β ( τ α ) | ν ( supp ( θ g x ) ) θ f x = sup α Ω , β Δ | g β ( τ α ) | ν ( supp ( θ g x ) ) x .
Therefore
1 sup α Ω , β Δ | g β ( τ α ) | ν ( supp ( θ g x ) ) .
Corollary 2
(Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle for 1-Schauder Frames). Let ( { f j } j = 1 n , { τ j } j = 1 n ) and ( { g k } k = 1 m , { ω k } k = 1 m ) be 1-Schauder frames for a finite dimensional Banach space X . Then for every x X { 0 } , we have
θ f x 0 1 max 1 j n , 1 k m | f j ( ω k ) | and θ g x 0 1 max 1 j n , 1 k m | g k ( τ j ) | .
In particular,
θ f x 0 θ g x 0 1 max 1 j n , 1 k m | f j ( ω k ) | max 1 j n , 1 k m | g k ( τ j ) |
and
θ f x 0 + θ g x 0 1 max 1 j n , 1 k m | f j ( ω k ) | + 1 max 1 j n , 1 k m | g k ( τ j ) | .
Remark 1.
We note that the L 1 -norm uncertainty principles derived in [1,6,8] differ from our result.

References

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