Uncertainty principle for vector-valued functions

1.   Introduction

The uncertainty principle is one of the most famous theories of quantum mechanics. Gabor's work ^[10] is considered to be the foundation for the study of uncertainty principle in the area of signal analysis. Since then, rich forms of the uncertainty principle have appeared in mathematical forms. In ^{[1,2,3,4,6,7,16,17]}, the uncertainty principle for signal functions defined on the real line, on the circle, on the Euclidean space, and on the sphere, etc., were intensively studied. In ^[4], the so called phase and amplitude derivatives of signal function are defined in the sense of the Fourier transform, so it provides us a method for studying the uncertainty principle based on the Fourier transform of signal functions. In ^[5,13], the theory of uncertainty principle is generalized from the complex domain to the hypercomplex domain using quaternion algebras, associated with the quaternion Fourier transform. The algorithm in ^[5] provides us a method to estimate the probability by the data and predict whether the missing signals can be recovered. In ^[8,15], the uncertainty principle for doubly periodic signal functions was studied. A doubly periodic signal function is regarded as a function of $L^2({\mathbb{R}}^2)$. The uncertainty principle for signal functions belonging to $L^{2}(\mathbb{R}^{n})$ was studied in ^[11], and the result is the following theorem:

Theorem 1.1. ^[11] Let $h\in L^{2}(\mathbb{R}^{n})$ and let $\hat h$ be the Fourier transformation of $h$. Then for any $x_{0}, \xi_{0}\in \mathbb{R}^{n}$

The equality holds if and only if $h(x) = ce^{2\pi ix\cdot\xi_{0}}e^{-\alpha|x-x_{0}|^{2}/2}$, where $\alpha > 0$ and $c\in \mathbb{C}$.

Here, $\hat{h}$, the Fourier transform of $h$, is given by

Let $t = (t_{1}, t_{2}, \cdots, t_{n})\in \mathbb{R}^{n}$, and let $f_j \in L^2({\mathbb{R}}^n)$, $j = 1, \dots, m$. Denote by $f = (f_{1}, \cdots, f_{m})$ a vector-valued function from ${\mathbb{R}}^n$ to ${\mathbb{R}}^m$. Thus, $f\in L^2({\mathbb{R}}^n, {\mathbb{R}}^m)$, and the norm of $f$ can be written as

Since each entry $f_j$ of $f$ is in $L^2({\mathbb{R}}^n)$, the Fourier transform of $f_j$ is well defined. As a consequence, the Fourier transform of $f$ is denoted by

where $\hat{f_{j}}$ are given by (1.2). In the following definition, we define several concepts which are frequently used in the rest of this paper.

Definition 1.2. Let $f \in L^{2}(\mathbb{R}^{n}, {\mathbb{R}}^m)$. The mean of time $t$ and of Fourier frequency $\omega$ is defined by

and by

Let $\langle t_{k}\rangle = \sum^{m}_{j = 1}\int_{\mathbb{R}^{n}}t_{k}|f_{j}(t)|^{2}dt$ and let $\langle\omega_{k}\rangle = \sum^{m}_{j = 1}\int_{\mathbb{R}^{n}}\omega_{k}|\hat{f_{j}}(\omega)|^{2}d\omega$ for $k = 1, 2, \cdots, n$. The variance of $t$ and of $\omega$ is defined by

and by

In Theorem 1.1, inequality (1.1) is the uncertainty principle for $f\in L^2({\mathbb{R}}^n, {\mathbb{R}}^m)$ with the special case when $n\ge 2$ and $m = 1$. In our notations, it can be represented as

When $m = 1$, the definitions of the means and the variances about time and about frequency of the signal function $f(t) = \rho(t)e^{i\varphi(t)}\in L^{2}({\mathbb{R}}^n)$ can be found in ^[14]. Meanwhile, the following theorem proposes a stronger form of the uncertainty principle for $f\in L^{2}(\mathbb{R}^{n})$ when $n\ge 2$.

Theorem 1.3. ^[14] Let $f(t) = \rho(t)e^{i\varphi(t)}\in L^{2}(\mathbb{R}^{n})$ with $\|f\|_{2} = 1$. Suppose that the gradients $\nabla\rho$, $\nabla\varphi$, and $\nabla f$ all exist and that $\frac{\partial f}{\partial t_{k}}, \; t_{k} f\in L^{2}(\mathbb{R}^{n})$ for $k = 1, 2, \cdots, n$. Then

If $\frac{\partial\varphi}{\partial t_{k}}$ are continuous and $\rho(t)\not = 0$ almost everywhere, then the equality of (1.4) holds if and only if $f$ is in one of the following $2^{n}$ forms

where $\lambda_{1} > 0$, $\lambda_{2} > 0$, $\ell_{k}\in\mathbb{N}_{+}$, and $d_{1}$, $\lambda_{1}$ satisfy the equation $d_{1}^{\frac{2}{n}}\sqrt{\frac{\pi}{\lambda_{1}}} = 1$.

In this paper, we propose a form of uncertainty principle for $f\in L^2({\mathbb{R}}^n, {\mathbb{R}}^m)$ with $n, m\ge 2$:

It is straightforward to verify that (1.5) reduces to (1.4) if $m = 1$. Hence, (1.5) can be regarded as an appropriate generalization of (1.4) into the case of vector-valued functions.

In Theorem 1.2, it is required that $\nabla\rho$, $\nabla\varphi$, and $\nabla f$ all exist. However, general signal functions do not have such good properties. We establish Fourier partial derivatives of $f$ and prove a form of the uncertainty principle, which is

where $D_k \varphi(t)$ are properly defined in our proof. Easy verification shows that (1.6) reduces to (1.4) if $\nabla\rho$, $\nabla\varphi$, and $\nabla f$ all exist. Therefore, (1.6) can be regard as a generalization of (1.4).

This paper is organized as follows. In Section 2, we prove a form of uncertainty principle for vector-valued signal functions $f\in L^2({\mathbb{R}}^n, {\mathbb{R}}^m)$ with conditions that the classical first order partial derivatives of $f_j$, $\rho_j$, and $\varphi_j$ exist at all points, and that $\partial f_j/ \partial t_k, t_k f_j\in L^2({\mathbb{R}}^n)$ for $j = 1, \dots, m$ and $k = 1, \dots, n$. This result generalizes the uncertainty principle obtained in ^[14]. In Section 3, we assume that $t_jf(t), \omega_j\hat f(\omega)\in L^2({\mathbb{R}}^n)$, $j = 1, \dots, n$ for signal functions $f\in L^2({\mathbb{R}}^n)$. The Fourier phase and amplitude derivatives $\nabla_{\mathscr F} \varphi(t)$ and $\nabla_{\mathscr F}\rho(t)$ are properly defined. We prove a form of the uncertainty principle based on these Fourier transform derivatives, which also generalizes the result in ^[14].

2.   Uncertainty principle for vector-valued functions

In this section, we study uncertainty principle for functions $f\in L^2({\mathbb{R}}^n, {\mathbb{R}}^m)$. Each entry of $f = (f_{1}, f_{2}, \cdots, f_{m})$ can be written as $f_j(t) = \rho_j(t) e^{i \varphi_j(t)}, \; j = 1, \dots, m$. We assume that the classical first order partial derivatives of $f_j$, $\rho_j$, and $\varphi_j$ exist at all points, and that $\partial f_j/ \partial t_k, t_k f_j\in L^2({\mathbb{R}}^n)$ for $j$$=$$1, \dots, m$ and $k = 1, \dots, n$. The main theorem of this section is Theorem 2.1. We will use the relation

in the proof of the main theorem. Equality (2.1) holds because of

Here, we have used the fact that if $f_j\in L^2({\mathbb{R}}^n)$, then $\rho_j = |f_j|\to 0$ when $|t|\to \infty$.

Lemma 2.1. Let $f = (f_{1}, f_{2}, \cdots, f_{m})\in L^2({\mathbb{R}}^n, {\mathbb{R}}^m)$ and write $f_{j}(t) = \rho_{j}(t)e^{\varphi_{j}(t)}, \ j = 1, 2, \cdots, m$. Suppose that $\langle\omega\rangle = 0$ and that $\nabla \rho_{j}(t)$, $\nabla\varphi_{j}(t)$, and $\nabla f_{j}(t)$ all exist and $\frac{\partial f_{j}(t)}{\partial t_{k}}\in L^{2}(\mathbb{R}^{n})$ for $j = 1, 2, \cdots, m,$ $k = 1, 2, \cdots, n$. Then,

Proof. By the definition of $\sigma^{2}_{\omega}$ and by the assumption of $\langle \omega\rangle = 0$, it follows that

This completes the proof. □

The following lemma proves a form of uncertainty for $f\in L^2({\mathbb{R}}^n, {\mathbb{R}}^m)$ with the extra assumption that $\langle t\rangle = 0$ and $\langle\omega\rangle = 0$. The main result of Theorem 2.1 can be derived easily from the following lemma.

Lemma 2.2. Let $f = (f_{1}, f_{2}, \cdots, f_{m})\in L^2({\mathbb{R}}^n, {\mathbb{R}}^m)$ with $\|f\|_2 = 1$, and write $f_{j}(t) = \rho_{j}(t)e^{\varphi_{j}(t)}$, $j$$=$$1, 2, \cdots, m$. Suppose that $\langle t\rangle = 0$, $\langle\omega\rangle = 0$, $\nabla\rho_{j}(t)$, $\nabla\varphi_{j}(t)$, and $\nabla f_{j}(t)$ all exist, and that $\frac{\partial f_{j}(t)}{\partial t_{k}}, t_{k}f_{j}(t)\in L^{2}(\mathbb{R}^{n})$, $j = 1, 2, \cdots, m, \; k = 1, 2, \cdots, n$. Then

If $\frac{ \partial \varphi_j}{ \partial t_{k}}$ are continuous and $\rho_j \not = 0$ almost everywhere, then the equality of (2.2) holds if and only if

Here, $\lambda_{j_{1}}, \lambda_{j_{2}}$, and $\{d_j\}_{j = 1}^m$ are positive real numbers, while $\{\ell_k \}_{k = 1}^n$ are positive integers.

Proof. By Lemma 2.1, it follows that

So, in order to prove (2.2), we could prove two separate inequalities. The first inequality can be proved as follows:

Equality (2.1) shows that

Thus,

The second inequality holds because of

By (2.5) and (2.6), inequality (2.2) follows.

Next, we discuss the conditions such that the equality of (2.2) holds. The second inequality of (2.4) is going to be an equality if and only if there exists $\lambda_1\in {\mathbb{R}}$ with $\lambda_1 > 0$ such that

for all $t\in {\mathbb{R}}^n$. The first inequality of (2.4) is going to be an equality if and only if either

or

is true. If the first one is true, it follows that

Obviously, the function $\rho_j(t) = d_j e^{ \lambda_1 |t|^2/2}$ is not in $L^2({\mathbb{R}}^n)$. Therefore, we must have

and then

Here, $d_j$ are positive real numbers.

The equality of (2.6) holds if and only if there exists $\lambda_2\in {\mathbb{R}}$ with $\lambda_2 > 0$ such that

for all $t\in {\mathbb{R}}^n$. Since $\rho_j(t)\not = 0$ almost everywhere and since $\frac{ \partial \varphi_j}{ \partial t_k}$ are continuous for $\; k = 1, \dots, n,$ $j = 1, \dots, m$, we have

When $k = 1$, we have

Thus,

When $k = 2$, we have

Plugging (2.7) into (2.8) implies that

and then

Continue this process, when $k = n$, we have

where $\ell_1, \dots, \ell_n$ are positive integers. Combining the formulas of $\rho_j$ we have obtained, then

Therefore, the equality of (2.2) holds if and only if every $f_{j}(t)$ is in one of the forms of (2.3). This completes the proof. □

By Lemma 2.2, we can prove a form of uncertainty principle for $f\in L^2({\mathbb{R}}^n, {\mathbb{R}}^m)$ without the assumption that $\langle t\rangle = 0$ and $\langle\omega\rangle = 0$, which is the main result of this section. In the proof of the following theorem, the notations $\rho^g$, $\varphi^g$, $\langle t\rangle_g$, $\langle \omega \rangle_g$, $\sigma_{t}^g$, and $\sigma_{\omega}^g$ represent the corresponding notation with respect to the function $g$.

Theorem 2.1. Let $f = (f_{1}, f_{2}, \cdots, f_{m})\in L^2({\mathbb{R}}^n, {\mathbb{R}}^m)$ with $\|f\|_2 = 1$, and write $f_{j}(t) = \rho_{j}(t)e^{\varphi_{j}(t)}$, $j$$=$$1, 2, \cdots, m$. Suppose that $\nabla\rho_{j}(t)$, $\nabla\varphi_{j}(t)$, and $\nabla f_{j}(t)$ all exist and $\frac{\partial f_{j}(t)}{\partial t_{k}}, t_{k}f_{j}(t)\in L^{2}(\mathbb{R}^{n})$, $j$$=$$1, 2, \cdots, m,$ $k = 1, 2, \cdots, n$. Then

If $\frac{ \partial \varphi_j}{ \partial t_{k}}$ are continuous and $\rho_j \not = 0$ almost everywhere, then the equality of (2.9) holds if and only if

Here, $\lambda_{j_{1}}, \lambda_{j_{2}}$, and $\{d_j\}_{j = 1}^m$ are positive real numbers, while $\{\ell_k \}_{k = 1}^n$ are positive integers.

Proof. Now the quantities $\langle t\rangle$ and $\langle \omega\rangle$ are not $0$. Let

for $j = 1, 2, \cdots, m$. Then $g = (g_1, \dots, g_m)\in L^2({\mathbb{R}}^n, {\mathbb{R}}^m)$. The mean of time $t$ of the signal $g$ is

Also, it is straightforward to obtain that

and then

Therefore, the vector-valued function $g$ satisfies the conditions of Lemma 2.2. Then we have

Because of

and

we have $(\sigma_{t}^g)^{2} = \sigma_t^2$ and $(\sigma_{\omega}^g)^{2} = \sigma_ \omega^2$. Also, since

and

we obtain (2.9), which is

The equality of (2.9) holds if and only if the equality of (2.10) holds. By Lemma 2.2, the equality of (2.10) holds if and only if

By the relationship between $f$ and $g$, i.e.,

the equality of (2.9) holds if and only if

This completes the proof. □

3.   Fourier gradient and uncertainty principle

In this section, we go back to the situation of $m = 1$. The so called Fourier phase and amplitude derivatives of $f\in L^2({\mathbb{R}}^n)$ are defined. Because, in general, the signal functions may not have ideal smoothness conditions, such as that $\nabla\rho$, $\nabla\varphi$, and $\nabla f$ all exist, which are assumed in Theorem 1.3. Lemma 3.1 guarantees that the Fourier transform derivative of $f\in L^2({\mathbb{R}})$ is valid once $tf(t), \; \omega \hat f(\omega)\in L^2({\mathbb{R}})$. This lemma is also fundamental for Definition 3.2.

Lemma 3.1. ^[4] Assume that $f(t)$, $tf(t)$, and $\omega \hat f(\omega)\in L^2({\mathbb{R}})$. Then $\hat f\in L^1({\mathbb{R}})$, and $f(t)$ is almost everywhere equal to a function in $C_0({\mathbb{R}})$. Moreover, there exists the Fourier transform derivative $(Df)(t)\in L^2({\mathbb{R}})$ of $f$ such that $(Df)^\wedge(\omega) = i \omega \hat f(\omega)\in L^2({\mathbb{R}})$ and

Therefore,

holds almost everywhere on ${\mathbb{R}}$. If, in particular, $f$ has classical derivatives $f'$ almost everywhere on ${\mathbb{R}}$, then $(Df)(t) = f'$ almost everywhere on ${\mathbb{R}}$.

It is worth noting that the definition of the Fourier transform in ^[4] is slightly different from our definition. They define the Fourier transform of $f$ to be

However, we define the Fourier transform by (1.2). Under our definition, $\widehat{f'}(\omega) = 2\pi i \omega \hat f(\omega)$, and then the definition of Fourier derivative should be slightly changed. It should be a function $(Df)(t)\in L^2({\mathbb{R}})$ such that $(Df)^\wedge(\omega) = 2\pi i \omega \hat f(\omega)$. Now we can introduce our definition of the Fourier partial derivative.

Definition 3.2. Let $f\in L^2({\mathbb{R}}^n)$. If $t_jf(t), \; \omega_j \hat f(\omega)\in L^2({\mathbb{R}}^n)$, $j = 1, \dots, n$, and denote $g_j(\omega):$$=$$2\pi i \omega_j \hat f (\omega)$, $j = 1, \dots, n$. Then the Fourier transform partial derivative of $f$ with respect to $t_j$ is defined by

Here, $\mathscr{F}^{-1}$ is the inverse Fourier transform operator.

Call $\nabla_\mathscr F: = (D_1, D_2, \cdots, D_n)$ the Fourier gradient operator, and we have

Definition 3.3. Let $f(t)\in L^2({\mathbb{R}}^n)$. Suppose that $\omega_j\hat f(\omega)\in L^2({\mathbb{R}}^n)$, $j = 1, \dots, n$. Rewrite $f(t) = \rho(t)e^{i \varphi(t)}$. The Fourier transform phase and amplitude derivatives are defined to be

and

Here, $j = 1, \dots, n$.

The following lemma proves that $f(t)$ is identical with an absolutely continuous function almost everywhere. It is crucial in one proof, which concerns the uncertainty principle studied by the Fourier transform of ^[4].

Lemma 3.4. ^[4] Assume that $1\le p_1\le 2, \; 1\le p_2\le 2, \; f(t)\in L^{p_1}({\mathbb{R}}),$ and $h(\omega) = i \omega\hat f(\omega)\in L^{p_2}({\mathbb{R}})$. Let

where $a$ is a Lebesgue point of $f$. Then $f(t)$ is identical almost everywhere with the absolutely continuous function $g(t)$, and

The following lemma generalizes Lemma 3.4 to a higher dimensional case.

Lemma 3.5. Assume that $f(t)\in L^2({\mathbb{R}}^n),$ and $h_j(\omega) = i \omega_j\hat f(\omega)\in L^2({\mathbb{R}}^n), \; j = 1, \dots, n$. Let

where $a$ is a Lebesgue point of $f$. Then $f(t)$ is identical almost everywhere with $g(t)$, and

Moreover, $g$ is absolutely continuous in each argument.

Proof. Let $a = (a_1, \dots, a_n)$ be a Lebesgue point of $f$. It can be observed that

By Lemma 3.4, $f(a_1, \dots, t_j, \dots, a_n)$ is identical almost everywhere with the absolutely continuous function $g(a_1, \dots, t_j, \dots, a_n)$, and

Then we have $g(a) = f(a)$ and

Since the points of ${\mathbb{R}}^n$ are almost everywhere Lebesgue points of $f$, we conclude that $g(t) = f(t)$ almost everywhere; meanwhile,

This completes the proof. □

In the following lemma, the variance of $\omega$, which is $\sigma^2_ \omega$, of a signal function is represented by its Fourier phase and amplitude derivatives.

Lemma 3.6. Assume that $f(t)\in L^2({\mathbb{R}}^n),$ and $h_j(\omega) = i \omega_j\hat f(\omega)\in L^2({\mathbb{R}}^n), \; j = 1, \dots, n$. Then

Proof. Since $f(t) \in L^2({\mathbb{R}}^n)$ and $\omega_j\hat f(\omega)\in L^2({\mathbb{R}}^n), \; j = 1, \dots, n$, $\sigma^2_ \omega$ is well defined. By the definition of $\sigma^{2}_{\omega}$, we know that

For each fixed $k$, we obtain that

The last equality follows from the Plancherel theorem, which states that $\|\hat{f}\|_{L^2} = \| f\|_{L^2}$, see [12, p.156] for details. Also,

where $E: = \{t\in {\mathbb{R}}^n:f(t) = 0\}$. Then we have

Because of

and

It follows that

Therefore we obtain that

Then, we have finished the proof. □

Theorem 3.1. Let $f\in L^2({\mathbb{R}}^n)$ with $\|f\|_2 = 1$. Suppose that $t_jf(t), \; \omega_j\hat f(\omega)\in L^2({\mathbb{R}}^n)$, $j = 1, \dots, n$. Write $f(t) = \rho(t)e^{i \varphi (t)}$, then

Under the extra assumptions that $f(t) = \rho(t)e^{i \varphi(t)}$ has the classical partial derivatives $\frac{ \partial f}{ \partial t_j}$, $\frac{ \partial \varphi}{ \partial t_j}$, $\frac{ \partial \rho}{ \partial t_j}$ for $j = 1, \dots, n$, where $\frac{ \partial \varphi}{ \partial t_j}$ are continuous and $\rho$ is non-zero almost everywhere, then the equality of (3.1) is attained if and only if $f(t)$ has one of the following $2^{n}$ forms:

where $\lambda_{1} > 0$, $\lambda_{2} > 0$, $\ell_{k}\in\mathbb{N}_{+}$, and $d_{1}$, $\lambda_{1}$ satisfy equation $d_{1}^{\frac{2}{n}}\sqrt{\frac{\pi}{\lambda_{1}}} = 1.$

Proof. By Definition 1.2, the variance of time is

Here, in our considering section, $m = 1$. Lemma 3.6 provides that

In order to prove inequality (3.1), it suffices to prove two separate inequalities. The first one is

and the second one is

It is obvious that (3.2) is equivalent to (3.4)

and that (3.3) is equivalent to (3.5)

Now, we prove (3.4). By Lemma 3.5, we may assume that $g(t)$ is a function that is equal to $f(t)$ almost everywhere and is absolutely continuous in each argument. Let $M_n$ and $N_n$ be two particular sequences of numbers tending to infinity as $n\to \infty$. In the following computation, let $(t', t_k)$ represents the tuple $(t_1, ..., t_k, ..., t_n)$. Then we have

where $E = \{t\in {\mathbb{R}}^n: f(t) = 0\}$.

By Hölder's inequality of vector-valued functions ^[9], it implies that

Thus we proved (3.5). Therefore, inequality (3.1) holds.

It is not hard to verify that the $2^n$ types of functions in the statement of the theorem make (3.1) equalities. When we consider the necessity of the $2^n$ types, we assumed that the classical partial derivatives $\frac{ \partial f}{ \partial t_j}$, $\frac{ \partial \varphi}{ \partial t_j}$, $\frac{ \partial \rho}{ \partial t_j}$ for $j = 1, \dots, n$ all exist, $\frac{ \partial \varphi}{ \partial t_j}$ are continuous, and $\rho$ is almost everywhere non-zero. In this case,

The same proof as in that of Theorem 1.3 is valid. □

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by NSFC (No.12101453) and by the Science & Technology Development Fund of Tianjin Education Commission for Higher Education (No.2019KJ140).

Conflict of interest

The authors declared that they have no conflicts of interest to this work.