Mean-Dispersion Principles and the Wigner Transform

Abstract

Given a function $f\in L^2(\mathbb {R})$, we consider means and variances associated to f and its Fourier transform $\hat{f}$, and explore their relations with the Wigner transform W(f), obtaining, as particular cases, a simple new proof of Shapiro’s mean-dispersion principle, as well as a stronger result due to Jaming and Powell. Uncertainty principles for orthonormal sequences in $L^2(\mathbb {R})$ involving linear partial differential operators with polynomial coefficients and the Wigner distribution, or different Cohen class representations, are obtained, and an extension to the case of Riesz bases is studied.

1 Introduction

This paper treats uncertainty principles for families of orthonormal functions in $L^2(\mathbb {R})$ in connection with time-frequency analysis. When talking about uncertainty principles, in harmonic analysis, one refers to a class of theorems giving limitations on how much a function and its Fourier transform can be both localized at the same time. Different meanings of the word “localized” give rise to different uncertainty principles. For instance, referring to the most classical results (see [12] for a survey), in the Heisenberg uncertainty principle the localization of f and its Fourier transform $\hat{f}$ has to do with their associated variances, in Benedicks [1] it has to do with the measure of their supports, in Donoho-Stark [10] with the concept of $\varepsilon$-concentration, in Hardy [15] with (exponential) decay at infinity, and so on. Over the years many other authors studied uncertainty principles involving various time-frequency distributions (see, for instance, [2, 5, 9, 11, 13, 14]). There are, moreover, uncertainty principles giving not only limitations on the localization of a single function and its Fourier transform, but on how such limitations behave, becoming stronger and stronger, when adding more and more elements of an orthonormal system in $L^2$. In this paper we focus in particular on results of this type involving means and variances. For $f\in L^2(\mathbb {R})$ we define the associated mean

$$\begin{aligned} \mu (f):=\frac{1}{\Vert f\Vert ^2}\int _\mathbb {R}t|f(t)|^2dt \end{aligned}$$

(1.1)

and the associated variance

$$\begin{aligned} \Delta ^2(f):=\frac{1}{\Vert f\Vert ^2}\int _\mathbb {R}|t-\mu (f)|^2|f(t)|^2dt; \end{aligned}$$

(1.2)

observe that, for $\Vert f\Vert _2=1$, such quantities are the mean and the variance of $|f|^2$. The dispersion associated with f is $\Delta (f):=\sqrt{\Delta ^2(f)}$.

Our aim is to study uncertainty principles of mean-dispersion type involving quadratic time-frequency representations applied to the elements of an orthonormal system in $L^2(\mathbb {R})$. In order to state our main results we need some basic definitions. The classical cross-Wigner distribution is defined as

$$\begin{aligned} W(f,g)(x,\xi )=\frac{1}{\sqrt{2\pi }}\int _\mathbb {R}f\left( x+\frac{t}{2}\right) \overline{g\left( x-\frac{t}{2}\right) } e^{-it\xi }dt, \quad f,g\in L^2(\mathbb {R}),\nonumber \\ \end{aligned}$$

(1.3)

and we set for convenience $W(f):=W(f,f)$. Let moreover $\hat{L}$ be the linear partial differential operator in $\mathbb {R}^2$ defined as

$$\begin{aligned} \hat{L}:=\left( \frac{1}{2} D_\xi +x\right) ^2+\left( \frac{1}{2} D_x-\xi \right) ^2. \end{aligned}$$

(1.4)

The following result (that we prove in Theorem 4.4 and Corollary 4.5 below) constitutes a Mean-Dispersion uncertainty principle associated to the Wigner transform.

Theorem 1.1

Let $\{ f_k\}_{k\in \mathbb {N}_0}$ be an orthonormal sequence in $L^2(\mathbb {R})$. Then for every $n\ge 0$

$$\begin{aligned} \sum _{k=0}^n \langle \hat{L} W(f_k),W(f_k)\rangle \ge (n+1)^2, \end{aligned}$$

(1.5)

where as usual $\langle \cdot ,\cdot \rangle$ indicates the inner product in $L^2$ (see Sect. 3 for a discussion on the domain of $\hat{L}$ and the corresponding meaning of $\langle \hat{L} W(f_k),W(f_k)\rangle$). Equality in (1.5) holds for every $0\le n\le n_0$, $n_0\in \mathbb {N}_0$, if and only if there exist $c_k\in \mathbb {C}$ with $|c_k|=1$ such that $f_k=c_k h_k$, $k=0,\dots ,n_0$, where $h_k$ are the Hermite functions (1.8).

This result comes motivated by an uncertainty principle for orthonormal sequences due to Shapiro. We shall use throughout the paper the notation $\mathbb {N}_0:=\mathbb {N}\cup \{0\}$, and adopt the following normalization of the Fourier transform:

$$\begin{aligned} \hat{f}(\xi )=\frac{1}{\sqrt{2\pi }}\int _\mathbb {R}f(t)e^{-it\xi }dt, \qquad \xi \in \mathbb {R}. \end{aligned}$$

(1.6)

Theorem 1.2

(Shapiro’s Mean-Dispersion Principle) There does not exist an infinite orthonormal sequence $\{f_k\}_{k\in \mathbb {N}_0}$ in $L^2(\mathbb {R})$ such that all $\mu (f_k)$, $\mu (\hat{f}_k)$, $\Delta (f_k)$, $\Delta (\hat{f}_k)$ are uniformly bounded.

This theorem appeared in an unpublished manuscript of Shapiro from 1991; in [19] a stronger result has been proved, namely, there does not exist an orthonormal basis $\{ f_k\}_{k\in \mathbb {N}_0}$ of $L^2(\mathbb {R})$ such that

$$\begin{aligned} \Delta (f_k),\ \Delta (\hat{f}_k),\ \mu (f_k) \end{aligned}$$

are uniformly bounded, while there exists an orthonormal basis $\{f_k\}_{k\in \mathbb {N}_0}$ of $L^2(\mathbb {R})$ such that

$$\begin{aligned} \mu (f_k),\ \mu (\hat{f}_k),\ \Delta (f_k) \end{aligned}$$

are uniformly bounded (see also [18]). Moreover, the following quantitative version of Shapiro’s Mean-Dispersion Principle is proved in [16].

Theorem 1.3

[16, Theorem 2.3] Let $\{ f_k\}_{k\in \mathbb {N}_0}$ be an orthonormal sequence in $L^2(\mathbb {R})$. Then for every $n\ge 0$

$$\begin{aligned} \sum _{k=0}^n \left( \Delta ^2(f_k)+\Delta ^2(\hat{f}_k) +|\mu (f_k)|^2+|\mu (\hat{f}_k)|^2\right) \ge (n+1)^2. \end{aligned}$$

(1.7)

Equality holds for every $0\le n\le n_0$, $n_0\in \mathbb {N}_0$, if and only if there exist $c_k\in \mathbb {C}$ with $|c_k|=1$ such that $f_k=c_k h_k$ for $k=0,\dots ,n_0$, where $h_k$ are the Hermite functions on $\mathbb {R}$ defined as follows:

$$\begin{aligned} h_k(t)=\frac{1}{(2^kk!\sqrt{\pi })^{1/2}}e^{-t^2/2}H_k(t),\qquad t\in \mathbb {R}, \end{aligned}$$

(1.8)

where $H_k$ is the Hermite polynomial of degree k given by

$$\begin{aligned} H_k(t)=(-1)^ke^{t^2}\frac{d^k}{dt^k}e^{-t^2}, \qquad t\in \mathbb {R}. \end{aligned}$$

Observe that (1.7) differs for a constant from the result in [16], due to a different normalization of the Fourier transform. Theorem 1.2 is an easy consequence of Theorem 1.3; moreover, Theorem 1.3 also says that the limitation on the concentration of $f_k$ and $\hat{f}_k$ become stronger and stronger by adding more and more elements from the orthonormal system, as the lower bound $(n+1)^2$ increases faster than the number of involved functions.

We show that Theorem 1.1 implies Theorem 1.3 (and then also Theorem 1.2), and in this sense it can be interpreted as a Mean-Dispersion principle associated to the Wigner transform. Theorems 1.1 and 1.3 are in fact intimately connected, and this is linked to a deep relation between (1.4) and the Hermite operator; such a relation has many consequences and applications in different fields, see for instance [8] where dispersive estimates of the wave flow for the twisted Laplacian (and the Hermite operator) are investigated, or [3, 6, 22] where it has been used to study global regularity properties of partial differential equations. The advantage of the formulation of Theorem 1.1 on the side of the Wigner transform is twofold. First, the proof is simpler than the one of Theorem 1.3 in [16]. In particular, it does not need the Rayleigh-Ritz technique used there. Moreover, $\hat{L}$ is not the only operator that can be used in (1.5) in order to have Mean-Dispersion principles of the kind of Theorem 1.1. In Sects. 4 and 5 we give more details on this fact. Here, we just point out that we can use instead of $\hat{L}$ the multiplication operator by $x^2+\xi ^2$, obtaining that (see Theorem 5.1 below) if $\{ f_k\}_{k\in \mathbb {N}_0}$ is an orthonormal sequence in $L^2(\mathbb {R})$, then for every $n\ge 0$

$$\begin{aligned} \sum _{k=0}^n \int _{\mathbb {R}^2} (x^2+\xi ^2) |W(f_k)(x,\xi )|^2 dx d\xi \ge \frac{(n+1)^2}{2}, \end{aligned}$$

(1.9)

and equality is characterized as in Theorem 1.1. We show that if $f_k$ satisfies $\mu (f_k)=\mu (\hat{f}_k)=0$, then the quantity

$$\begin{aligned} \int _{\mathbb {R}^2} (x^2+\xi ^2) |W(f_k)(x,\xi )|^2 dx d\xi \end{aligned}$$

is the trace of the covariance matrix of $|W(f_k)(x,\xi )|^2$; then, comparing (1.9) with (1.7) (in the case $\mu (f_k)=\mu (\hat{f}_k)=0$) we observe that we have replaced the two variances associated with $f_k$ and $\hat{f}_k$ in (1.7), with (a constant times) the trace of the covariance matrix associated with $W(f_k)$, which reflects the fact that $W(f_k)$ includes at the same time both information on $f_k$ and on $\hat{f}_k$.

Other extensions of Theorem 1.1 are also studied. Since there are many different time-frequency representations besides the classical Wigner, we consider the so-called Cohen class, given by all the representations Q(f, g) of the form

$$\begin{aligned} Q(f,g)=\frac{1}{\sqrt{2\pi }}\,\sigma * W(f,g),\qquad \sigma \in \mathcal {S}'(\mathbb {R}^2),\ f,g\in \mathcal {S}(\mathbb {R}); \end{aligned}$$

(1.10)

such class contains all the most used time-frequency representations. A natural question is if in Theorem 1.1 one can substitute $W(f_k)$ with $Q(f_k):=Q(f_k,f_k)$, and which operators can be considered instead of $\hat{L}$. We prove in Sect. 6 that for a suitable class of kernels $\sigma$ in (1.10) a result of the kind of Theorem 1.1 can be formulated for representations Q in the Cohen class. Finally, the Mean-Dispersion principle for the Wigner transform can be extended to Riesz bases instead of orthonormal bases.

The paper is organized as follows. In Sects. 2 and 3 we give basic results on the Wigner transform and on the action of the Wigner transform on Hermite functions. In Sect. 4 we prove Theorem 1.1. Section 5 is devoted to the study of the case of the covariance matrix associated with $W(f_k)$ and to the proof of (1.9). In Sects. 6 and 7 we extend the results to the Cohen class and Riesz bases.

2 The Wigner Distribution

Besides the classical cross-Wigner distribution W(f, g) for $f,g\in L^2(\mathbb {R})$ defined in (1.3) we also consider the following Wigner-like transform introduced in [6]

$$\begin{aligned} \textrm{Wig}[u](x,\xi )=\frac{1}{\sqrt{2\pi }}\int _\mathbb {R}u \left( x+\frac{t}{2},x-\frac{t}{2}\right) e^{-it\xi }dt, \qquad u\in L^2(\mathbb {R}^2), \end{aligned}$$

with standard extensions to $f,g\in \mathcal {S}'(\mathbb {R})$ and $u\in \mathcal {S}'(\mathbb {R}^2)$. Such operators are strictly related since

$$\begin{aligned} W(f,g)=\textrm{Wig}[f\otimes \bar{g}]. \end{aligned}$$

However, the second one has the advantage, with respect to the classical Wigner transform, that

$$\begin{aligned}{} & {} \textrm{Wig}:\ \mathcal {S}(\mathbb {R}^2) \longrightarrow \mathcal {S}(\mathbb {R}^2)\\{} & {} \textrm{Wig}:\ \mathcal {S}'(\mathbb {R}^2) \longrightarrow \mathcal {S}'(\mathbb {R}^2) \end{aligned}$$

is a linear invertible operator, being composition of a linear invertible change of variables and a partial Fourier transform. Indeed, denoting by $\mathcal {F}(f)(\xi )=\hat{f}(\xi )$ the classical Fourier transform (1.6), by

$$\begin{aligned} \mathcal {F}_2(u)(x,\xi )=\frac{1}{\sqrt{2\pi }}\int _\mathbb {R}u(x,t)e^{-it\xi }dt,\qquad (x,\xi )\in \mathbb {R}^2, \end{aligned}$$

the partial Fourier transform with respect to the second variable, and by

$$\begin{aligned} \tau _s u(x,t)=u\left( x+\frac{t}{2},x-\frac{t}{2}\right) , \end{aligned}$$

we have that

$$\begin{aligned} \textrm{Wig}[u]=\mathcal {F}_2\tau _s u. \end{aligned}$$

The inverses of the operators above are

$$\begin{aligned} \mathcal {F}^{-1}(F)(x)=\frac{1}{\sqrt{2\pi }}\int _\mathbb {R}F(\xi )e^{ix\xi }d\xi \end{aligned}$$

and

$$\begin{aligned} \tau _s^{-1}F(x,t)=F\left( \frac{x+t}{2},x-t\right) . \end{aligned}$$

Moreover, denoting by

$$\begin{aligned}{} & {} M_1u(x,y)=xu(x,y),\quad M_2u(x,y)=yu(x,y),\\{} & {} D_1u(x,y)=D_xu(x,y),\quad D_2u(x,y)=D_yu(x,y), \end{aligned}$$

for $D_x=-i\partial _x$ and $D_y=-i\partial _y$, a straightforward computation (see also [6]) shows that

$$\begin{aligned}{} & {} D_1\textrm{Wig}[u]=\textrm{Wig}[(D_1+D_2)u] \end{aligned}$$

(2.1)

$$\begin{aligned}{} & {} D_2\textrm{Wig}[u]=\textrm{Wig}[(M_2-M_1)u]\end{aligned}$$

(2.2)

$$\begin{aligned}{} & {} M_1\textrm{Wig}[u]=\textrm{Wig}\left[ \frac{1}{2}(M_1+M_2)u\right] \end{aligned}$$

(2.3)

$$\begin{aligned}{} & {} M_2\textrm{Wig}[u]=\textrm{Wig}\left[ \frac{1}{2}(D_1-D_2)u\right] \end{aligned}$$

(2.4)

for all $u\in \mathcal {S}(\mathbb {R}^2)$.

We write M and D for the multiplication and differentiation operators when just one variable is involved, so for $u\in \mathcal {S}(\mathbb {R})$

$$\begin{aligned} Mu(t)=tu(t),\quad Du(t)=-iu'(t). \end{aligned}$$

Moreover, we also adopt, for convenience, the following notations. First, we write $\langle \cdot ,\cdot \rangle$ to indicate both the inner product in $L^2$, the duality $\mathcal {S}'$-$\mathcal {S}$ (we consider here distributions as conjugate-linear functionals), and in general the integral

$$\begin{aligned} \langle g,h\rangle =\int _\mathbb {R} g(t)\overline{h(t)}\,dt \end{aligned}$$

each time such integral is finite, even though g, h are not $L^2$ functions (in particular, we use $\langle \cdot ,\cdot \rangle$ for the duality $L^2_v-L^2_{1/v}$ between weighted $L^2$-spaces with weight v). Second, we write

$$\begin{aligned} \langle D^{n}f, D^{m}g\rangle \end{aligned}$$

(2.5)

for the integral

$$\begin{aligned} \int _\mathbb {R} \xi ^{n+m}\hat{f}(\xi )\overline{\hat{g}(\xi )}\,d\xi \end{aligned}$$

(2.6)

when the last one makes sense and is finite. It coincides with

$$\begin{aligned} \int _\mathbb {R} D^nf(t)\overline{D^mg(t)}\,dt \end{aligned}$$

if $D^nf, D^mg\in L^2$ by Parseval’s formula

$$\begin{aligned} \langle f, g\rangle =\langle \hat{f},\hat{g}\rangle , \qquad \forall f,g\in L^2(\mathbb {R}). \end{aligned}$$

(2.7)

We use the symbol $\langle \cdot ,\cdot \rangle$ with analogous meaning in dimension greater than 1.

With this notation, formulas (2.1)-(2.4) hold also for $u\in \mathcal {S}'(\mathbb {R}^2)$. Let us prove, for instance, (2.1). Since it’s valid in $\mathcal {S}(\mathbb {R}^2)$, then for all $u,\varphi \in \mathcal {S}(\mathbb {R}^2)$:

$$\begin{aligned} \langle D_1\textrm{Wig}[u],\varphi \rangle= & {} \langle \textrm{Wig}[(D_1+D_2)u],\varphi \rangle =\langle \mathcal {F}_2\tau _s(D_1+D_2)u,\varphi \rangle \nonumber \\= & {} \langle \tau _s(D_1+D_2)u,\mathcal {F}_2^{-1}\varphi \rangle =\langle (D_1+D_2)u,\tau _s^{-1}\mathcal {F}_2^{-1}\varphi \rangle \nonumber \\= & {} \langle u,(D_1+D_2)(\tau _s^{-1}\mathcal {F}_2^{-1}\varphi )\rangle \end{aligned}$$

(2.8)

by Parseval’s formula and

$$\begin{aligned} \langle \tau _s u,\tau _s v\rangle =\langle u, v\rangle , \qquad \forall u,v\in L^2(\mathbb {R}^2). \end{aligned}$$

(2.9)

On the other hand, for all $u,\varphi \in \mathcal {S}(\mathbb {R}^2)$,

$$\begin{aligned} \langle D_1\textrm{Wig}[u],\varphi \rangle =\langle \textrm{Wig}[u],D_1\varphi \rangle =\langle \mathcal {F}_2\tau _s u,D_1\varphi \rangle =\langle u, \tau _s^{-1}\mathcal {F}_2^{-1}(D_1\varphi )\rangle , \end{aligned}$$

which yields, together with (2.8),

$$\begin{aligned} \tau _s^{-1}\mathcal {F}_2^{-1}(D_1\varphi )=(D_1+D_2)(\tau _s^{-1}\mathcal {F}_2^{-1}\varphi ). \end{aligned}$$

Therefore, if $u\in \mathcal {S}'(\mathbb {R}^2)$ and $\varphi \in \mathcal {S}(\mathbb {R}^2)$:

$$\begin{aligned} \langle D_1\textrm{Wig}[u],\varphi \rangle= & {} \langle \textrm{Wig}[u],D_1\varphi \rangle =\langle \mathcal {F}_2\tau _s u,D_1\varphi \rangle =\langle u, \tau _s^{-1}\mathcal {F}_2^{-1}(D_1\varphi )\rangle \\= & {} \langle u,(D_1+D_2)(\tau _s^{-1}\mathcal {F}_2^{-1}\varphi )\rangle =\langle (D_1+D_2)u,\tau _s^{-1}\mathcal {F}_2^{-1}\varphi \rangle \\= & {} \langle \textrm{Wig}[(D_1+D_2)u], \varphi \rangle , \end{aligned}$$

so that (2.1) is valid also for $u\in \mathcal {S}'(\mathbb {R}^2)$.

Similarly also (2.2)-(2.4) hold for $u\in \mathcal {S}'(\mathbb {R}^2)$.

More generally, we have the following result (proved in [3] for $u\in \mathcal {S}(\mathbb {R}^2)$):

Proposition 2.1

Let $P(x,y,D_x,D_y)$ be a linear partial differential operator with polynomial coefficients. Then for all $u\in \mathcal {S}'(\mathbb {R}^2)$:

$$\begin{aligned}{} & {} P(M_1,M_2,D_1,D_2)\textrm{Wig}[u]\nonumber \\{} & {} \quad =\textrm{Wig}\left[ P\left( \frac{1}{2}(M_1+M_2), \frac{1}{2}(D_1-D_2),D_1+D_2,M_2-M_1\right) u\right] , \end{aligned}$$

(2.10)

$$\begin{aligned}{} & {} \textrm{Wig}[P(M_1,M_2,D_1,D_2)u]\nonumber \\{} & {} \quad =P\left( M_1-\frac{1}{2}D_2,M_1+\frac{1}{2}D_2, \frac{1}{2}D_1+M_2,\frac{1}{2}D_1-M_2\right) \textrm{Wig}[u]. \end{aligned}$$

(2.11)

The above proposition will be useful to relate the classical Wigner distribution W(f) to the mean (1.1) and the variance (1.2) associated with a function $f\in L^2(\mathbb {R})$ and its Fourier transform $\hat{f}\in L^2(\mathbb {R})$.

Proposition 2.2

Given $f\in L^2(\mathbb {R})$ with $\Vert f\Vert =1$ and finite associated means and variances of f and $\hat{f}$, the following properties hold:

• (a) $\langle M^2 f,f\rangle =\mu ^2(f)+\Delta ^2(f)$

• (b) $\langle D^2 f,f\rangle =\mu ^2(\hat{f})+\Delta ^2(\hat{f})$

• (c) $\langle M_1 W(f),W(f)\rangle =\mu (f)$

• (d) $\langle M_2 W(f),W(f)\rangle =\mu (\hat{f})$

• (e) $\langle D_1 W(f),W(f)\rangle =0$

• (f) $\langle D_2 W(f),W(f)\rangle =0$

• (g) $\langle D^2_1 W(f),W(f)\rangle =2\Delta ^2(\hat{f})$

• (h) $\langle D^2_2 W(f),W(f)\rangle =2\Delta ^2(f)$

• (i) $\langle M_1D_1 W(f),W(f)\rangle =\frac{i}{2}$

  $\langle D_1M_1 W(f),W(f)\rangle =-\frac{i}{2}$

• (j) $\langle M_2D_2 W(f),W(f)\rangle =\frac{i}{2}$

  $\langle D_2M_2 W(f),W(f)\rangle =-\frac{i}{2}$

• (k) $\langle M_1^2 W(f),W(f)\rangle =\mu ^2(f)+\frac{1}{2}\Delta ^2(f)$

• (l) $\langle M_2^2 W(f),W(f)\rangle =\mu ^2(\hat{f}) +\frac{1}{2}\Delta ^2(\hat{f})$.

Proof

Let us first recall that (2.7) and (2.9) imply the following Moyal’s formula for the cross-Wigner distribution (cf. [13, p. 66])

$$\begin{aligned} \langle W(f_1,g_1),W(f_2,g_2)\rangle =\langle f_1,f_2\rangle \overline{\langle g_1,g_2\rangle },\qquad \forall f_1,f_2,g_1,g_2\in L^2(\mathbb {R}).\nonumber \\ \end{aligned}$$

(2.12)

Note that the assumption that f has finite associated mean and variance implies that $Mf\in L^2(\mathbb {R})$. Indeed,

$$\begin{aligned} \langle Mf,Mf\rangle= & {} \int _\mathbb {R}y^2|f(y)|^2dy =\int _\mathbb {R}(y-\mu (f)+\mu (f))^2|f(y)|^2dy\nonumber \\= & {} \int _\mathbb {R}|y-\mu (f)|^2|f(y)|^2dy \nonumber \\{} & {} +2\mu (f)\int _\mathbb {R}(y-\mu (f))|f(y)|^2dy +\mu ^2(f)\Vert f\Vert ^2\nonumber \\= & {} \Vert f\Vert ^2\Delta ^2(f)+2\mu ^2(f)\Vert f\Vert ^2-2\mu ^2(f)\Vert f\Vert ^2 +\mu ^2(f)\Vert f\Vert ^2 \nonumber \\= & {} \Delta ^2(f)+\mu ^2(f), \end{aligned}$$

(2.13)

which is the well-known relation between the variance of a random variable and the expectations of the randon variable and its square.

In the same way, the fact that $\hat{f}$ has finite associated mean and variance implies that $Df\in L^2(\mathbb {R})$. This means that Moyal’s formula (2.12) can be applied when, in its left-hand side, Mf or Df appear in the arguments of the Wigner transform.

Now we analyze the case when in the left-hand side of (2.12) the expression $W(f,M^2g)$ appears, for $f,g\in L^2(\mathbb {R})$ with finite associated means and variances of $f, g, \hat{f}, \hat{g}$.

From (2.3) we have that

$$\begin{aligned} 2M_1W(f,g)(x,\xi )=W(f,Mg)(x,\xi )+W(Mf,g)(x,\xi )\in L^2(\mathbb {R}^2), \end{aligned}$$

since $Mf,Mg\in L^2(\mathbb {R})$. Therefore $W(f,g)\in L^2_v(\mathbb {R}^2)$ for $v(x,\xi )=1+|x|$. On the other hand, for $f,g\in \mathcal {S}(\mathbb {R})$,

$$\begin{aligned} W(f,M^2g)(x,\xi )= & {} \int _{\mathbb {R}} f\left( x+\frac{t}{2}\right) \overline{\left( x-\frac{t}{2}\right) ^2 g\left( x-\frac{t}{2}\right) } e^{-it\xi }\,dt \\= & {} \int _\mathbb {R} \left[ 2x-\left( x+\frac{t}{2}\right) \right] f\left( x+\frac{t}{2}\right) \overline{\left( x-\frac{t}{2}\right) g\left( x-\frac{t}{2}\right) } e^{-it\xi }\,dt \\= & {} 2xW(f,Mg)(x,\xi )-W(Mf,Mg)(x,\xi ). \end{aligned}$$

Such an equality holds in fact for $f,g\in \mathcal {S}'(\mathbb {R})$ and for tempered distributions it reads

$$\begin{aligned} W(f,M^2g)=2M_1 W(f,Mg)-W(Mf,Mg). \end{aligned}$$

(2.14)

Being $Mf, Mg\in L^2(\mathbb {R})$, it follows from (2.14) that $W(f,M^2g)\in L^2_{1/v}(\mathbb {R}^2)$. Then, by the duality between $L^2_{1/v}(\mathbb {R}^2)$ and $L^2_{v}(\mathbb {R}^2)$, we have that

$$\begin{aligned} \langle W(f,M^2g),W(f,g)\rangle = \int _{\mathbb {R}^2} W(f,M^2g)(x,\xi )\overline{W(f,g)(x,\xi )}\,dx\,d\xi \end{aligned}$$

is convergent.

Using again (2.14) and (2.3), we can then write

$$\begin{aligned} \langle W(f,M^2g),W(f,g)\rangle= & {} \langle 2M_1 W(f,Mg),W(f,g)\rangle -\langle W(Mf,Mg),W(f,g)\rangle \\= & {} \langle W(f,Mg), 2M_1W(f,g)\rangle -\langle W(Mf,Mg),W(f,g)\rangle \\= & {} \langle W(f,Mg),W(Mf,g)\rangle +\langle W(f,Mg),W(f,Mg)\rangle \\{} & {} -\langle W(Mf,Mg),W(f,g)\rangle . \end{aligned}$$

Now, all functions are in $L^2(\mathbb {R})$ and applying Moyal’s formula (2.12) we finally get

$$\begin{aligned} \langle W(f,M^2g),W(f,g)\rangle= & {} \langle f, Mf \rangle \overline{\langle Mg,g\rangle } +\langle f,f\rangle \overline{\langle Mg,Mg\rangle } -\langle Mf,f\rangle \overline{\langle Mg,g\rangle }\nonumber \\= & {} \langle f,f\rangle \overline{ \langle M^2g,g\rangle }. \end{aligned}$$

(2.15)

Recall now that for every $u_1,u_2\in \mathcal {S}'(\mathbb {R})$ the following formula holds

$$\begin{aligned} W(\hat{u}_1,\hat{u}_2)(x,\xi )=W(u_1,u_2)(-\xi ,x); \end{aligned}$$

(2.16)

then, since $\hat{f}$ and $\hat{g}$ have finite associated means and variances, the same procedure can be applied when we have $W(f,D^2g)$ instead of $W(f,M^2g)$ obtaining that, with the notation (2.5)-(2.6),

$$\begin{aligned} \langle W(f,D^2g),W(f,g)\rangle =\langle W(\hat{f},M^2\hat{g}),W(\hat{f},\hat{g})\rangle =\langle f,f\rangle \overline{\langle D^2g,g\rangle }. \end{aligned}$$

(2.17)

Similar considerations can be done for MDf, obtaining that $W(MDf,g)\in L^2_{1/v}(\mathbb {R}^2)$ because

$$\begin{aligned} W(MDf,g)= & {} \int \left( x+\frac{t}{2}\right) Df\left( x+\frac{t}{2}\right) \overline{g\left( x-\frac{t}{2}\right) }e^{-it\xi }dt\\= & {} \int \left[ 2x-\left( x-\frac{t}{2}\right) \right] Df\left( x+\frac{t}{2}\right) \overline{g\left( x-\frac{t}{2}\right) }e^{-it\xi }dt\\= & {} 2xW(Df,g)-W(Df,Mg), \end{aligned}$$

with $Df,Mg\in L^2(\mathbb {R})$ under the assumptions of finite associated means and variances. Arguing as for $M^2f$ we then have

$$\begin{aligned} \langle W(MDf,g),W(f,g)\rangle =\langle MDf,f\rangle \overline{\langle g,g\rangle } =\langle Df,Mf\rangle \overline{\langle g,g\rangle }. \end{aligned}$$

(2.18)

All the above considerations will be implicit from now on.

Let us now prove point (a): it follows from (2.13) since $\langle M^2f,f\rangle =\langle Mf,Mf\rangle$.

(b): Let us first remark that by Parseval’s formula (2.7) and the assumption $\Vert f\Vert =1$ we also have that $\Vert \hat{f}\Vert =1$. With the notations (2.5)-(2.6), by point (a) applied to $\hat{f}$ we thus obtain:

$$\begin{aligned} \langle D^2f,f\rangle =\langle \xi ^2\hat{f},\hat{f}\rangle =\mu ^2(\hat{f})+\Delta ^2(\hat{f}). \end{aligned}$$

(c): From (2.3) and Moyal’s formula (2.12):

$$\begin{aligned} \langle M_1 W(f),W(f)\rangle= & {} \langle M_1\textrm{Wig}[f\otimes \bar{f}],W(f)\rangle \\= & {} \langle \textrm{Wig}[\frac{1}{2}(M_1+M_2)(f\otimes \bar{f})],W(f)\rangle \\= & {} \frac{1}{2}(\langle W(Mf,f),W(f,f)\rangle +\langle W(f,Mf),W(f,f)\rangle )\\= & {} \frac{1}{2}(\langle Mf,f\rangle \overline{\langle f,f\rangle } +\langle f,f\rangle \overline{\langle Mf,f\rangle }) =\mu (f), \end{aligned}$$

since $\mu (f)\in \mathbb {R}$.

(d): From (2.4), Moyal’s and Parseval’s formulas (2.12) and (2.7):

$$\begin{aligned} \langle M_2 W(f),W(f)\rangle= & {} \langle \textrm{Wig}[\frac{1}{2}(D_1-D_2)f\otimes \bar{f}],W(f)\rangle \\= & {} \frac{1}{2}(\langle W(Df,f),W(f,f)\rangle +\langle W(f,Df),W(f,f)\rangle )\\= & {} \frac{1}{2}(\langle Df,f\rangle \overline{\langle f, f\rangle } +\langle f,f\rangle \overline{\langle Df,f\rangle })\\= & {} \frac{1}{2}(\langle \xi \hat{f},\hat{f}\rangle +\overline{\langle \xi \hat{f},\hat{f}\rangle }) =\mu (\hat{f}), \end{aligned}$$

since $\mu (\hat{f})\in \mathbb {R}$.

(e): From (2.1), (2.12) and (2.7):

$$\begin{aligned} \langle D_1 W(f),W(f)\rangle= & {} \langle \textrm{Wig}[(D_1+D_2)f\otimes \bar{f}],W(f)\rangle \\= & {} \langle W(Df,f)-W(f,Df),W(f,f)\rangle \\= & {} \langle Df,f\rangle \overline{\langle f,f\rangle }-\langle f,f\rangle \overline{\langle Df,f\rangle }\\= & {} \langle \xi \hat{f}, \hat{f}\rangle -\overline{\langle \xi \hat{f},\hat{f}\rangle } =0. \end{aligned}$$

(f): From (2.2) and (2.12):

$$\begin{aligned} \langle D_2W(f),W(f)\rangle= & {} \langle \textrm{Wig}[(M_2-M_1)f\otimes \bar{f}],W(f)\rangle \\= & {} \langle W(f,Mf)-W(Mf,f),W(f,f)\rangle \\= & {} \langle f,f\rangle \overline{\langle Mf,f\rangle }-\langle Mf,f\rangle \overline{\langle f,f\rangle } =0. \end{aligned}$$

(g): From (2.1), (2.12), (2.17), (2.7) and point (a):

$$\begin{aligned} \langle D_1^2W(f),W(f)\rangle= & {} \langle \textrm{Wig}[(D_1+D_2)^2f\otimes \bar{f}],W(f)\rangle \\= & {} \langle W(D^2f,f)-2W(Df,Df)+W(f,D^2f),W(f,f)\rangle \\= & {} \langle D^2f,f\rangle \overline{\langle f,f\rangle } -2\langle Df,f\rangle \overline{\langle Df,f\rangle } +\langle f,f\rangle \overline{\langle D^2f,f\rangle }\\= & {} \langle \xi ^2\hat{f}, \hat{f}\rangle -2|\langle \xi \hat{f}, \hat{f}\rangle |^2+ \overline{\langle \xi ^2\hat{f},\hat{f}\rangle }\\= & {} 2(\mu ^2(\hat{f})+\Delta ^2(\hat{f}))-2\mu ^2(\hat{f}) =2\Delta ^2(\hat{f}). \end{aligned}$$

(h): From (2.2), (2.12), (2.15) and point (a):

$$\begin{aligned} \langle D_2^2 W(f),W(f)\rangle= & {} \langle \textrm{Wig}[(M_2-M_1)^2f\otimes \bar{f}],W(f)\rangle \\= & {} \langle W(f,M^2f)-2W(Mf,Mf)+W(M^2f,f),W(f,f)\rangle \\= & {} \langle f,f\rangle \overline{\langle M^2f,f\rangle } -2\langle Mf,f\rangle \overline{\langle Mf,f\rangle } +\langle M^2f,f\rangle \overline{\langle f,f\rangle }\\= & {} 2(\mu ^2(f)+\Delta ^2(f))-2\mu ^2(f)=2\Delta ^2(f). \end{aligned}$$

(i): From (2.1), (2.3), (2.12), (2.18) and (2.7):

$$\begin{aligned}{} & {} \langle M_1D_1W(f),W(f)\rangle \\{} & {} \quad =\left\langle \textrm{Wig}\left[ \frac{1}{2}(M_2+M_1)(D_1+D_2) f\otimes \bar{f}\right] ,W(f)\right\rangle \\{} & {} \quad =\frac{1}{2}\langle \textrm{Wig}[(M_2D_1+M_1D_1+M_2D_2+M_1D_2)f \otimes \bar{f}],W(f)\rangle \\{} & {} \quad =\frac{1}{2}\langle W(Df,Mf)+W(MDf,f)-W(f,MDf) -W(Mf,Df),W(f,f)\rangle \\{} & {} \quad =\frac{1}{2}(\langle Df,f\rangle \overline{\langle Mf,f\rangle } +\langle Df,Mf\rangle \overline{\langle f,f\rangle }-\langle f,f\rangle \overline{\langle Df,Mf\rangle } -\langle Mf,f\rangle \overline{\langle Df,f\rangle })\\{} & {} \quad =\frac{1}{2}(\langle \xi \hat{f},\hat{f}\rangle \overline{\mu (f)} +\langle Df,Mf\rangle -\overline{\langle Df,Mf\rangle } -\mu (f)\overline{\langle \xi \hat{f},\hat{f}\rangle }). \end{aligned}$$

Since $\mu (f)\in \mathbb {R}$, $\langle \xi \hat{f},\hat{f}\rangle =\mu (\hat{f})\in \mathbb {R}$ and

$$\begin{aligned} \langle Df,Mf\rangle= & {} \langle f,DMf\rangle =i\langle f,f\rangle +\langle f,MDf\rangle \nonumber \\= & {} i+\langle Mf,Df\rangle =i+\overline{\langle Df,Mf\rangle }, \end{aligned}$$

(2.19)

we finally have that

$$\begin{aligned} \langle M_1D_1 W(f),W(f)\rangle =\frac{i}{2}. \end{aligned}$$

Therefore

$$\begin{aligned} \langle D_1M_1W(f),W(f)\rangle= & {} \langle M_1W(f),D_1W(f)\rangle =\langle W(f),M_1D_1 W(f)\rangle \\= & {} \overline{\langle M_1D_1 W(f),W(f)\rangle } =-\frac{i}{2}. \end{aligned}$$

(j): From (2.2), (2.4), (2.12), (2.18), (2.7) and (2.19):

$$\begin{aligned}{} & {} \langle M_2D_2 W(f),W(f)\rangle \\{} & {} \quad =\left\langle \textrm{Wig}\left[ \frac{1}{2}(D_1-D_2)(M_2-M_1) f\otimes \bar{f}\right] ,W(f)\right\rangle \\{} & {} \quad =\frac{1}{2} \langle \textrm{Wig}[(D_1M_2-D_2M_2-D_1M_1+D_2M_1)f \otimes \bar{f}],W(f)\rangle \\{} & {} \quad =\frac{1}{2}\langle W(Df,Mf)-\frac{1}{i}W(f,f)+W(f,MDf),W(f,f)\rangle \\{} & {} \qquad -\frac{1}{2}\langle \frac{1}{i}W(f,f) +W(MDf,f) +W(Mf,Df),W(f,f)\rangle \\{} & {} \quad =\frac{1}{2}(\langle Df,f\rangle \overline{\langle Mf,f\rangle } -\frac{1}{i}\langle f,f\rangle \overline{\langle f,f\rangle } +\langle f,f\rangle \overline{\langle Df,Mf\rangle })\\{} & {} \qquad -\frac{1}{2}\left( \frac{1}{i}+ \langle Df,Mf\rangle \overline{\langle f,f\rangle } +\langle Mf,f\rangle \overline{\langle Df,f\rangle }\right) \\{} & {} \quad =\frac{1}{2}\langle \xi \hat{f},\hat{f}\rangle \mu (f)+i+\frac{1}{2} (\overline{\langle Df,Mf\rangle }-\langle Df,Mf\rangle ) -\frac{1}{2}\mu (f)\overline{\langle \xi \hat{f},\hat{f}\rangle }\\{} & {} \quad =i-\frac{i}{2}=\frac{i}{2}. \end{aligned}$$

It follows that

$$\begin{aligned} \langle D_2M_2W(f),W(f)\rangle =\frac{1}{i}\langle W(f,f), W(f,f)\rangle +\langle M_2D_2W(f),W(f)\rangle =-\frac{i}{2}. \end{aligned}$$

(k): From (2.3), (2.12) and point (a):

$$\begin{aligned}{} & {} \langle M_1^2W(f),W(f)\rangle \\{} & {} \quad =\langle M_1\textrm{Wig}[f\otimes \bar{f}],M_1\textrm{Wig}[f\otimes \bar{f}]\rangle \\{} & {} \quad =\left\langle \textrm{Wig}\left[ \frac{1}{2}(M_2+M_1)f\otimes \bar{f}\right] , \textrm{Wig}\left[ \frac{1}{2}(M_2+M_1)f\otimes \bar{f}\right] \right\rangle \\{} & {} \quad =\frac{1}{4}\langle W(f,Mf)+W(Mf,f),W(f,Mf)+W(Mf,f)\rangle \\{} & {} \quad =\frac{1}{4}(\langle f,f\rangle \overline{\langle Mf,Mf\rangle } +\langle Mf,f\rangle \overline{\langle f,Mf\rangle } +\langle f,Mf\rangle \overline{\langle Mf,f\rangle } +\langle Mf,Mf\rangle \overline{\langle f,f\rangle })\\{} & {} \quad =\frac{1}{4}(\overline{\langle M^2f,f\rangle } +2\mu ^2(f)+\langle M^2 f,f\rangle )\\{} & {} \quad =\frac{1}{4}\cdot 2(\mu ^2(f)+\Delta ^2(f)) +\frac{1}{2}\mu ^2(f) =\frac{1}{2}\Delta ^2(f)+\mu ^2(f). \end{aligned}$$

(l): From (2.4), (2.12), (2.7) and point (b):

$$\begin{aligned}{} & {} \langle M_2^2W(f),W(f)\rangle \\{} & {} \quad =\langle M_2\textrm{Wig}[f\otimes \bar{f}], M_2\textrm{Wig}[f\otimes \bar{f}]\rangle \\{} & {} \quad =\left\langle \textrm{Wig}\left[ \frac{1}{2}(D_1-D_2)f\otimes \bar{f}\right] , \textrm{Wig}\left[ \frac{1}{2}(D_1-D_2)f\otimes \bar{f}\right] \right\rangle \\{} & {} \quad =\frac{1}{4}\langle W(Df,f)+W(f,Df),W(Df,f)+W(f,Df)\rangle \\{} & {} \quad =\frac{1}{4}(\langle Df,Df\rangle \overline{\langle f,f\rangle } +\langle f,Df\rangle \overline{\langle Df,f\rangle } +\langle Df,f\rangle \overline{\langle f,Df\rangle } +\langle f,f\rangle \overline{\langle Df,Df\rangle })\\{} & {} \quad =\frac{1}{4}(\langle D^2 f,f\rangle +\langle \hat{f}, \xi \hat{f}\rangle \overline{\langle \xi \hat{f},\hat{f}\rangle } +\langle \xi \hat{f},\hat{f}\rangle \overline{\langle \hat{f}, \xi \hat{f}\rangle } +\overline{\langle D^2f,f\rangle })\\{} & {} \quad =\frac{1}{2}\Delta ^2(\hat{f})+\mu ^2(\hat{f}). \end{aligned}$$

The proof is complete. $\square$

Let us remark that the identities of Proposition 2.2 can be rewritten for any $f\in L^2(\mathbb {R})$, without the restriction that $\Vert f\Vert =1$, by considering $\frac{f}{\Vert f\Vert }$ instead of f. For instance, points (a), (b) of Proposition 2.2 would be replaced by

$$\begin{aligned} \langle M^2 f,f\rangle= & {} \Vert f\Vert ^2(\mu ^2(f)+\Delta ^2(f)), \end{aligned}$$

(2.20)

$$\begin{aligned} \langle D^2 f,f\rangle= & {} \Vert f\Vert ^2(\mu ^2(\hat{f})+\Delta ^2(\hat{f})). \end{aligned}$$

(2.21)

3 The Hermite Basis

For $k\in \mathbb {N}_0=\mathbb {N}\cup \{0\}$, let $h_k$ be the Hermite functions on $\mathbb {R}$ defined by (1.8). It is well known that $h_k$ are eigenfunctions of the Fourier transform and form an orthonormal basis in $L^2(\mathbb {R})$. Moreover, they are an absolute basis in $\mathcal {S}(\mathbb {R})$ (see [17]).

Denoting by

$$\begin{aligned} h_{j,k}:=\mathcal {F}^{-1}W(h_j,h_k), \end{aligned}$$

by [22, Thms. 3.2 and 3.4] we have that the functions $\{h_{j,k}\}_{j,k\in \mathbb {N}_0}$ form an orthonormal basis in $L^2(\mathbb {R}^2)$ and are eigenfunctions of the twisted Laplacian:

$$\begin{aligned} Lh_{j,k}(y,t)=(2k+1)h_{j,k}(y,t),\qquad j,k\in \mathbb {N}_0, \end{aligned}$$

for

$$\begin{aligned} L:=\left( D_y-\frac{1}{2} t\right) ^2+\left( D_t+\frac{1}{2} y\right) ^2. \end{aligned}$$

By Fourier transform (see [4, Ex. 3.20])

$$\begin{aligned} \hat{h}_{j,k}(x,\xi )=W(h_j,h_k)(x,\xi ) \end{aligned}$$

(3.1)

are eigenfunctions of the operator $\hat{L}$ defined in (1.4), with the same eigenvalues as before, in the sense that

$$\begin{aligned} \hat{L}\hat{h}_{j,k}=(2k+1)\hat{h}_{j,k}. \end{aligned}$$

(3.2)

Note that also $\{\hat{h}_{j,k}\}_{j,k\in \mathbb {N}_0}$ are in $\mathcal {S}(\mathbb {R}^2)$ and form an orthonormal basis in $L^2(\mathbb {R}^2)$.

More in general, following the same ideas as in [21, Thm. 21.2], we can prove:

Theorem 3.1

If $\{f_k\}_{k\in \mathbb {N}_0}$ is an orthonormal basis in $L^2(\mathbb {R})$, then $\{W(f_j,f_k)\}_{j,k\in \mathbb {N}_0}$ is an orthonormal basis in $L^2(\mathbb {R}^2)$.

Proof

Let us first remark that if $\{f_k\}_k$ is an othonormal sequence in $L^2(\mathbb {R})$, then $\{W(f_j,f_k)\}_{j,k}$ is an orthonormal sequence in $L^2(\mathbb {R}^2)$ since, by (2.12),

$$\begin{aligned} \langle W(f_j,f_k),W(f_i,f_h)\rangle= & {} \langle f_j,f_i\rangle \overline{\langle f_k,f_h\rangle }\\= & {} \delta _{j,i}\cdot \delta _{k,h} ={\left\{ \begin{array}{ll} 1&{} \text{ if } \ (j,k)=(i,h) \\ 0&{} \text{ if }\ (j,k)\ne (i,h). \end{array}\right. } \end{aligned}$$

In order to prove that $\{W(f_j,f_k)\}_{j,k\in \mathbb {N}_0}$ is a basis for $L^2(\mathbb {R}^2)$, by [7, Thm. 3.4.2], it is enough to prove that if $F\in L^2(\mathbb {R}^2)$ is such that

$$\begin{aligned} \int _{\mathbb {R}^2}F(x,\xi )W(f_j,f_k)(x,\xi )dxd\xi =0, \qquad \forall j,k\in \mathbb {N}_0, \end{aligned}$$

(3.3)

then $F=0$ a.e. in $\mathbb {R}^2$.

By [21, Thms. 4.4 and 7.5] the operator

$$\begin{aligned}{} & {} L^2(\mathbb {R}^2) \longrightarrow \mathcal {L}(L^2(\mathbb {R}),L^2(\mathbb {R}))\\{} & {} F \longmapsto W_F \end{aligned}$$

defined by

$$\begin{aligned} \langle W_F\varphi ,\psi \rangle =\frac{1}{\sqrt{2\pi }} \int _{\mathbb {R}^2}F(x,\xi )W(\varphi ,\psi )(x,\xi )dxd\xi \end{aligned}$$

is a bounded linear operator satisfying

$$\begin{aligned} \Vert W_F\Vert _{\mathcal {L}(L^2,L^2)}\le \frac{1}{\sqrt{2\pi }} \Vert F\Vert _{L^2(\mathbb {R}^2)}=\Vert W_F\Vert _{HS}, \end{aligned}$$

(3.4)

where $\Vert \cdot \Vert _{HS}$ is the Hilbert-Schmidt norm defined by (see [21, formula (7.1)]):

$$\begin{aligned} \Vert W_F\Vert ^2_{HS}:=\sum _{j=0}^{+\infty }\Vert W_F f_j\Vert _{L^2(\mathbb {R})}^2 \end{aligned}$$

(3.5)

for an orthonormal basis $\{f_j\}_{j\in \mathbb {N}_0}$ of $L^2(\mathbb {R})$. The operator $W_F$ is in fact the classical Weyl operator with symbol F. Then

$$\begin{aligned} \langle W_F f_j,f_k\rangle =\frac{1}{\sqrt{2\pi }} \int _{\mathbb {R}^2}F(x,\xi )W(f_j,f_k)(x,\xi )dxd\xi =0, \qquad \forall j,k\in \mathbb {N}_0, \end{aligned}$$

by assumption, which implies that

$$\begin{aligned} W_Ff_j=0,\qquad \forall j\in \mathbb {N}_0, \end{aligned}$$

(3.6)

since $\{f_j\}_{j\in \mathbb {N}_0}$ is an orthonormal basis in $L^2(\mathbb {R})$.

From (3.4) and (3.5) we finally have that $F=0$ a.e. in $\mathbb {R}^2$. $\square$

The operator $\hat{L}$ defined in (1.4) is unbounded on $L^2(\mathbb {R}^2)$ (see Remark 6.6 below) and defined (at least) in $\mathcal {S}(\mathbb {R}^2)\subset L^2(\mathbb {R}^2)$. Now, since the functions (3.1) are an orthonormal basis for $L^2(\mathbb {R}^2)$, every element $F\in L^2(\mathbb {R}^2)$ can be written as

$$\begin{aligned} F=\sum _{j,k=0}^{+\infty } c_{j,k} \hat{h}_{j,k} \end{aligned}$$

where $c_{j,k}=\langle F,\hat{h}_{j,k}\rangle$. Then, writing

$$\begin{aligned} F_N=\sum _{j,k=0}^{N} c_{j,k} \hat{h}_{j,k}\in \mathcal {S}(\mathbb {R}^2) \end{aligned}$$

we have from (3.2)

$$\begin{aligned} \hat{L}F_N(x,\xi ) = \sum _{j,k=0}^{N} c_{j,k} (2k+1)\hat{h}_{j,k}(x,\xi ). \end{aligned}$$

The operator $\hat{L}$ is then the unbounded and densely defined operator with domain

$$\begin{aligned} D(\hat{L})=\left\{ F\in L^2(\mathbb {R}^2) : \sum _{j,k=0}^{+\infty } c_{j,k} (2k+1)\hat{h}_{j,k}\ \text {converges\ in}\ L^2(\mathbb {R}^2)\right\} \end{aligned}$$

for $c_{j,k}=\langle F,\hat{h}_{j,k}\rangle$, acting on $F\in D(\hat{L})$ as

$$\begin{aligned} \hat{L}F=\sum _{j,k=0}^{+\infty } c_{j,k} (2k+1) \hat{h}_{j,k}\in L^2(\mathbb {R}^2). \end{aligned}$$

In this case

$$\begin{aligned} \langle \hat{L} F,F\rangle =\sum _{j,k=0}^{+\infty }|c_{j,k}|^2(2k+1). \end{aligned}$$

(3.7)

In general, we shall write (3.7) for all $F\in L^2(\mathbb {R}^2)$, meaning that $\langle \hat{L} F,F\rangle =+\infty$ if the series diverges. Note that, being $\{\hat{h}_{j,k}\}_{j,k\in \mathbb {N}_0}$ an orthonormal basis for $L^2(\mathbb {R}^2)$, we have that $F\in D(\hat{L})$ if and only if $\{c_{j,k}(2k+1)\}_{j,k\in \mathbb {N}_0}\in \ell ^2$. This implies that the series (3.7) converges (but not vice versa).

4 Mean-Dispersion Principle

From the results of the previous sections we obtain now an alternative formulation and a simple proof of the Shapiro’s Mean-Dispersion Principle (see [16] and the references therein). To this aim let us first prove some preliminary results.

Lemma 4.1

Let $\{h_k\}_{k\in \mathbb {N}_0}$ be the Hermite functions defined in (1.8) and $\hat{L}$ as in (1.4). Then for every $j\in \mathbb {N}_0$ we have

$$\begin{aligned} \sum _{k=0}^n\langle \hat{L} W(h_j,h_k),W(h_j,h_k)\rangle =(n+1)^2, \qquad \forall n\in \mathbb {N}_0. \end{aligned}$$

Proof

From (3.2) for all $j,k\in \mathbb {N}_0$ we have

$$\begin{aligned} \langle \hat{L} W(h_j,h_k),W(h_j,h_k)\rangle =\langle \hat{L}\hat{h}_{j,k},\hat{h}_{j,k}\rangle =\langle (2k+1)\hat{h}_{j,k},\hat{h}_{j,k}\rangle =2k+1, \end{aligned}$$

since $\{\hat{h}_{j,k}\}_{j,k}$ is an orthonormal basis in $L^2(\mathbb {R}^2)$.

It follows that

$$\begin{aligned} \sum _{k=0}^n\langle \hat{L}W(h_j,h_k),W(h_j,h_k)\rangle =\sum _{k=0}^n(2k+1)=(n+1)^2, \end{aligned}$$

where the last equality is the formula for the sum of all odd numbers from 1 to $2n+1$. $\square$

Lemma 4.2

Let $\hat{L}$ be the operator in (1.4). Then for all $f,g\in L^2(\mathbb {R})$ with finite associated means and variances of $f,g,\hat{f},\hat{g}$:

• (i) $\displaystyle \hat{L} W(f,g)=W(f,(M^2+D^2)g)$,

• (ii) $\displaystyle \langle \hat{L}W(f,g),W(f,g)\rangle =\Vert f\Vert ^2\Vert g\Vert ^2(\Delta ^2(g)+\Delta ^2(\hat{g})+\mu ^2(g)+\mu ^2(\hat{g}))$.

In particular, $\langle \hat{L}W(f,g),W(f,g)\rangle \in \mathbb {R}$ and if $\Vert f\Vert =\Vert g\Vert =1$, then

$$\begin{aligned} \langle \hat{L}W(f,g),W(f,g)\rangle =\Delta ^2(g)+\Delta ^2(\hat{g})+\mu ^2(g)+\mu ^2(\hat{g}). \end{aligned}$$

Proof

(i): From (2.10) we have

$$\begin{aligned} \hat{L}W(f,g)= & {} \left[ \left( \frac{1}{2}D_2+M_1\right) ^2 +\left( \frac{1}{2}D_1-M_2\right) ^2\right] \textrm{Wig}[f\otimes \bar{g}]\\= & {} \textrm{Wig}\left[ \left( \left( \frac{1}{2}(M_2-M_1) +\frac{1}{2}(M_2+M_1)\right) ^2\right. \right. \\{} & {} \qquad \left. \left. +\left( \frac{1}{2} (D_1+D_2) -\frac{1}{2}(D_1-D_2)\right) ^2\right) f \otimes \bar{g}\right] \\= & {} \textrm{Wig}[(M_2^2+D_2^2)f\otimes \bar{g}]=W(f,(M^2+D^2)g). \end{aligned}$$

(ii): From (i), (2.15), (2.17), (2.20) and (2.21):

$$\begin{aligned} \langle \hat{L}W(f,g),W(f,g)\rangle= & {} \langle W(f,(M^2+D^2)g),W(f,g)\rangle \\= & {} \langle W(f,M^2g),W(f,g)\rangle +\langle W(f,D^2g),W(f,g)\rangle \\= & {} \langle f,f\rangle \overline{\langle M^2g,g\rangle } +\langle f,f\rangle \overline{\langle D^2g,g\rangle }\\= & {} \Vert f\Vert ^2\Vert g\Vert ^2(\Delta ^2(g)+\mu ^2(g))+\Vert f\Vert ^2\Vert g\Vert ^2 (\Delta ^2(\hat{g})+\mu ^2(\hat{g}))\\= & {} \Vert f\Vert ^2\Vert g\Vert ^2(\Delta ^2(g)+\Delta ^2 (\hat{g})+\mu ^2(g)+\mu ^2(\hat{g})). \end{aligned}$$

$\square$

Lemma 4.3

Let $0\le \alpha _\ell \le 1$ satisfy $\sum _{\ell =0}^{+\infty }\alpha _\ell =n+1$ for some $n\in \mathbb {N}_0$. Then

$$\begin{aligned} \sum _{\ell =0}^{+\infty }\alpha _\ell (2\ell +1)\ge (n+1)^2. \end{aligned}$$

Moreover, the inequality is an identity if and only if $\alpha _0=\dots =\alpha _n=1$.

Proof

By assumption we can write

$$\begin{aligned} n+1=\sum _{\ell =0}^{+\infty }\alpha _\ell =\alpha _0+\dots +\alpha _n +R_n \end{aligned}$$

for a reminder

$$\begin{aligned} R_n=\sum _{\ell =n+1}^{+\infty }\alpha _\ell . \end{aligned}$$

(4.1)

Note that $\alpha _0=\dots =\alpha _n=1$ if $R_n=0$.

For all $0\le k\le n$ we set

$$\begin{aligned} c_k= {\left\{ \begin{array}{ll} 0, &{} \text{ if }\ R_n=0\\ \frac{1-\alpha _k}{R_n}, &{}\text{ if } \ R_n>0. \end{array}\right. } \end{aligned}$$

(4.2)

Then

$$\begin{aligned} \alpha _k+c_kR_n=1\qquad \forall 0\le k\le n \end{aligned}$$

(4.3)

and $(c_0+\dots +c_n)R_n=R_n$, so that

$$\begin{aligned} c_0+\dots +c_n= {\left\{ \begin{array}{ll} 1 &{} \text{ if }\ R_n>0\\ 0&{} \text{ if }\ R_n=0 \end{array}\right. } \end{aligned}$$

and we can write

$$\begin{aligned} (c_0+\dots +c_n)\sum _{\ell =n+1}^{+\infty }\alpha _\ell (2\ell +1) =\sum _{\ell =n+1}^{+\infty }\alpha _\ell (2\ell +1), \end{aligned}$$

(4.4)

being $R_n=0$ iff $\alpha _\ell =0$ for all $\ell \ge n+1$.

By (4.4) and (4.3)

$$\begin{aligned} \sum _{\ell =0}^{+\infty }\alpha _\ell (2\ell +1)= & {} \sum _{\ell =0}^n\alpha _\ell (2\ell +1)+(c_{0}+\dots +c_{n}) \sum _{\ell =n+1}^{+\infty } \alpha _\ell (2\ell +1)\nonumber \\= & {} \sum _{\ell =0}^n\alpha _\ell (2\ell +1)+c_0\sum _{\ell =n+1}^{+\infty } \alpha _\ell \underbrace{(2\ell +1)}_{>1} +c_1\sum _{\ell =n+1}^{+\infty } \alpha _\ell \underbrace{(2\ell +1)}_{>3}\nonumber \\{} & {} \dots +c_{n-1}\sum _{\ell =n+1}^{+\infty } \alpha _\ell \underbrace{(2\ell +1)}_{>2n-1} +c_{n}\sum _{\ell =n+1}^{+\infty } \alpha _\ell \underbrace{(2\ell +1)}_{>2n+1} \nonumber \\\ge & {} \left( \alpha _0+c_{0}\sum _{\ell =n+1}^{+\infty }\alpha _\ell \right) +\left( \alpha _1\cdot 3+c_{1}\sum _{\ell =n+1}^{+\infty } \alpha _\ell \cdot 3\right) \nonumber \\{} & {} \dots +\left( \alpha _{n-1}\cdot (2n-1) +c_{n-1}\sum _{\ell =n+1}^{+\infty }\alpha _\ell \cdot (2n-1)\right) \nonumber \\{} & {} +\left( \alpha _{n}\cdot (2n+1)+c_{n}\sum _{\ell =n+1}^{+\infty } \alpha _\ell \cdot (2n+1)\right) \nonumber \\= & {} \sum _{k=0}^n\underbrace{\left( \alpha _k+c_kR_n\right) }_{=1} (2k+1)=\sum _{k=0}^n(2k+1)=(n+1)^2. \end{aligned}$$

(4.5)

Finally note that the inequality in (4.5) is strict unless $c_0=\dots =c_n=0$, that is $\alpha _0=\dots =\alpha _n=1$. $\square$

Theorem 4.4

Let $\{f_k\}_{k\in \mathbb {N}_0}$ be such that $\Vert f_k\Vert =1$ for every $k\in \mathbb {N}_0$, and let $\{g_k\}_{k\in \mathbb {N}_0}$ be an orthonormal sequence in $L^2(\mathbb {R})$. Then

$$\begin{aligned} \sum _{k=0}^n\langle \hat{L}W(f_k,g_k),W(f_k,g_k)\rangle \ge (n+1)^2, \qquad \forall n\in \mathbb {N}_0. \end{aligned}$$

(4.6)

Moreover, the estimate is optimal, in the sense that if $g_k$ are the Hermite functions, then equality holds in (4.6) and, conversely, given $n_0\in \mathbb {N}$, if equality holds in (4.6) for all $n\le n_0$, then there exist $c_k\in \mathbb {C}$ with $|c_k|=1$ such that $g_k=c_kh_k$ for all $0\le k\le n_0$.

Proof

Since $W(f_k,g_k)\in L^2(\mathbb {R}^2)$ and the sequence $\{\hat{h}_{j,\ell }\}=\{W(h_j,h_\ell )\}$ defined in (3.1) is an orthonormal basis in $L^2(\mathbb {R}^2)$, we can write

$$\begin{aligned} W(f_k,g_k)=\sum _{j,\ell =0}^{+\infty }c^{(k)}_{j,\ell }W(h_j,h_\ell ) \end{aligned}$$

with

$$\begin{aligned} c^{(k)}_{j,\ell }=\langle W(f_k,g_k),W(h_j,h_\ell )\rangle =\langle f_k,h_j\rangle \overline{\langle g_k,h_\ell \rangle }, \end{aligned}$$

(4.7)

by (2.12). As in (3.7) we have

$$\begin{aligned} \sum _{k=0}^n\langle \hat{L}W(f_k,g_k),W(f_k,g_k)\rangle =\sum _{k=0}^n\sum _{j,\ell =0}^{+\infty }|c^{(k)}_{j,\ell }|^2 (2\ell +1), \end{aligned}$$

(4.8)

and we can assume that for every $0\le k\le n$ the series in (4.8) converges, otherwise (4.6) would be trivial, being the left-hand side equal to $+\infty$.

By (4.7) and (4.8), we get

$$\begin{aligned}{} & {} \sum _{k=0}^n\langle \hat{L}W(f_k,g_k),W(f_k,g_k)\rangle =\sum _{k=0}^n\sum _{j,\ell =0}^{+\infty }|\langle f_k,h_j\rangle |^2|\langle g_k,h_\ell \rangle |^2 (2\ell +1)\nonumber \\{} & {} \quad = \sum _{k=0}^n\left( \sum _{j=0}^{+\infty }|\langle f_k,h_j\rangle |^2\right) \sum _{\ell =0}^{+\infty }|\langle g_k,h_\ell \rangle |^2(2\ell +1) =\sum _{\ell =0}^{+\infty }\left( \sum _{k=0}^n|\langle g_k,h_\ell \rangle |^2\right) (2\ell +1),\nonumber \\ \end{aligned}$$

(4.9)

since $\Vert f_k\Vert ^2=1$. Setting

$$\begin{aligned} \alpha _\ell :=\sum _{k=0}^n|\langle g_k,h_\ell \rangle |^2, \end{aligned}$$

we remark that

$$\begin{aligned} \sum _{\ell =0}^{+\infty }\alpha _\ell =\sum _{\ell =0}^{+\infty } \sum _{k=0}^n|\langle g_k,h_\ell \rangle |^2 =\sum _{k=0}^n\sum _{\ell =0}^{+\infty } |\langle g_k,h_\ell \rangle |^2 =\sum _{k=0}^n\Vert g_k\Vert ^2=n+1. \end{aligned}$$

(4.10)

But for each $\ell \in \mathbb {N}_0$

$$\begin{aligned} \alpha _\ell \le \sum _{k=0}^{+\infty }|\langle g_k,h_\ell \rangle |^2 \le \Vert h_\ell \Vert ^2=1, \end{aligned}$$

so that we can apply Lemma 4.3 to obtain from (4.9) that

$$\begin{aligned} \sum _{k=0}^n\langle \hat{L}W(f_k,g_k),W(f_k,g_k)\rangle =\sum _{\ell =0}^{+\infty }\alpha _\ell (2\ell +1)\ge (n+1)^2, \end{aligned}$$

and equality holds if and only if

$$\begin{aligned} \alpha _\ell =\sum _{k=0}^n|\langle g_k,h_\ell \rangle |^2=1,\qquad \forall \ell =0,\dots , n. \end{aligned}$$

If $g_k=h_k$ for every $k\in \mathbb {N}_0$, then $\alpha _\ell =1$ for all $\ell \in \mathbb {N}_0$ and equality holds in (4.6) for all $n\in \mathbb {N}_0$.

Conversely, if equality holds in (4.6) for all $n\le n_0$, then

$$\begin{aligned} \sum _{k=0}^n|\langle g_k,h_\ell \rangle |^2=1, \qquad \forall \ell =0,\dots ,n,\ \forall n\le n_0. \end{aligned}$$

This implies $h_\ell \in \textrm{span}(g_0,\dots ,g_\ell )$ for all $\ell \le n_0$. Therefore $g_0=c_0h_0$ for some $c_0\in \mathbb {C}$ with $|c_0|=1$ and then, recursively, $g_k=c_kh_k$ for some $c_k\in \mathbb {C}$ with $|c_k|=1$ and for $k=1,\ldots ,n_0$. $\square$

Let us remark that we can choose in Theorem 4.4 a constant sequence $f_k=f\in L^2(\mathbb {R})$ with $\Vert f\Vert =1$, obtaining that if $\{g_k\}_{k\in \mathbb {N}_0}$ is an orthonormal sequence in $L^2(\mathbb {R})$, then

$$\begin{aligned} \sum _{k=0}^n\langle \hat{L}W(f,g_k),W(f,g_k)\rangle \ge (n+1)^2, \qquad \forall n\in \mathbb {N}_0,\ \forall f\in L^2(\mathbb {R}),\,\Vert f\Vert =1. \end{aligned}$$

Choosing $g_k=f_k$ in Theorem 4.4, we immediately get:

Corollary 4.5

If $\{f_k\}_{k\in \mathbb {N}_0}$ is an orthonormal sequence in $L^2(\mathbb {R})$, then

$$\begin{aligned} \sum _{k=0}^n\langle \hat{L}W(f_k),W(f_k)\rangle \ge (n+1)^2, \qquad \forall n\in \mathbb {N}_0, \end{aligned}$$

(4.11)

and the estimate is optimal, in the sense that if $f_k$ are the Hermite functions, then equality holds in (4.11) and, conversely, given $n_0\in \mathbb {N}$, if equality holds in (4.11) for all $n\le n_0$, then there exist $c_k\in \mathbb {C}$ with $|c_k|=1$ such that $f_k=c_kh_k$ for all $0\le k\le n_0$.

From Corollary 4.5 we have that if

$$\begin{aligned} \langle \hat{L}W(f_j),W(f_j)\rangle \le A,\qquad \forall j\in J, \end{aligned}$$

then J must be finite.

Moreover, since

$$\begin{aligned} \langle \hat{L}W(f_k),W(f_k)\rangle =\mu ^2(f_k) +\mu ^2(\hat{f}_k)+\Delta ^2(f_k)+\Delta ^2(\hat{f}_k) \end{aligned}$$

(4.12)

by Lemma 4.2, we have obtained a simple proof of Theorem 1.3 (the sharp Mean-Dispersion Principle [16, Thm. 2.3]), and then also of Theorem 1.2 (the original Shapiro’s Mean-Dispersion Principle).

Formula (4.12) says that Corollary 4.5 is exactly a reformulation of Theorem 1.3, and in this sense Theorem 4.4 and Corollary 4.5 can be seen as Mean-Dispersion principles related with the Wigner transform. On the other hand we observe that working with the Wigner transform gives several advantages. First of all we have more generality since in Theorem 4.4 we can consider different arguments $f_k, g_k$ in the cross-Wigner distribution; moreover the proofs with the Wigner transform are simpler and more self-contained with respect to [16]. Another advantage is that we have information on the Wigner transform of an orthonormal sequence $\{ f_k\}_{k\in \mathbb {N}_0}$ rather than on $f_k$ and $\hat{f}_k$ themselves, and this gives more possibilities on how such information can be treated and written. In Sect. 5 we give a Mean-Dispersion principle on the trace of the covariance matrix associated to the Wigner transform; here we start by noting that, from Proposition 2.2, the quantity $\mu ^2(f_k)+\mu ^2(\hat{f}_k)+\Delta ^2(f_k)+\Delta ^2(\hat{f}_k)$ in (4.12) can be written not only as $\langle \hat{L}W(f_k),W(f_k)\rangle$, but also through many other operators, as we can see in the following examples.

Example 4.6

For all $f\in L^2(\mathbb {R})$ with $\Vert f\Vert =1$ and finite associated mean and variance of f and $\hat{f}$

$$\begin{aligned} \mu ^2(f)+\mu ^2(\hat{f})+\Delta ^2(f)+\Delta ^2(\hat{f}) =\langle M^2f,f\rangle +\langle D^2f,f\rangle \end{aligned}$$

by Proposition 2.2(a), (b). Therefore formula (1.7) for an orthonormal sequence $\{ f_k\}_{k\in \mathbb {N}_0}$ in $L^2(\mathbb {R})$ can be rewritten as

$$\begin{aligned} \sum _{k=0}^n\langle (M^2+D^2)f_k,f_k\rangle \ge (n+1)^2, \qquad \forall n\in \mathbb {N}_0. \end{aligned}$$

Example 4.7

For all $f\in L^2(\mathbb {R})$ with $\Vert f\Vert =1$ and finite associated mean and variance of f and $\hat{f}$ we have from Proposition 2.2(g), (h), (k), (l):

$$\begin{aligned}{} & {} \left\langle \left[ \frac{1}{4}(D_1^2+D_2^2)+(M_1^2+M_2^2)\right] W(f),W(f)\right\rangle \\{} & {} \quad =\Delta ^2(f)+\Delta ^2(\hat{f})+\mu ^2(f)+\mu ^2(\hat{f}) \end{aligned}$$

and hence for an orthonormal sequence $\{f_k\}_{k\in \mathbb {N}_0}\subset L^2(\mathbb {R})$

$$\begin{aligned} \sum _{k=0}^n\langle PW(f_k),W(f_k)\rangle \ge (n+1)^2, \qquad \forall n\in \mathbb {N}_0, \end{aligned}$$

for $P=\frac{1}{4}(D_1^2+D_2^2)+(M_1^2+M_2^2)$, by Theorem 1.3.

We can also combine, for example, the operators of Examples 4.6 and 4.7, or add combinations of $D_1$, $D_2$, $M_1D_1-M_2D_2$, by Proposition 2.2(e), (f), (i), (j).

5 Covariance

In this section we give an uncertainty principle involving the trace of the covariance matrix of the square of the Wigner distribution $|W(f)(x,\xi )|^2$, and explore its relations with Theorem 1.3.

To this aim, let us first recall some notions about mean and covariance for a function of two variables $\rho (x,y)\in L^1(\mathbb {R}^2)$. We set

$$\begin{aligned} \rho _X(x):=\int _\mathbb {R}\rho (x,y)dy,\qquad \rho _Y(y):=\int _\mathbb {R}\rho (x,y)dx, \end{aligned}$$

(5.1)

and then consider the means

$$\begin{aligned} M(X):=\int _\mathbb {R}x\rho _X(x)dx,\qquad M(Y):=\int _\mathbb {R}y\rho _Y(y)dy, \end{aligned}$$

(5.2)

and the covariances

$$\begin{aligned} C(X,Y):= & {} \int _{\mathbb {R}^2}(x-M(X))(y-M(Y))\rho (x,y)dxdy=C(Y,X)\\ C(X,X)= & {} \int _{\mathbb {R}^2}(x-M(X))^2\rho (x,y)dxdy\\ C(Y,Y)= & {} \int _{\mathbb {R}^2}(y-M(Y))^2\rho (x,y)dxdy. \end{aligned}$$

The covariance matrix

$$\begin{aligned} \left( \begin{matrix} C(X,X)&{}C(X,Y)\\ C(Y,X)&{}C(Y,Y) \end{matrix}\right) \end{aligned}$$

is symetric and its trace is given by

$$\begin{aligned} C(X,X)+C(Y,Y)= & {} \int _{\mathbb {R}^2} \left( (x-M(X))^2+(y-M(Y))^2\right) \rho (x,y)dxdy\nonumber \\= & {} \int _{\mathbb {R}^2}(x^2+y^2)\rho (x,y)dxdy\nonumber \\{} & {} -2M(X)\int _{\mathbb {R}^2}x\rho (x,y)dxdy -2M(Y)\int _{\mathbb {R}^2}y\rho (x,y)dxdy\nonumber \\{} & {} +(M^2(X)+M^2(Y))\int _{\mathbb {R}^2}\rho (x,y)dxdy. \end{aligned}$$

(5.3)

If $\rho (x,y)$ has null means $M(X)=M(Y)=0$, then (5.3) represents the trace of the covariance matrix of $\rho (x,y)$.

For $f\in L^2(\mathbb {R})$ we can consider $\rho (x,\xi )=|W(f)(x,\xi )|^2\in L^1(\mathbb {R}^2)$ since $W(f)\in L^2(\mathbb {R}^2)$. It is then interesting to consider the quantity in (5.3)

$$\begin{aligned} \int _{\mathbb {R}^2}(x^2+\xi ^2)|W(f)(x,\xi )|^2dxd\xi , \end{aligned}$$

which is related to means and variances of f and $\hat{f}$; indeed, if $f\in L^2(\mathbb {R})$ with $\Vert f\Vert =1$, by Proposition 2.2(k), (l) we have

$$\begin{aligned}{} & {} \int _{\mathbb {R}^2}(x^2+\xi ^2)|W(f)(x,\xi )|^2dxd\xi \nonumber \\= & {} \langle (M_1^2+M_2^2)W(f),W(f)\rangle \end{aligned}$$

(5.4)

$$\begin{aligned}= & {} \mu ^2(f)+\frac{1}{2}\Delta ^2(f)+\mu ^2(\hat{f}) +\frac{1}{2}\Delta ^2(\hat{f}) \end{aligned}$$

(5.5)

$$\begin{aligned}\ge & {} \frac{1}{2}(\mu ^2(f)+\mu ^2(\hat{f})+\Delta ^2(f)+\Delta ^2(\hat{f})) \end{aligned}$$

(5.6)

and the equality in (5.6) holds if and only if $\mu (f)=\mu (\hat{f})=0$. In particular, since the Hermite functions satisfy $\mu (h_k)=\mu (\hat{h}_k)=0$ by [16, Ex. 2.4], from Theorem 1.3 we have the following:

Theorem 5.1

If $\{f_k\}_{k\in \mathbb {N}_0}$ is an orthonormal sequence in $L^2(\mathbb {R})$, then

$$\begin{aligned} \sum _{k=0}^n\int _{\mathbb {R}^2}(x^2+\xi ^2)|W(f_k)(x,\xi )|^2dxd\xi \ge \frac{(n+1)^2}{2},\qquad \forall n\in \mathbb {N}_0. \end{aligned}$$

(5.7)

Moreover, given $n_0\in \mathbb {N}$, the equality holds for all $n\le n_0$ if and only if there exist $c_k\in \mathbb {C}$ with $|c_k|=1$ such that $f_k=c_kh_k$ for all $0\le k\le n_0$.

Proof

The inequality (5.7) immediately follows from (5.6) and Theorem 1.3. If $f_k$ are multiples of the Hermite functions $c_k h_k$ with $|c_k|=1$, then the equality holds because of (5.5), the fact that $\mu (h_k)=\mu (\hat{h}_k)=0$, and Theorem 1.3.

In the other direction, if the equality holds in (5.7) for all $n\le n_0$, then from (5.5) we have, for $n\le n_0$,

$$\begin{aligned} \sum _{k=0}^n(\mu ^2(f_k)+\mu ^2(\hat{f}_k)+\frac{1}{2}\Delta ^2(f_k) +\frac{1}{2}\Delta ^2(\hat{f}_k)) =\frac{(n+1)^2}{2} \end{aligned}$$

and hence, from Theorem 1.3:

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \mu (f_k)=\mu (\hat{f}_k)=0&{}\forall \,0\le k\le n \\ \displaystyle \sum _{k=0}^n(\Delta ^2(f_k)+\Delta ^2(\hat{f}_k))=(n+1)^2. \end{array}\right. } \end{aligned}$$

Then we conclude from Theorem 1.3. $\square$

Let us remark that from Theorem 5.1 we immediately get the following uncertainty principle for the covariance matrix:

Corollary 5.2

If $\{f_j\}_{j\in J}$ is an orthonormal sequence in $L^2(\mathbb {R})$ with zero means $\mu (f_j)=\mu (\hat{f}_j)=0$, and if the trace of the covariance matrix of $|W(f_j)(x,\xi )|^2$ is uniformly bounded in j, then J is finite.

Proof

From Proposition 2.2(c), (d) we have that

$$\begin{aligned} M(X)= & {} \int _{\mathbb {R}^2}x|W(f_k)(x,\xi )|^2dxd\xi =\langle M_1W(f_k),W(f_k)\rangle =\mu (f_k)=0\\ M(Y)= & {} \langle M_2W(f_k),W(f_k)\rangle =\mu (\hat{f}_k)=0 \end{aligned}$$

by assumption, and hence from (5.3):

$$\begin{aligned} C(X,X)+C(Y,Y)=\int _{\mathbb {R}^2}(x^2+\xi ^2)|W(f_k)(x,\xi )|^2dxd\xi . \end{aligned}$$

The thesis thus immediately follows from Theorem 5.1. $\square$

Note that Corollary 5.2 can be stated also in terms of the variances of $|W(f_j)(x,\xi )|^2$ since, in general, the variances

$$\begin{aligned} V(X)= & {} \int _\mathbb {R}(x-M(X))^2\rho _X(x)dx,\\ V(Y)= & {} \int _\mathbb {R}(y-M(Y))^2\rho _Y(y)dy, \end{aligned}$$

for $\rho _X,\rho _Y,M(X),M(Y)$ defined as in (5.1)-(5.2), satisfy:

$$\begin{aligned} C(X,X)=V(X),\qquad C(Y,Y)=V(Y), \end{aligned}$$

if $\rho (x,y)\in L^1(\mathbb {R}^2)$.

6 Cohen Classes

Infinitely many operators playing the same role as in the previous sections may be constructed by means of the Cohen class

$$\begin{aligned} Q(f,g)=\frac{1}{\sqrt{2\pi }}\,\sigma * W(f,g),\qquad f,g\in \mathcal {S}(\mathbb {R}), \end{aligned}$$

for some tempered distribution $\sigma \in \mathcal {S}'(\mathbb {R}^2)$. For $f,g\in \mathcal {S}(\mathbb {R})$ we have $W(f,g)\in \mathcal {S}(\mathbb {R}^2)$, and then Q(f, g) is well-defined for every $\sigma \in \mathcal {S}'(\mathbb {R}^2)$. As for the Wigner we define

$$\begin{aligned} Q[w]=\frac{1}{\sqrt{2\pi }}\,\sigma *\textrm{Wig}[w],\qquad w\in \mathcal {S}(\mathbb {R}^2). \end{aligned}$$

If $\sigma =\mathcal {F}^{-1}(e^{-iP(\xi ,\eta )})\in \mathcal {S}'(\mathbb {R}^2)$ for some polynomial $P\in \mathbb {R}[\xi ,\eta ]$, we have the following result (see [3, Thms. 3.1 and 3.2]):

Theorem 6.1

Let $B(x,y,D_x,D_y)$ be a linear partial differential operator with polynomial coefficients and let $\sigma =\mathcal {F}^{-1}(e^{-iP(\xi ,\eta )})\in \mathcal {S}'(\mathbb {R}^2)$ for some $P\in \mathbb {R}[\xi ,\eta ]$. Then for every $w\in \mathcal {S}(\mathbb {R}^2)$:

• (i)
  $$\begin{aligned}{} & {} Q[B(M_1,M_2,D_1,D_2)w]\\{} & {} \quad =B\left( M_1-\frac{1}{2}D_2-P_1,M_1+\frac{1}{2} D_2 -P_1,\frac{1}{2}D_1+M_2-P_2, \frac{1}{2}D_1-M_2+P_2\right) Q[w] \end{aligned}$$

  for

  $$\begin{aligned} P_1=(iD_1P)(D_1,D_2), \quad P_2=(iD_2P)(D_1,D_2). \end{aligned}$$

  (6.1)
• (ii)
  $$\begin{aligned}{} & {} B(M_1,M_2,D_1,D_2)Q[w]\\{} & {} \quad =Q\left[ B\left( \frac{M_2+M_1}{2}+P_1^*, \frac{D_1-D_2}{2}+P_2^*,D_1+D_2,M_2-M_1\right) w\right] \end{aligned}$$

  for

  $$\begin{aligned} P_1^*=(iD_1P)(D_1+D_2,M_2-M_1), \quad P_2^*=(iD_2P)(D_1+D_2,M_2-M_1). \end{aligned}$$

Let us remark that if $\sigma =\mathcal {F}^{-1}(e^{-iP(\xi ,\eta )})$, then $|\hat{\sigma }|=1$ and hence, for all $f_1,f_2,g_1,g_2\in \mathcal {S}(\mathbb {R})$, from (2.7) and (2.12):

$$\begin{aligned} \langle Q(f_1,g_1),Q(f_2,g_2)\rangle= & {} \frac{1}{{2\pi }}\langle \sigma * W(f_1,g_1),\sigma *W(f_2,g_2)\rangle \nonumber \\= & {} \frac{1}{{2\pi }}\langle \mathcal {F}^{-1}(\sqrt{2\pi } \,\hat{\sigma }\cdot \widehat{W(f_1,g_1)}), \mathcal {F}^{-1}(\sqrt{2\pi }\,\hat{\sigma } \cdot \widehat{W(f_2,g_2)})\rangle \nonumber \\= & {} \langle \hat{\sigma }\cdot \widehat{W(f_1,g_1)}, \hat{\sigma }\cdot \widehat{W(f_2,g_2)}\rangle \nonumber \\= & {} \langle |\hat{\sigma }|^2\widehat{W(f_1,g_1)},\widehat{W(f_2,g_2)}\rangle =\langle \widehat{W(f_1,g_1)},\widehat{W(f_2,g_2)}\rangle \nonumber \\= & {} \langle W(f_1,g_1),W(f_2,g_2)\rangle =\langle f_1,f_2\rangle \overline{\langle g_1,g_2\rangle }, \end{aligned}$$

(6.2)

since $\widehat{f*g}=\sqrt{2\pi }\hat{f}\cdot \hat{g}$.

Moreover:

Theorem 6.2

Let $\{f_k\}_{k\in \mathbb {N}_0}\subset \mathcal {S}(\mathbb {R})$ be an orthonormal basis in $L^2(\mathbb {R})$ and $Q,\sigma$ as in Theorem 6.1. Then $\{Q(f_j,f_k)\}_{j,k\in \mathbb {N}_0}$ is an orthonormal basis in $L^2(\mathbb {R}^2)$.

Proof

Let us first remark that $Q(f_j,f_k)\in \mathcal {S}(\mathbb {R}^2)\subset L^2(\mathbb {R}^2)$. Moreover $\{Q(f_j,f_k)\}_{j,k\in \mathbb {N}_0}$ is an orthonormal sequence by (6.2).

We only have to prove that if $F\in L^2(\mathbb {R}^2)$ satisfies

$$\begin{aligned} \langle F,Q(f_j,f_k)\rangle =0,\qquad \forall j,k\in \mathbb {N}_0, \end{aligned}$$

then $F=0$ a.e. in $\mathbb {R}^2$ (see [7, Thm. 3.4.2]). Let $G=\mathcal {F}^{-1}(\hat{F}/\hat{\sigma })\in L^2(\mathbb {R}^2)$, so that from (2.7)

$$\begin{aligned} 0= & {} \langle F,Q(f_j,f_k)\rangle = \langle \hat{F}, \widehat{Q(f_j,f_k)}\rangle =\langle \hat{G}\cdot \hat{\sigma }, \hat{\sigma } \cdot \widehat{W(f_j,f_k)}\rangle \\= & {} \langle |\hat{\sigma }|^2\hat{G},\widehat{W(f_j,f_k)}\rangle =\langle \hat{G},\widehat{W(f_j,f_k)}\rangle =\langle G,{W(f_j,f_k)}\rangle ,\qquad \forall j,k\in \mathbb {N}_0, \end{aligned}$$

which implies $G=0$ a.e. in $\mathbb {R}^2$ since $\{W(f_j,f_k)\}_{j,k\in \mathbb {N}_0}$ is a basis in $L^2(\mathbb {R}^2)$ by Theorem 3.1.

Then $\hat{F}=\hat{G}\cdot \hat{\sigma }=0$, i.e. $F=0$ a.e. in $\mathbb {R}^2$. $\square$

Let us remark that if $f,g\in \mathcal {S}(\mathbb {R})\subset L^2(\mathbb {R})$ with $\Vert f\Vert =\Vert g\Vert =1$ then, by Lemma 4.2, (6.2) and Theorem 6.1, we have

$$\begin{aligned}{} & {} \mu ^2(g)+\mu ^2(\hat{g})+\Delta ^2(g)+\Delta ^2(\hat{g}) \nonumber \\{} & {} \quad =\langle \hat{L} W(f,g),W(f,g)\rangle \nonumber \\{} & {} \quad =\langle W(f,(M^2+D^2)g),W(f,g)\rangle =\langle Q(f,(M^2+D^2)g),Q(f,g)\rangle \nonumber \\{} & {} \quad =\left\langle \left[ \left( M_1+\frac{1}{2}D_2-P_1\right) ^2 +\left( \frac{1}{2}D_1-M_2+P_2\right) ^2\right] Q(f,g),Q(f,g)\right\rangle \end{aligned}$$

(6.3)

for $P_1,P_2$ as in (6.1).

Then Theorem 4.4 can be rephrased as follows, for any choice of $P\in \mathbb {R}[\xi ,\eta ]$:

Theorem 6.3

Let $\{f_k\}_{k\in \mathbb {N}_0},\{g_k\}_{k\in \mathbb {N}_0}\subset \mathcal {S}(\mathbb {R})$ be two orthonormal sequences in $L^2(\mathbb {R})$. Then

$$\begin{aligned} \sum _{k=0}^n\langle \tilde{L}Q(f_k,g_k),Q(f_k,g_k)\rangle \ge (n+1)^2, \qquad \forall n\in \mathbb {N}_0, \end{aligned}$$

(6.4)

for any linear partial differential operator $\tilde{L}$ of the form

$$\begin{aligned} \tilde{L}(M_1,M_2,D_1,D_2)=\left( M_1+\frac{1}{2}D_2-P_1\right) ^2 +\left( \frac{1}{2}D_1-M_2+P_2\right) ^2 \end{aligned}$$

with

$$\begin{aligned}{} & {} P_1=(iD_1P)(D_1,D_2),\qquad P_2=(iD_2P)(D_1,D_2),\\{} & {} P\in \mathbb {R}[\xi ,\eta ],\qquad \sigma =\mathcal {F}^{-1}(e^{-iP(\xi ,\eta )}),\\{} & {} Q(f_k,g_k)=\frac{1}{\sqrt{2\pi }}\,\sigma * W(f_k,g_k). \end{aligned}$$

Example 6.4

Let $P(D_1,D_2)=\frac{1}{2} D_1D_2$. Then

$$\begin{aligned} P_1= & {} iD_1P(\xi _1,\xi _2)\big |_{(\xi _1,\xi _2)=(D_1,D_2)}=\frac{1}{2}D_2\\ P_2= & {} iD_2P(\xi _1,\xi _2)\big |_{(\xi _1,\xi _2)=(D_1,D_2)}=\frac{1}{2}D_1 \end{aligned}$$

and hence

$$\begin{aligned} \tilde{L}= & {} \left( M_1+\frac{1}{2}D_2-\frac{1}{2}D_2\right) ^2 +\left( \frac{1}{2}D_1-M_2+\frac{1}{2}D_1\right) ^2\\= & {} M_1^2+(D_1-M_2)^2. \end{aligned}$$

Therefore, by Theorem 6.3, we obtain

$$\begin{aligned} \sum _{k=1}^n\langle (M_1^2+(D_1-M_2)^2)Q(f_k,f_k),Q(f_k,f_k)\rangle \ge (n+1)^2,\qquad \forall n\in \mathbb {N}_0. \end{aligned}$$

Example 6.5

Similar results can be obtained considering the operator $P(M_1,M_2)=M_1^2+M_2^2$ in (5.4) instead of $\hat{L}$ and then Theorem 5.1 instead of Corollary 4.5. Indeed, for $f\in \mathcal {S}(\mathbb {R})$ with $\Vert f\Vert =1$ we can write, by Proposition 2.1, (6.2) and Theorem 6.1:

$$\begin{aligned}{} & {} \langle (M_1^2+M_2^2)W(f),W(f)\rangle \\{} & {} \quad =\left\langle \textrm{Wig}\left[ \left( \frac{1}{4}(M_1+M_2)^2 +\frac{1}{4}(D_1-D_2)^2\right) f\otimes \bar{f}\right] ,W(f)\right\rangle \\{} & {} \quad =\frac{1}{4}\langle Q[((M_1+M_2)^2+(D_1-D_2)^2) f\otimes \bar{f}], Q(f)\rangle \\{} & {} \quad =\frac{1}{4}\left\langle \left( M_1-\frac{1}{2}D_2-P_1+M_1 +\frac{1}{2}D_2-P_1\right) ^2Q(f),Q(f)\right\rangle \\{} & {} \qquad +\frac{1}{4}\left\langle \left( \frac{1}{2}D_1+M_2-P_2 -\frac{1}{2}D_1+M_2-P_2\right) ^2Q(f),Q(f)\right\rangle \\{} & {} \quad =\langle ((M_1-P_1)^2+(M_2-P_2)^2)Q(f),Q(f)\rangle \end{aligned}$$

for any $P_1,P_2$ as in (6.1).

It follows that if $\{f_k\}_{k\in \mathbb {N}_0}\subset \mathcal {S}(\mathbb {R})$ is an orthonormal sequence in $L^2(\mathbb {R})$, then, from Theorem 5.1,

$$\begin{aligned} \sum _{k=0}^n\langle L^*Q(f_k),Q(f_k)\rangle \ge \frac{(n+1)^2}{2}, \qquad \forall n\in \mathbb {N}_0, \end{aligned}$$

(6.5)

for any linear partial differential operator $L^*$ of the form

$$\begin{aligned} L^*(M_1,M_2,D_1,D_2)=(M_1-P_1)^2+(M_2-P_2)^2 \end{aligned}$$

with

$$\begin{aligned}{} & {} P_1=(iD_1P)(D_1,D_2),\qquad P_2=(iD_2P)(D_1,D_2),\\{} & {} P\in \mathbb {R}[\xi ,\eta ],\qquad \sigma =\mathcal {F}^{-1}(e^{-iP(\xi ,\eta )}),\\{} & {} Q(f_k)=\frac{1}{\sqrt{2\pi }}\,\sigma * W(f_k). \end{aligned}$$

Remark 6.6

Any linear operator $T:\ L^2(\mathbb {R}^2)\rightarrow L^2(\mathbb {R}^2)$ (not necessarily everywhere defined) satisfying, for some orthonormal sequence $\{f_k\}_{k\in \mathbb {N}_0}\subset L^2(\mathbb {R})$,

$$\begin{aligned} \sum _{k=0}^n\langle TW(f_k,f_k),W(f_k,f_k)\rangle \ge (n+1)^2,\qquad \forall n\in \mathbb {N}_0, \end{aligned}$$

(6.6)

cannot be a bounded operator on $L^2(\mathbb {R}^2)$. Indeed, assuming by contradiction that T is bounded, by Theorem 3.1 we would have, for all $n\in \mathbb {N}_0$:

$$\begin{aligned} (n+1)^2\le & {} \sum _{k=0}^n\langle TW(f_k,f_k),W(f_k,f_k)\rangle \\\le & {} \sum _{k=0}^n\Vert T\Vert _{\mathcal {L}(L^2,L^2)}\Vert W(f_k,f_k)\Vert _{L^2}^2 =(n+1)\Vert T\Vert _{\mathcal {L}(L^2,L^2)} \end{aligned}$$

which gives a contradiction for large n. The above considerations can be applied to the partial differential operators with polynomial coefficients appearing in the various results where we have proved estimates of the kind of (6.6). This is not surprising since all non-constant differential operators with polynomial coefficients are in fact unbounded in $L^2(\mathbb {R}^n)$.

7 Riesz Bases

In this section we consider a general Riesz basis of $L^2(\mathbb {R})$ instead of an orthonormal basis. We recall that a Riesz basis in a Hilbert space H is the image of an orthonormal basis for H under an invertible linear bounded operator. In particular, if $\{u_k\}_{k\in \mathbb {N}_0}$ is a Riesz basis for $L^2(\mathbb {R})$, we can find an invertible linear bounded operator $U_1:\,L^2(\mathbb {R})\rightarrow L^2(\mathbb {R})$ such that

$$\begin{aligned} U_1(u_k)=h_k,\qquad \forall k\in \mathbb {N}_0, \end{aligned}$$

for the Hermite functions $\{h_k\}_{k\in \mathbb {N}_0}$. Moreover (see [7, Lemma 3.6.2])

$$\begin{aligned} 0<C_1:=\inf _{k\in \mathbb {N}_0}\Vert u_k\Vert ^2\le \sup _{k\in \mathbb {N}_0}\Vert u_k\Vert ^2=:C_2<+\infty . \end{aligned}$$

(7.1)

We can thus generalize Theorem 4.4 to Riesz bases:

Theorem 7.1

If $\{u_k\}_{k\in \mathbb {N}_0}$ and $\{v_k\}_{k\in \mathbb {N}_0}$ are Riesz bases for $L^2(\mathbb {R})$ and $\hat{L}$ is the operator in (1.4), then for all $n\in \mathbb {N}_0$

$$\begin{aligned} \sum _{k=0}^n\langle \hat{L}W(u_k,v_k),W(u_k,v_k)\rangle \ge \frac{\Vert U_2^{-1}\Vert _{\mathcal {L}(L^2,L^2)}^2}{\Vert U_1\Vert _{\mathcal {L}(L^2,L^2)}^2} \left[ \frac{n+1}{\Vert U_2^{-1}\Vert _{\mathcal {L}(L^2,L^2)}^2 \Vert U_2\Vert _{\mathcal {L}(L^2,L^2)}^2}\right] ^2,\quad \end{aligned}$$

(7.2)

where $U_j:\,L^2(\mathbb {R})\rightarrow L^2(\mathbb {R})$, $j=1,2$, are such that $U_1(u_k)=h_k$ and $U_2(v_k)=h_k$, for the Hermite functions $h_k$ defined in (1.8), and [x] denotes the integer part of x.

Proof

As in (4.8)-(4.9) we obtain that

$$\begin{aligned} \sum _{k=0}^n \langle \hat{L}W(u_k,v_k),W(u_k,v_k)\rangle= & {} \sum _{k=0}^n \left( \sum _{j=0}^{+\infty }| \langle u_k,h_j\rangle |^2\right) \sum _{\ell =0}^{+\infty }| \langle v_k,h_\ell \rangle |^2 (2\ell +1) \\ ={} & {} \sum _{\ell =0}^{+\infty } \sum _{k=0}^n \Vert u_k\Vert ^2| \langle v_k,h_\ell \rangle |^2 (2\ell +1), \end{aligned}$$

and we can suppose that the series in the right-hand side is convergent, otherwise (7.2) would be trivial. We thus obtain, for the constant $C_1$ defined in (7.1):

$$\begin{aligned} \sum _{k=0}^n\langle \hat{L}W(u_k,v_k),W(u_k,v_k)\rangle \ge C_1\sum _{\ell =0}^{+\infty } \sum _{k=0}^n|\langle v_k,h_\ell \rangle |^2(2\ell +1)=C_1\sum _{\ell =0}^{+\infty }\alpha _\ell (2\ell +1)\nonumber \\ \end{aligned}$$

(7.3)

for

$$\begin{aligned} \alpha _\ell :=\sum _{k=0}^n|\langle v_k,h_\ell \rangle |^2\le \sum _{k=0}^{+\infty }|\langle v_k,h_\ell \rangle |^2\le B \Vert h_\ell \Vert ^2=B \end{aligned}$$

for $B=\Vert U_2^{-1}\Vert ^2_{\mathcal {L}(L^2,L^2)}$ because of [7, Prop. 3.6.4].

We have

$$\begin{aligned} \sum _{\ell =0}^{+\infty }\alpha _\ell ={} & {} \sum _{\ell =0}^{+\infty } \sum _{k=0}^n|\langle v_k,h_\ell \rangle |^2 =\sum _{k=0}^n\sum _{\ell =0}^{+\infty }|\langle v_k,h_\ell \rangle |^2 =\sum _{k=0}^n\Vert v_k\Vert ^2\ge \tilde{C}_1(n+1), \end{aligned}$$

for $\tilde{C}_1:=\inf _{k\in \mathbb {N}_0} \Vert v_k\Vert ^2$. Note that $0<\tilde{C}_1\le \sup _{k\in \mathbb {N}_0}\Vert U_2^{-1}\Vert ^2_{\mathcal {L} (L^2,L^2)}\Vert h_k\Vert ^2=B$.

Let us now assume $n\ge \frac{B}{\tilde{C}_1}-1$ so that

$$\begin{aligned} m:=\left[ \frac{n+1}{B}\tilde{C}_1\right] -1\in \mathbb {N}_0 \end{aligned}$$

and write

$$\begin{aligned} \tilde{C}_1(n+1)\le \sum _{\ell =0}^{+\infty }\alpha _\ell =\alpha _0+\dots +\alpha _m+R_m \end{aligned}$$

with

$$\begin{aligned} R_m:=\sum _{\ell \ge m+1}\alpha _\ell . \end{aligned}$$

If $R_m=0$, then $\frac{n+1}{B}\tilde{C}_1\in \mathbb {N}$ and $\alpha _0=\dots =\alpha _m=B$ because otherwise or $m+1<\frac{n+1}{B}\tilde{C}_1$ or $\alpha _k<B$ for some $k=0,\dots ,m$ and

$$\begin{aligned} \tilde{C}_1(n+1)\le \alpha _0+\dots +\alpha _m<B \cdot \frac{n+1}{B}\tilde{C}_1=(n+1)\tilde{C}_1 \end{aligned}$$

would give a contradiction. It follows that setting

$$\begin{aligned} c_k:= {\left\{ \begin{array}{ll} 0 &{} \text{ if }\ R_m=0 \\ \frac{B-\alpha _k}{R_m}&{} \text{ if }\ R_m>0, \end{array}\right. } \end{aligned}$$

we have $c_k\ge 0$ and $\alpha _k+c_kR_m=B$ for all $0\le k\le m$.

Moreover, if $R_m>0$

$$\begin{aligned} (c_0+\dots +c_m)R_m={} & {} B\left[ \frac{n+1}{B}\tilde{C}_1\right] -(\alpha _0+\dots +\alpha _m)\\ \le{} & {} \tilde{C}_1(n+1)-(\alpha _0+\dots +\alpha _m)\le R_m. \end{aligned}$$

It follows that $c_0+\dots +c_m\le 1$ and hence, for all $n\ge \frac{B}{\tilde{C}_1}-1$,

$$\begin{aligned} \sum _{\ell =0}^{+\infty }\alpha _\ell (2\ell +1)= & {} \sum _{\ell =0}^{m}\alpha _\ell (2\ell +1) +\sum _{\ell \ge m+1}\alpha _\ell (2\ell +1)\\\ge & {} \sum _{\ell =0}^{m}\alpha _\ell (2\ell +1) +(c_0+\dots +c_m) \sum _{\ell \ge m+1}\alpha _\ell (2\ell +1)\\= & {} \sum _{\ell =0}^{m}\alpha _\ell (2\ell +1)+ c_{0} \sum _{\ell \ge m+1} \alpha _\ell \underbrace{(2\ell +1)}_{>1}+\dots + c_{m}\sum _{\ell \ge m+1}\alpha _\ell \underbrace{(2\ell +1)}_{> 2m+1}\\\ge & {} \underbrace{(\alpha _0 + c_{0}R_m)}_{=B}\cdot 1 +\dots +\underbrace{(\alpha _{m} + c_{m}R_m)}_{=B}\cdot (2m+1)\\= & {} B\sum _{k=0}^{m}(2k+1)=B(m+1)^2=B \left[ \frac{n+1}{B}\tilde{C}_1\right] ^2. \end{aligned}$$

Note that the above inequality is trivial if $\frac{n+1}{B}\tilde{C}_1<1$ so that from (7.3) we have, for all $n\in \mathbb {N}_0$,

$$\begin{aligned} \sum _{k=0}^n\langle \hat{L}W(u_k,v_k),W(u_k,v_k)\rangle \ge C_1B\left[ \frac{n+1}{B}\tilde{C}_1\right] ^2. \end{aligned}$$

(7.4)

Let us now remark that, from the continuity of $U_j:\,L^2(\mathbb {R})\rightarrow L^2(\mathbb {R})$, $j=1,2$,

$$\begin{aligned} 1=\Vert h_k\Vert _{L^2}\le \Vert U_1\Vert _{\mathcal {L}(L^2,L^2)} \cdot \Vert u_k\Vert _{L^2}, \quad 1=\Vert h_k\Vert _{L^2} \le \Vert U_2\Vert _{\mathcal {L}(L^2,L^2)}\cdot \Vert v_k\Vert _{L^2} \end{aligned}$$

for every $k\in \mathbb {N}_0$, and therefore

$$\begin{aligned} C_1=\inf _{k\in \mathbb {N}_0}\Vert u_k\Vert _{L^2}^2\ge \frac{1}{\Vert U_1\Vert _{\mathcal {L}(L^2,L^2)}^2}, \quad \tilde{C}_1=\inf _{k\in \mathbb {N}_0}\Vert v_k\Vert _{L^2}^2 \ge \frac{1}{\Vert U_2\Vert _{\mathcal {L}(L^2,L^2)}^2}. \end{aligned}$$

Since $B=\Vert U_2^{-1}\Vert _{\mathcal {L}(L^2,L^2)}^2$ we finally have from (7.4) that

$$\begin{aligned} \sum _{k=0}^n\langle \hat{L}W(u_k,v_k),W(u_k,v_k)\rangle \ge \frac{\Vert U_2^{-1}\Vert _{\mathcal {L}(L^2,L^2)}^2}{\Vert U_1 \Vert _{\mathcal {L}(L^2,L^2)}^2} \left[ \frac{n+1}{\Vert U_2^{-1}\Vert _{\mathcal {L}(L^2,L^2)}^2 \Vert U_2\Vert _{\mathcal {L}(L^2,L^2)}^2}\right] ^2 \end{aligned}$$

for all $n\in \mathbb {N}_0$.

From Theorem 7.1 and Lemma 4.2 we have the mean-dispersion principle for Riesz bases:

Corollary 7.2

Let $\{u_k\}_{k\in \mathbb {N}_0}$ be a Riesz basis in $L^2(\mathbb {R})$, with $U(u_k)=h_k$, for the Hermite functions $\{h_k\}_{k\in \mathbb {N}_0}$ defined in (1.8). Then for all $n\in \mathbb {N}_0$

$$\begin{aligned} \sum _{k=0}^n(\Delta ^2(u_k)+\Delta ^2(\hat{u}_k) +\mu ^2(u_k)+\mu ^2(\hat{u}_k)) \ge \frac{1}{\Vert U^{-1}\Vert ^2\Vert U\Vert ^2} \left[ \frac{n+1}{\Vert U^{-1}\Vert ^2\Vert U\Vert ^2}\right] ^2, \end{aligned}$$

where $\Vert \cdot \Vert =\Vert \cdot \Vert _{\mathcal {L}(L^2,L^2)}$.

Proof

From Lemma 4.2 we have that

$$\begin{aligned} \langle \hat{L}W(u_k),W(u_k)\rangle =\Vert u_k\Vert ^4 (\Delta ^2(u_k)+\Delta ^2(\hat{u}_k)+\mu ^2(u_k)+\mu ^2(\hat{u}_k)). \end{aligned}$$

Since $\Vert u_k\Vert \le \Vert U^{-1}\Vert \cdot \Vert h_k\Vert =\Vert U^{-1}\Vert$, by Theorem 7.1 we obtain:

$$\begin{aligned}{} & {} \sum _{k=0}^n(\Delta ^2(u_k)+\Delta ^2(\hat{u}_k) +\mu ^2(u_k)+\mu ^2(\hat{u}_k))\\{} & {} \quad \ge \frac{1}{\Vert U^{-1}\Vert ^4}\sum _{k=0}^n\Vert u_k\Vert ^4 (\Delta ^2(u_k)+\Delta ^2(\hat{u}_k)+\mu ^2(u_k)+\mu ^2(\hat{u}_k))\\{} & {} \quad \ge \frac{1}{\Vert U^{-1}\Vert ^2\Vert U\Vert ^2} \left[ \frac{n+1}{\Vert U^{-1}\Vert ^2\Vert U\Vert ^2}\right] ^2. \end{aligned}$$

$\square$

Note that if the Riesz basis $\{u_k\}_{k\in \mathbb {N}_0}$ is orthonormal, then $\Vert U\Vert =1$ since

$$\begin{aligned} U(f)=\sum _{k=0}^{+\infty }\langle f,u_k\rangle U(u_k) =\sum _{k=0}^{+\infty }\langle f,u_k\rangle h_k,\qquad f\in L^2(\mathbb {R}), \end{aligned}$$

and hence $\Vert U(f)\Vert _{L^2}=\Vert f\Vert _{L^2}$ for all $f\in L^2(\mathbb {R})$.

From Corollary 7.2 in the case of orthonormal Riesz bases we thus find again (1.7), i.e. Shapiro’s Mean Dispersion principle. This improves [16, Cor. 2.8] for $\Vert U\Vert =1$, where a weaker estimate is obtained with respect to Shapiro’s Mean Dispersion principle [16, Thm. 2.3].