Spectral clumping for functions and distributions decreasing rapidly on a half-line

This note studies a certain manifestation of the uncertainty principle in Fourier analysis, where a smallness condition on a function $f$ forces its Fourier transform $\widehat{f}$ to be, in some sense, large. Vice versa, smallness of $\widehat{f}$ forces $f$ to be large. In our context, the smallness is defined in terms of a one-sided decay condition, and the largeness in terms of the existence of a clump. This will be our moniker for an interval on which the function has an integrable logarithm. We emphasize that our results concern functions with a spectrum which might vanish on an interval (commonly referred to as functions with a spectral gap), but for which the spectrum should be large on some other interval.

We will use the following definition of the transform:

(1.1)

$$\widehat{f}(\zeta):=\int_{\mathbb{R}}f(x)e^{-ix\zeta}\,d\lambda(x),\quad\zeta\in\mathbb{R}.$$

Here $d\lambda(x)=dx/\sqrt{2\pi}$ is a normalization of the Lebesgue measure $dx$ on $\mathbb{R}$. Then, the inversion formula is given by

(1.2)

$$f(x):=\int_{\mathbb{R}}\widehat{f}(\zeta)e^{i\zeta x}\,d\lambda(\zeta),\quad x\in\mathbb{R}.$$

For $p>0$, let $\mathcal{L}^{p}(\mathbb{R},dx)$ be the usual Lebesgue space of functions $f$ for which $|f|^{p}$ is integrable with respect to $dx$. The formula (1.1) can be interpreted literally only for $f\in\mathcal{L}^{1}(\mathbb{R},dx)$. It is interpreted in terms of Plancherel’s theorem in the case $f\in\mathcal{L}^{2}(\mathbb{R},dx)$, and in order to state our most general results we will later need to interpret the transform in the sense of distribution theory. The spectrum $\sigma(f)$ of a function $f$ is the subset of $\mathbb{R}$ on which $\widehat{f}$ lives. Since $\widehat{f}\in\mathcal{L}^{2}(\mathbb{R},dx)$ is defined only up to a set of Lebesgue measure zero, so is the spectrum $\sigma(f)$ in this case. If we accept making errors of measure zero (which we will), we may define the spectrum as

$$\sigma(f):=\{\zeta\in\mathbb{R}:|\widehat{f}(\zeta)|>0\},\quad f\in\mathcal{L}^{1}(\mathbb{R},dx)\cup\mathcal{L}^{2}(\mathbb{R},dx).$$

Note specifically that our definition of $\sigma(f)$ might not coincide with the usual notion of closed support of the distribution $\widehat{f}$.

The uncertainty principle in Fourier analysis presents itself in plenty of ways, and the excellent monograph [6] of Havin and Jöricke describes many of its most interesting interpretations. One of them is the following statement, well-known to function theorists. If $f\in\mathcal{L}^{2}(\mathbb{R},dx)$ is non-zero and $\mathbb{R}_{-}$ is the negative half-axis, then we have the implication

(1.3)

$$f(x)\equiv 0\text{ on }\mathbb{R}_{-}\quad\Rightarrow\quad\int_{\mathbb{R}}\frac{\log|\widehat{f}(\zeta)|}{1+\zeta^{2}}\,d\zeta>-\infty.$$

Here the extreme decay (indeed, vanishing) of $f$ on a half-axis implies global integrability of $\log|\widehat{f}|$ against the Poisson measure $d\zeta/(1+\zeta^{2})$. A fortiori, $\log|\widehat{f}|$ is integrable on every interval $I$ of $\mathbb{R}$. Naturally, this is not typical. By Plancherel’s theorem, every function in $\mathcal{L}^{2}(\mathbb{R},dx)$ is the Fourier transform of some other function in the same space. So on the other extreme, plenty of functions $f\in\mathcal{L}^{2}(\mathbb{R},dx)$ have a Fourier transform which lives on sparse sets containing no intervals. This forces the divergence of the logarithmic integral of $\widehat{f}$ over any interval. In other words, plenty of functions admit no spectral clumps. The results of this note give conditions under which such clumps form.

We shall introduce our results at first in the context of the Hilbert space $\mathcal{L}^{2}(\mathbb{R},dx)$. Here we can prove a claim which symmetric in $f$ and $\widehat{f}$, and also we can argue for near-optimality of the result. This is the content of \threfCondensationTheorem and \threfSparsenessTheorem. The more general distributional clumping result is presented in \threfDistributionalClumpingTheorem.

In other words, the one-sided decay condition (1.4) implies that $\widehat{f}$ lives on the union of the spectral clumps of $f$. Since the Fourier transform is a unitary operation on $\mathcal{L}^{2}(\mathbb{R},dx)$, the roles of $f$ and $\widehat{f}$ may obviously be interchanged in the statement of \threfCondensationTheorem. Thus a one-sided spectral decay condition of $f$ implies local integrability properties of $\log|f|$ on the set where $f$ lives. In this form, the result encourages us to extend it to tempered distributions. We shall do so in a moment.

The integrand in (1.4) may seem a bit unnatural in the context of square-integrable functions $f$. It is more natural in the context of functions of tempered growth appearing in \threfDistributionalClumpingTheorem. Anyhow, we note that one can prove that an estimate of the form $\int_{x}^{\infty}|f(t)|^{2}\,dt=\mathcal{O}\big{(}e^{-c\sqrt{x}}\big{)}$ in fact implies (1.4) for some slightly smaller $c$.

We can prove also that the condition (1.4) on the decay of $\rho_{f}$ appearing in \threfCondensationTheorem is close to optimal. We do so by exhibiting a non-zero function with rapid one-sided decay but sparse spectrum.

After an initial reduction, the proof of this result follows ideas of Khrushchev from [8]. Note that the function $f$ appearing in \threfSparsenessTheorem is entire by the virtue of having a spectrum $\sigma(f)$ of compact support. More importantly, the condition on $E$ implies that $I\setminus E$ has positive Lebesgue measure for every interval $I$, so we obtain

$$\int_{I}\log|\widehat{f}(t)|\,dt=-\infty$$

for every interval $I\subset\mathbb{R}$. This is in contrast to the conclusion of \threfCondensationTheorem. It follows that the exponent $a=1/2$ in estimates of the form $\rho_{f}(x)=\mathcal{O}\big{(}e^{-x^{a}}\big{)}$ is critical for the spectral clumping phenomenon.

As mentioned above, clumping statements makes sense for objects in a class much wider than $\mathcal{L}^{2}(\mathbb{R},dx)$. Here is our distrbutional result.

For instance, the result shows that a function $f\in\mathcal{L}^{1}(\mathbb{R},dx)$ which lives on a sparse set containing no intervals cannot satisfy even a one-sided spectral decay condition of the form $\rho_{\widehat{f}}(\zeta)\lesssim e^{-c\sqrt{\zeta}}$. Note also that in this extended form, our result includes the trivial but important examples such as $f=1$ and $\widehat{f}=\delta_{0}$ (Dirac delta), the trigonometric functions and the polynomials.

The Beurling-Malliavin theory implies a partial converse result. If $f$ is a locally integrable function on $\mathbb{R}$ which has a clump $I$ as in (1.5), and a constant $c>0$ is given, then a bounded multiplier $m$ exists for which $mf$ has a Fourier transform satisfying $\rho_{\widehat{mf}}(\zeta)=\mathcal{O}\big{(}e^{-c\sqrt{\zeta}}\big{)}$ for $\zeta>0$. To see this, recall that a smooth function $g$ supported in $I$ exists which satisfies the bilateral spectral decay $|\widehat{g}(\zeta)|\leq e^{-c\sqrt{|\zeta|}}$, $\zeta\in\mathbb{R}$ (this simpler version of the famous Beurling-Malliavin theorem is proved in [6, p. 276-277], and in fact we may ensure an even faster bilateral spectral decay of $g$). There exists also a bounded function $h\in\mathcal{L}^{1}(\mathbb{R},dx)$ which satisfies $\sigma(h)\subset(0,\infty)$ and $|h(x)|=\min(|f(x)|,1)$ on $I$ (we use the assumption that $I$ is a clump for $f$ and construct $h$ as in (2.7) below). Then an argument similar to the one used in the proof of \threfConvolutionFourierDecayLemma below shows that the function $\overline{h}g$ will satisfy the desired one-sided spectral decay, and clearly $\overline{h}g=mf$ for some bounded function $m$ supported in $I$.

The motivation for the research presented in this note was a desire to produce a self-contained exposition of the clumping phenomenon which was observed in two other contexts, both somewhat more esoteric than Fourier analysis on the real line.

The first of these is a polynomial approximation problem in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$. Here we are presented with a measure

$$d\mu=G(1-|z|)dA(z)+w(z)d\textit{m}(z),$$

where $dA$ and $d\textit{m}$ are the area and arc-length measures on $\mathbb{D}$ and $\mathbb{T}:=\partial\mathbb{D}=\{z\in\mathbb{C}:|z|=1\}$. The functions $G$ and $w$ are non-negative weights, and one would like to understand under which conditions splitting occurs. Namely, when is the weighted space $\mathcal{L}^{2}(\mathbb{T},w\,d\textit{m})$ contained in the closure of analytic polynomials in the $\mathcal{L}^{2}$-norm induced by the measure $\mu$? In the case that $G(1-|z|)$ decays exponentially as $|z|\to 1^{-}$, the necessary and sufficient condition is that $w$ has no clumps, or in other words that the integral of $\log w$ diverges over any arc on $\mathbb{T}$. The lack-of-clumping condition was conjectured by Kriete and MacCluer in [9] and confirmed in [11]. Some of the techniques used in the proofs of the results in the present note are adaptations of the ideas from [11].

The other context is a circle of ideas surrounding the Aleksandrov-Clark measures appearing in spectral theory, and spaces $\mathcal{H}(b)$ defined by de Branges and Rovnyak, well-known to operator theorists. To any positive finite Borel measure $\nu$ on $\mathbb{T}$ we may associate a so-called Clark operator $\mathcal{C}_{\nu}$ which takes a function $g\in\mathcal{L}^{2}(\mathbb{T},d\nu)$ to the analytic function in $\mathbb{D}$ given by the formula

$$\mathcal{C}_{\nu}g(z):=\frac{\int_{\mathbb{T}}\frac{g(x)}{1-\overline{x}z}d\nu(x)}{\int_{\mathbb{T}}\frac{1}{1-\overline{x}z}d\nu(x)},\quad z\in\mathbb{D}.$$

The operator $\mathcal{C}_{\nu}$ maps $\mathcal{L}^{2}(\mathbb{T},d\nu)$ onto a space of analytic functions denoted by $\mathcal{H}(b)$, the symbol function $b:\mathbb{D}\to\mathbb{D}$ itself being related to $\nu$ by the formula

$$\frac{1}{1-b(z)}=\int_{\mathbb{T}}\frac{1}{1-\overline{x}z}d\nu(x),\quad z\in\mathbb{D}$$

in the case that $\nu$ is a probability measure, with a similar formula in the general case. For many choices of $\nu$ (or equivalently, choices of $b$), the space $\mathcal{H}(b)$ is somewhat mysterious, with the distinctive feature of containing very few functions extending analytically to a disk larger than $\mathbb{D}$. This extension property is characterized by the exponential decay of the Taylor series of the function, and the clumping of the absolutely continuous part of $\nu$ is decisive for existence and density of functions in $\mathcal{H}(b)$ which have a Taylor series decaying just a bit slower than exponentially. Results of this nature are contained in [12]. In fact, a Fourier series version of \threfCondensationTheorem is a consequence of the results in [12].

The implication (1.3) has a well-known Fourier series version. If a function $f$ defined on the circle $\mathbb{T}:=\{z\in\mathbb{C}:|z|=1\}$ is integrable with respect to arc-length $ds$ on $\mathbb{T}$, and the negative portion of the Fourier series of $f$ vanishes, then $\int_{\mathbb{T}}\log|f|ds>-\infty$, unless $f$ is the zero function.. Volberg derived the same conclusion from the weaker hypothesis of nearly-exponentially decaying negative portion of the Fourier series (see [15] and the exposition in [16]). Work of Borichev and Volberg [3] contains related results.

The decay condition (1.4) on $f\in\mathcal{L}^{2}(\mathbb{R},dx)$ prohibits $\widehat{f}$ from living on a set $S$ containing no intervals. Somewhat related are uniqueness statements in which one seeks to give examples of pairs of sets $(E,S)$ for which the following implication is valid: if $f$ in a certain class lives on $E$ and $\widehat{f}$ lives on $S$, then $f\equiv 0$. One says that $(E,S)$ is then a uniqueness pair for the corresponding class. A famous result of Benedicks presented in [2] (see also [1]) says that $(E,S)$ is a uniqueness pair for integrable $f$ if both sets have finite Lebesgue measure, and the result holds not only for the real line $\mathbb{R}$ but also for the $d$-dimensional Euclidean space $\mathbb{R}^{d}$. Hedenmalm and Montes-Rodríguez worked with the hyperbola $H=\{(x,y)\in\mathbb{R}^{2}:xy=1\}$ and th class of finite Borel measures $\mu$ supported on $H$ which are absolutely continuous with respect to arclength on $H$. They proved in [7] that if $\widehat{\mu}$ vanishes on certain types of discrete sets $\Lambda\subset\mathbb{R}^{2}$, then $\mu\equiv 0$, thus exhibiting interesting uniqueness pairs of the form $(H,\mathbb{\mathbb{R}}^{2}\setminus\Lambda)$. Recent work of Radchenko and Viazovska on interpolation formulas for Schwartz functions in [13] gives examples of pairs of discrete subsets $E$ and $S$ of $\mathbb{R}$ for which $(\mathbb{R}\setminus E,\mathbb{R}\setminus S)$ is a uniqueness pair for functions in the Schwartz class. Kulikov, Nazarov and Sodin exhibit similar interpolation formulas, and consequently new uniqueness pairs, in their recent work in [10].

For a set $E\subset\mathbb{R}$ and a measure $\mu$ defined on $\mathbb{R}$, the space $\mathcal{L}^{p}(E,d\mu)$ denotes the usual Lebesgue space consisting of equivalence classes of functions living only on $E$ and satisfying the integrability condition $\int_{E}|f(x)|^{p}d\mu(x)<\infty$. The containment $\mathcal{L}^{p}(E,d\mu)\subset\mathcal{L}^{p}(\mathbb{R},d\mu)$ is interpreted in the natural way. The symbols such as $dx$, $dt$ and $d\zeta$ denote the usual Lebesgue measure of the real line, while $d\lambda=dx/\sqrt{2\pi}$ will be the normalized version used in formulas involving Fourier transforms. If $E$ is a subset of $\mathbb{R}$, then $|E|$ denotes its usual Lebesgue measure. The positive half-axis of $\mathbb{R}$ is denoted by $\mathbb{R}_{+}:=\{x\in\mathbb{R}:x\geq 0\}$, and we set also $\mathbb{R}_{-}:=\mathbb{R}\setminus\mathbb{R}_{+}$. The notions of almost everywhere and of measure zero are always to be interpreted in the sense of Lebesgue measure on $\mathbb{R}$. The indicator function of a measurable set $E$ is denoted by $\mathbbm{1}_{E}$. Finally, we put $\log^{+}(x):=\max(\log(x),0)$.

For $p$ equal to $1$ or $2$, we denote by $\mathcal{H}^{p}(\mathbb{R})$ the subspace of $\mathcal{L}^{p}(\mathbb{R},dx)$ consisting of those functions $f$ for which the Fourier transform $\widehat{f}$ vanishes on the negative part of the real axis:

$$\mathcal{H}^{p}(\mathbb{R}):=\{f\in\mathcal{L}^{p}(\mathbb{R},dx):\widehat{f}|\mathbb{R}_{-}\equiv 0\}.$$

It is a well-known fact that functions in the Hardy classes $\mathcal{H}^{1}(\mathbb{R})$ and $\mathcal{H}^{2}(\mathbb{R})$ admit a type of analytic extension to the upper half-plane

$$\mathbb{H}:=\{x+iy\in\mathbb{C}:y>0\}.$$

We recall what exactly is meant by this extension and how it can be constructed. The Poisson kernel of the upper half-plane

$$\mathcal{P}(t,x+iy):=\frac{1}{\pi}\frac{y}{(x-t)^{2}+y^{2}},\quad y>0,$$

admits a decomposition

(2.1)

$$\mathcal{P}(t,z)=\operatorname{Re}\Bigg{(}\frac{1}{\pi i(t-z)}\Bigg{)}=\frac{1}{2\pi i}\Bigg{(}\frac{1}{t-z}-\frac{1}{t-\overline{z}}\Bigg{)},\quad z=x+iy\in\mathbb{H}.$$

Since

(2.2)

$$\frac{1}{t-\overline{z}}=-i\int_{0}^{\infty}e^{-i\overline{z}s}e^{its}\,ds$$

we may use Fubini’s theorem to compute, in the case $f\in\mathcal{H}^{1}(\mathbb{R})$, that

(2.3)

$$\int_{\mathbb{R}}\frac{f(t)}{t-\overline{z}}\,dt=-i\int_{0}^{\infty}\Bigg{(}\int_{\mathbb{R}}f(t)e^{its}\,dt\Bigg{)}e^{-i\overline{z}s}ds=0,$$

where the vanishing of the integral follows from

$$\int_{\mathbb{R}}f(t)e^{its}\,ds=\widehat{f}(-s)=0,\quad s>0,$$

which holds for any $f\in\mathcal{H}^{1}(\mathbb{R})$ by the definition of the class. In the case $f\in\mathcal{H}^{2}(\mathbb{R})$ this argument does not work, but what instead works is an application of Plancherel’s theorem and \threfFourierTransformCauchyKernel below to the first integral in (2.3), which again shows that this integral vanishes. Consequently, whenever $f\in\mathcal{H}^{p}(\mathbb{R})$ for $p=1,2$, the formula

$$f(z):=\int_{\mathbb{R}}f(t)\mathcal{P}(t,z)\,dt=\int_{\mathbb{R}}\frac{f(t)}{t-z}\frac{dt}{2\pi i},\quad z\in\mathbb{H}$$

defines, by the second integral expression above, an analytic extension of $f$ to $\mathbb{H}$. By the first expression, and classical properties of the Poisson kernel (see [4, Chapter I]), this extension satisfies

(2.4)

$$\lim_{y\to 0^{+}}f(x+iy)=f(x),\text{ for almost every }x\in\mathbb{R}.$$

Moreover, we have

(2.5)

$$\sup_{y>0}\int_{\mathbb{R}}|f(x+iy)|^{p}\,dx<\infty$$

and

(2.6)

$$\lim_{y\to 0^{+}}\int_{\mathbb{R}}|f(x+iy)-f(x)|^{p}\,dx=0.$$

if $f\in\mathcal{H}^{p}(\mathbb{R})$. The property (2.5) follows readily from the Poisson integral formula for the extension of $f$ and Fubini’s theorem. The property (2.6) is a bit tricky to establish, and is proved in [4, Chapter I, Theorem 3.1]. In fact, the above listed properties characterize the functions in the Hardy classes.

The proposition is not hard to derive from \threfH1FourierTransformFormula below. Anyway, a careful proof can be found in [6, p. 172].

The following restriction on smallness of the modulus $|f|$ of a function $f\in\mathcal{H}^{1}(\mathbb{R})$ will be of crucial importance to us.

A proof of the proposition can be found in [6, p. 35].

We shall also need to use the corresponding Hardy class of functions which are merely bounded on $\mathbb{R}$, and not necessarily integrable or square-integrable on $\mathbb{R}$. We use directly the complex interpretation of the class. Namely, we define $\mathcal{H}^{\infty}(\mathbb{R})$ to consist of those functions $f\in\mathcal{L}^{\infty}(\mathbb{R},dx)$ which can be realized as limits

$$\lim_{y\to 0^{+}}f(x+iy):=f(x)$$

for almost every $x\in\mathbb{R}$, where $f$ is bounded and analytic in $\mathbb{H}$. It can be checked that such $f$ has a distributional spectrum which vanishes on $\mathbb{R}_{-}$. Another important point is that if $f\in\mathcal{H}^{\infty}(\mathbb{R})$, then

$$\frac{f(x)}{(i+x)^{2}}\in\mathcal{H}^{1}(\mathbb{R}),$$

since we may apply \threfH1CharacterizationProp to the analytic function

$$z\mapsto\frac{f(z)}{(i+z)^{2}},\quad z\in\mathbb{H}.$$

A function $h\in\mathcal{H}^{\infty}(\mathbb{R})$ of a given (bounded, measurable) modulus $|h|=W$ on $\mathbb{R}$ may be constructed by setting

(2.7)

$$\log h(z):=\frac{1}{\pi i}\int_{\mathbb{R}}\Big{(}\frac{1}{t-z}-\frac{t}{1+t^{2}}\Big{)}\log W(t)\,dt,\quad z\in\mathbb{H},$$

and $h(z):=e^{\log h(z)}$. The integral above converges if

$$\int_{\mathbb{R}}\frac{\log W(t)}{1+t^{2}}\,dt>-\infty$$

which is a necessary condition for the construction to be possible. Then

$$\log|h(z)|=\int\mathcal{P}(t,z)\log W(t)\,dt,$$

so that the equality $\lim_{y\to 0^{+}}|h(x+iy)|=|h(x)|=W(x)$ for almost every $x\in\mathbb{R}$ is a consequence of the well-known properties of the Poisson kernel.

If $f\in\mathcal{H}^{1}(\mathbb{R})$, then the values of $\widehat{f}(\zeta)$ may be computed using a formula different from (1.1). To wit, denote by $f(z)$ the extension of $f$ to $\mathbb{H}$ which was discussed in Section 2.1. The function

$$G_{\zeta}(z):=f(z)e^{-iz\zeta}=f(z)e^{-ix\zeta+y\zeta},\quad z=x+iy\in\mathbb{H}$$

is analytic in $\mathbb{H}$, and for this reason Cauchy’s integral theorem implies

(2.8)

$$\int_{R(\epsilon,y,a)}G_{\zeta}(z)dz=0,$$

where $dz$ denotes the complex line integral, $\epsilon,y,a$ are all positive numbers, $\epsilon<y$, and $R(\epsilon,y,a)$ denotes the rectangular contour having as corners the four points with coordinates $(-a,\epsilon)$, $(a,\epsilon)$, $(a,y)$, $(-a,y)$, oriented counter-clockwise.

Fix $y>0$ and let $S_{y}$ denote the horizontal strip in $\mathbb{H}$ consisting of all complex numbers with imaginary part between $0$ and $y$. Then it follows from Fubini’s theorem and (2.5) that

$\displaystyle\int_{S}|G_{\zeta}(z)|dA(z)\leq e^{y|\zeta|}\int_{\mathbb{R}}\int_{0}^{y}|f(x+is)|dsdx<\infty,$

where $dA(z)$ denotes the area measure on the complex plane. This expresses the integrability on $\mathbb{R}$ of the continuous function

$$x\mapsto\int_{0}^{y}|G_{\zeta}(x+is)|ds.$$

Hence there exists a positive sequence $\{a_{n}\}_{n}$ which satisfies

$$\lim_{n\to\infty}a_{n}=+\infty$$

and for which

$$\lim_{n\to\infty}\int_{0}^{y}|G_{\zeta}(a_{n}+is)|ds+\int_{0}^{y}|G_{\zeta}(-a_{n}+is)|ds=0.$$

This means that

	$\displaystyle 0$	$\displaystyle=\lim_{n\to\infty}\int_{R(\epsilon,y,a_{n})}G_{\zeta}(z)\,dz$
		$\displaystyle=-\int_{\mathbb{R}}G_{\zeta}(x+iy)dx+\int_{\mathbb{R}}G_{\zeta}(x+i\epsilon)dx$

Moreover, equation (2.6) quite easily implies

$$\lim_{\epsilon\to 0^{+}}\int_{\mathbb{R}}G_{\zeta}(x+i\epsilon)dx=\int_{\mathbb{R}}f(x)e^{-ix\zeta}dx=\sqrt{2\pi}\widehat{f}(\zeta).$$

We have proven the following formula by combining the above two expressions.

This formula has the following simple corollary which will be of critical importance below.

If $w\in\mathcal{L}^{1}(\mathbb{R},dx)$ and $s\in\mathbb{R}$, the operator $U^{s}:\mathcal{L}^{2}(\mathbb{R},w\,dx)\to\mathcal{L}^{2}(\mathbb{R},w\,dx)$ given by

$$U^{s}f(x):=e^{isx}f(x)$$

is unitary on $\mathcal{L}^{2}(\mathbb{R},w\,dx)$. We shall be interested in subspaces of $\mathcal{L}^{2}(\mathbb{R},w\,dx)$ which are invariant for the operators in the semigroup $\{U^{s}\}_{s>0}$. Given any element $f\in\mathcal{L}^{2}(\mathbb{R},w\,dx)$, we denote by $[f]_{w}$ the smallest closed linear subspace of $\mathcal{L}^{2}(\mathbb{R},w\,dx)$ which contains $f$ and also all the functions $U^{s}f$, $s>0$.