Sbornik: Mathematics, 2022, Volume 213, Issue 8, Pages 1020–1040 DOI: https://doi.org/10.4213/sm9687e (Mi sm9687)

DOI: https://doi.org/10.4213/sm9687e

Funding agency	Grant number
Ministry of Education and Science of the Russian Federation	FZWU-2020-0027
This research was carried out in the framework of the state assignment of the Ministry of Science and Education of the Russian Federation (code of research theme FZWU-2020-0027).

Russian version:
Matematicheskii Sbornik, 2022, Volume 213, Number 8, Pages 3–25
DOI: https://doi.org/10.4213/sm9687

Bibliographic databases:

§ 1. Introduction

Let $(a,b)$ be a finite or infinite interval of the real line and let $\mathcal E(a,b) =C^{\infty} (a,b)$ be the Schwarz space with the metrizable topology of the inverse limit of Banach spaces $C^k [a_k,b_k]$, where $[a_1,b_1]\Subset [a_2,b_2]\Subset \dotsb$ is some sequence of intervals exhausting $(a,b)$. It is known that $\mathcal E(a,b)$ is a reflexive Fréchet space. Let $W$ be a closed subspace of this space that is invariant under the operator of differentiation $D={\mathrm{d}}/{\mathrm{d}t}$ (or $D$-invariant). Recall that a residual interval $I_W$ of $W$ is by definition a minimal relatively closed (in $(a,b)$) subinterval $I$ such that $W_{I}\subset W$, where

Let $\Lambda$ be a sequence of points (with multiplicities) in the complex plane that has the unique limit point at infinity, and let $\operatorname{Exp} (\Lambda)$ be the sequence of exponential monomials constructed from the set of exponents $(-\mathrm i\Lambda)$: to a point ${\lambda\in\Lambda}$ occurring with multiplicity $k$ in this sequence we assign the set of functions $e^{-\mathrm{i}\lambda t }, te^{-\mathrm{i}\lambda t},\dots,t^{k-1}e^{-\mathrm{i}\lambda t }$.

Recall that the completeness radius $\rho (\Lambda)$ of $\Lambda$ is the infimum of positive $d$ such that the system of functions $\operatorname{Exp} (\Lambda)$ is not complete in $\mathcal E (-d,d)$.

If $\rho (\Lambda)<(b-a)/2$, then $\mathcal E(a,b)$ contains nontrivial $D$-invariant subspaces $W$ such that the restriction $D\colon W\to W$ of the operator of differentiation has a discrete spectrum which coincides with $(-\mathrm{i}\Lambda )$. In this case the set of exponential polynomials lying in $W$ is $\operatorname{Exp} (\Lambda)$. For such a subspace $W$ the length $|I_W|$ of its residual interval $I_W$ satisfies $|I_W|\geqslant 2 \rho (\Lambda)$.

For the operation of differentiation in the Schwartz space the problem of spectral synthesis was stated in [1], § 6, as the question of whether the representation

The reason why for $D$-invariant subspaces we consider weak spectral synthesis (1.2), rather than classical synthesis (1.3), is the presence of subspaces of the form $W_I$ in $\mathcal E(a,b)$ (which are clearly $D$-invariant and nontrivial for $I\neq (a,b)$, but contain no exponential monomials).

We collect the results on spectral synthesis in the weak sense (1.2) in the following statement.

It follows from Theorem A that a $D$-invariant subspace $W$ with finite spectrum admits weak spectral synthesis (1.2). Moreover, in this case $W$ is the direct (algebraic and topological) sum of the finite-dimensional subspace $\operatorname{span}\operatorname{Exp} (\Lambda)$ and the residual subspace $W_{I_W}$:

Weak spectral synthesis (1.2) is a variant of the generalization of (1.4) to the case of an infinite discrete spectrum. On the other hand, it was asked in [1], § 6, whether the representation

We have been successful in showing that the conditions ensuring (1.5) for a nontrivial $D$-invariant subspace with infinite discrete spectrum have a similar form to the conditions ensuring weak spectral synthesis (1.2) in Theorem A. However, instead of the completeness radius $\rho(\Lambda)$ we have to use another characteristic of $\Lambda$. We define this new characteristic $D_{\mathrm{sd}} (\Lambda)$ in § 2. We announced recently [5] some of the results concerning the representation (1.5). Sections 2 and 3 of our paper contain, in particular, a detailed presentation of these results with full proofs and relevant comments. In the final section, § 4, we discuss threshold situations and examples illustrating them, as well as the properties of $D$-invariant subspaces representable as the direct sum (1.5).

§ 2. Statements of the main results

2.1. Auxiliary facts

Recall that the Schwarz algebra $\mathcal P$ is by definition the image of the strong dual $\mathcal E'$ of the space $\mathcal E=C^{\infty} (\mathbb R)$ under the Fourier-Laplace transform $\mathcal F$:

It is also known that $\mathcal P$ is the direct image of the countable sequence of Banach spaces $\{ P_k\}$, where $P_k$ consists of all entire functions $\varphi$ with finite norm

For an arbitrary interval $I\subset \mathbb R$ we introduce the space $\mathcal P(I)$ associated with $I$. It is defined to be the direct limit of the sequence of Banach spaces $\widetilde{P}_k$. In its turn the space $\widetilde{P}_k$, $k=1,2,\dots$, consists of the entire functions $\varphi$ with finite norm

The embeddings $\widetilde{P}_k\subset \widetilde{P}_{k+1}$ are completely continuous, so $\mathcal P(I)$ is a locally convex space of type $(\mathrm{LN}^*)$. Moreover, $\mathcal P(I)$ is a topological module over the polynomial ring $\mathbb C[z]$. The Schwartz algebra $\mathcal P$ is $\mathcal P(\mathbb R)$.

For an interval $I\subset\mathbb R$ let $\mathcal E(I)$ denote the space of infinitely differentiable functions on $I$ with the metrizable topology of an inverse limit of Banach spaces, similarly to how we did it above for $ I=(a,b)$. For example, if $I=[c,d]$, then $\mathcal E(I)$ is the inverse limit of the Banach spaces $C^k[c,d]$; if $I=[c,d)$, $d<+\infty$, then $\mathcal E(I)$ is the inverse limit of the Banach spaces $C^k[c,d_k]$, where $c<d_k\nearrow d$ as $k\to\infty$.

The space $\mathcal E(I) $ and each closed subspace $W$ of it, with the topology induced from $\mathcal E(I)$, are reflexive Fréchet spaces.

The strong dual $\mathcal E'(I)$ of $\mathcal E(I)$ consists of the distributions $S\in\mathcal E'$ with support in $I$. By the Paley-Wiener-Schwartz theorem (see [7], Theorem 7.3.1)

Let $W\subset \mathcal E (I)$ be a $D$-invariant subspace. Its annihilator submodule $\mathcal J$ in $\mathcal P(I)$ is defined by $\mathcal J=\mathcal F (W^0)$, where

Let $\Lambda\subset\mathbb C$ be a sequence such that the system of functions $\operatorname{Exp} (\Lambda)$ is not complete in $\mathcal E(I)$. Set

Let $\mathcal J(\Lambda, I)$ denote the set of functions $\varphi\in\mathcal P(I)$ that vanish on $\Lambda$. It is easy to see that $\mathcal J(\Lambda, I)$ is a localizable submodule of $\mathcal P(I)$ (see [8], § 1). It follows from the dual scheme presented in full detail in our papers [2] and [3] that $\mathcal J(\Lambda, I)$ is the annihilator submodule of the subspace $E(\Lambda,I)$.

Let

2.2. The characteristic $D_{\mathrm{sd}}(\Lambda)$ and main results

Given a complex sequence $\Lambda$, let $D_{\mathrm{BM}} (\Lambda)$ denote its Beurling-Malliavin density (for instance, see [9], § IX.D.2). By the well-known result on the completeness radius due to Beurling and Malliavin ([9], § X.B.3) we have

Recall that $\varphi\in\mathcal P$ is called a slowly decreasing function if there exists $ a>0$ such that

The slow decrease of $\psi\in\mathcal P$ is equivalent to the closedness of the principal ideal generated algebraically by this function in $\mathcal P$ (see [10]).

We present another equivalent definition from [11]: a function $\varphi\in\mathcal P$ is slowly decreasing if there exists a positive scalar $a_0$ such that the following two conditions hold:

(SD1) each connected component $L_{\alpha}$ of the set

(SD2) for each connected component $L_{\alpha}$ of $ L(\varphi, a_0)$ the inequality

For a sequence $\Lambda\subset \mathbb C$ such that $D_{\mathrm{BM}}(\Lambda)<+\infty$ we introduce a further characteristic, $D_{\mathrm{sd}} (\Lambda)$. If $\Lambda$ is not a subset of zeros of any slowly decreasing functions $\varphi\in\mathcal P$, then we set $D_{\mathrm{sd}}(\Lambda)=+\infty$. Otherwise $D_{\mathrm{sd}}(\Lambda)$ is by definition the set of all positive $c$ such that $\mathcal P$ contains a slowly decreasing function of exponential type $\pi c$ that vanishes on $\Lambda$.

As written in the introduction, in the problem of representing a $D$-invariant subspace with infinite discrete spectrum as the direct sum (1.5) the quantity $\pi D_{\mathrm{sd}}(\Lambda)$ plays a similar role to the completeness radius $\rho (\Lambda)$ in Theorem A.

The following criterion holds for $D$-invariant subspaces with discrete spectrum and residual interval with infinite length.

Theorems 1 and 2 imply a corollary.

§ 3. Proof of Theorems 1 and 2

3.1. A reduction to the dual interpolation problem

In the space $\mathcal E(a,b)$ consider a nontrivial $D$-invariant subspace $W$ with discrete spectrum $(-\mathrm{i}\Lambda) $ and residual interval $I_W$. Let $U\colon E(\Lambda, (a,b))\to E(\Lambda, I_W)$ be the restriction operator that assigns to each function $f\in E(\Lambda, (a,b))$ its restriction to $I_W$; here $E(\Lambda,I_W)$ is the subspace defined by (2.1) for the sequence $\Lambda $ and interval $I_W$.

Consider two intervals $I$, $\widetilde{I}\subset\mathbb R$, $I\subset \widetilde{I}$, and the adjoint operator

By [14], the corollary to Theorem 7, the restriction operator $U\colon E(\Lambda,\widetilde{I}) \to E(\Lambda, I)$ is a linear topological isomorphism if and only if

We summarize the above reasonings in the following proposition.

In what follows we call the interpolation problem in Proposition 2 the interpolation problem on $\Lambda$ for the pair of spaces $\mathcal P(\widetilde{I})$ and $\mathcal P(I)$.

This is a consequence of Propositions 1 and 2.

3.2. Solving the interpolation problem for the pair of spaces $\mathcal P(\widetilde{I})$ and $\mathcal P(I)$

Throughout what follows $I=\langle c,d\rangle$ is a finite or infinite interval, where ‘ $\langle$ ’ (and, in a similar way, ‘ $\rangle$ ’) means a round bracket ‘ ( ’, or a square bracket ‘ [ ’.

The simplest situation when this remark applies occurs when $\Lambda$ is the set of zeros of a slowly decreasing function of exponential type $\sigma$ and $I$ is an interval of length $2\pi\sigma$. In fact, in this case we have $2\pi D_{\mathrm{BM}}(\Lambda)=2\pi D_{\mathrm{sd}}(\Lambda)=|I|$.

In § 4.3 below we illustrate Remark 2 using a more interesting example, when $2\pi D_{\mathrm{BM}}(\Lambda)<|I|$, $2\pi D_{\mathrm{sd}}(\Lambda)=|I|$ and $\mathcal J(\Lambda,I)$ contains a slowly decreasing functions. On the other hand, in § 4.4 we present an example showing that the inequality $2\pi D_{\mathrm{sd}}(\Lambda)\leqslant |I|$ does not ensure the existence of a slowly decreasing function in the nontrivial submodule $J(\Lambda,I)$.

Combining parts I and II of Theorem 3 we obtain the following.

It follows from part II of Theorem 3 that the additional assumption concerning (2.5) in Corollary 2 holds a fortiori in the case when $I$ has a compact closure in $\widetilde{I}$. On the other hand, if the embedding $I\subset \widetilde{I}$ is not compact, then without assuming (2.5) we have the following weaker result.

Note that we can write part II of Theorem 3 in the following form.

From parts I, 2) and II of Theorem 3 we deduce a criterion for the solvability of the interpolation problem on $\Lambda$ for the pair of spaces $\mathcal P$ and $\mathcal P(I)$ in the case when $I$ is an infinite interval.

3.3. End of the proof of Theorems 1 and 2

Note first of all that, under the assumptions of part I, 1) of Theorem 1, as well as under the assumptions of Theorem 2 the $D$-invariant subspace $W$ with discrete spectrum $(-\mathrm{i}\Lambda)$ and residual interval $I_W$ has the form (1.2). This follows from the obvious relation $D_{\mathrm{BM}}(\Lambda) \leqslant D_{\mathrm{sd}} (\Lambda)$ and Theorem A stated in the introduction. Hence part I, 1) of Theorem 1 follows from Proposition 3 and part I, 1) of Theorem 3. On the other hand Theorem 2 follows from Proposition 3 and Corollary 4.

Now from Proposition 3 and part II of Theorem 3 for $\widetilde{I}=(a,b)$ and $I=I_W$, we obtain assertion I, 2) of Theorem 1.

To verify the affirmative part of assertion II of Theorem 1 consider a distribution $S\in\mathcal E' (a,b)$ whose Fourier-Laplace transform is an entire function $\varphi=e^{\mathrm{i}\gamma z}s$, where $s (z)=\widetilde{s}(t_0z)$ for some $t_0>0$, and $\widetilde{s}$ is a sine-type function. Consider the $D$-invariant subspace

The following facts on $W_S$ were established in [3], § 3, and [18], § 1:

• 1) this subspace has the discrete spectrum $(-\mathrm{i}\Lambda)$, where $\Lambda$ is the set of zeros of $\varphi$;
• 2) the residual interval of $W_S$ is a closed interval $[c,d]\subset (a,b)$ with length $2\pi D_{\mathrm{BM}} (\Lambda)$;
• 3) $W_S$ admits weak spectral synthesis (1.2).

By Proposition 3 the representability of $W_S$ in the form (1.5) is equivalent to the solvability of the interpolation problem on $\Lambda$ for the pair of spaces $\mathcal P(a,b)$ and $\mathcal P([c,d])$.

Note that $\mathcal J(\Lambda, [c,d]) $ is the principal submodule of the module $\mathcal P([c,d])$ which is generated by the slowly decreasing function $\varphi$. The sequence $\Lambda =\{\lambda_j\}$ coincides, up to a positive multiplicative constant, with the sequence of zeros of a sine-type function. Hence $\operatorname{Im} \lambda_j =O(1)$ as $j\to\infty$. By Remark 2 the interpolation problem on $\Lambda$ is solvable for the pair of spaces $\mathcal P(a,b)$ and $\mathcal P([c,d])$.

Now we present an example of a $D$-invariant subspace with discrete spectrum $(-\mathrm{i}\Lambda )$ and residual interval $I_W$ of length $2\pi D_{\mathrm{sd}}(\Lambda)$ that does not admit a representation (1.5).

Let $T\in \mathcal E' (-\pi,\pi)$ be the distribution with Fourier-Laplace transform

§ 4. Further properties of $D $-invariant subspaces representable as a direct sum and of the characteristic $D_{\mathrm{sd}}(\Lambda)$

4.1. An example of a $D$-invariant subspace of the form (1.5) with finite noncompact residual interval whose spectrum fails one of relations in (2.5)

Consider the function

It is well known that for each sufficiently small $\delta>0$ we have

Set $I=[-\pi,\pi+\varepsilon )$, $\varepsilon >0$. Since the sequence $\mathcal M$ fails the first relation in (2.5), the interpolation problem on $\mathcal M$ is not solvable for the pair of spaces $\mathcal P(\widetilde{I})$ and $\mathcal P(I)$, where $\widetilde{I}$ is any interval containing $\overline{I}=[-\pi, \pi+\varepsilon ]$. Nevertheless, the interpolation problem on $\mathcal M$ is solvable for the pair of spaces $\mathcal P(-\infty, \pi+\varepsilon )$ and $\mathcal P(I)$. Let us prove this.

We use the scheme of the proof of part I, 2) of Theorem 3. By (4.3) and the upper estimate for $\ln |\varphi' (z)|$ following from Bernstein’s theorem we can construct an infinitely differentiable function $\eta $ on $\mathbb C$ with the following properties:

Let $\Psi\in\mathcal P (-\infty,\pi+\varepsilon )$. Set

4.2. Expanding in a series of exponential polynomials with brackets

From [10], [11], Remark 2, Corollary 2 and also Proposition 3 we can deduce some additional properties of elements of a $D$-invariant subspace representable as a direct sum (1.5).

The above observations suggests the following questions concerning the expansion of elements of $D$-invariant subspaces in series of exponential polynomials: what conditions ensure that each function in $E(\Lambda, (a,b))$ or $E(\Lambda, I_W)$ can be represented by such a series which converges in the topology of $\mathcal E(a,b)$ or $\mathcal E(I_W)$, respectively, after grouping its terms in some way? These questions are equivalent to more general interpolation problems on $\Lambda$ for the spaces $\mathcal P(a,b) $ and $\mathcal P (I_W)$ than the problems treated here. We investigate these problems elsewhere.

4.3. Possible relations between the densities $D_{\mathrm{BM}}(\Lambda)$ and $D_{\mathrm{sd}}(\Lambda)$

In [5] we presented an example of a sequence $\Lambda$ such that

Here we show that for each $\delta\in (0,1)$ there exists a sequence $\Lambda$ such that

Fix $M_0\in\mathbb N$, $M_0>\delta^{-1}$, and set $\Lambda =\Lambda'\cup \Lambda''$, where $\Lambda'=\{\pm kM_0\}_{k=1}^{\infty}$ and $\Lambda''=\bigcup_{j=0}^{\infty}\Lambda''_j$, where

Relying, for instance, on [9], § IX.D, it is easy to see that $D_{\mathrm{BM}}(\Lambda)={1}/{M_0}$. It is also clear that $D_{\mathrm{sd}} (\Lambda)\leqslant 1$ because $\Lambda$ is a subset of zeros of the slowly decreasing function ${\sin\pi z}/(\pi z)$ in the space $\mathcal P([-\pi,\pi])$. We show that $\Lambda$ is not a subset of zeros of any slowly decreasing function of exponential type $\sigma<\pi$.

Suppose the converse is true: let $\varphi$ be a slowly decreasing function of exponential type $\sigma<\pi$ such that $\varphi (\Lambda) =0$. By Remark 1 in § 3.2 above and Lemma 1 in [19] we can assume that the set of zeros $Z_{\varphi}$ of $\varphi$ lies on the real axis. We introduce some notation: $M=Z_{\varphi}\setminus \Lambda$, $n_{Z_{\varphi}} (t)$ is the number of points in $Z_{\varphi }$ lying in the interval $(0,t]$ if $t>0$, and $(-n_{Z_{\varphi}} (t))$ is the number of points in $Z_{\varphi }$ lying in $[t,0)$ if $t<0$. The notation $n_{\Lambda} (t)$ and $n_{\mathcal M} (t)$ has similar meaning.

By Theorem 1 in [19] the relation

Setting

It follows from (4.6) and (4.7) that

Taking the definitions of $\widetilde t_j$ and $t_j$ into account we obtain

4.4. The (un)attainability of the infimum in the definition of $D_{\mathrm{sd}}(\Lambda)$

In the previous subsection and Remark 2 we understood that there is a ‘nontrivial gap’ between the sufficient and necessary conditions $2\pi D_{\mathrm{sd}} (\Lambda) <|I|$ and $2\pi D_{\mathrm{sd}} (\Lambda) \leqslant |I|$ for the solvability of the interpolation problem. This gap includes the following intermediate situation: there exists a slowly decreasing function $\varphi$ with exponential type at most $|I|/2$ such that $\varphi(\Lambda )=0$. This condition is sufficient for the solvability of the corresponding interpolation problem. It is obvious that this condition also implies the inequality $2\pi D_{\mathrm{sd}} (\Lambda) \leqslant |I|$.

In connection with the above the following question looks natural: does the inequality $2\pi D_{\mathrm{sd}} (\Lambda) \leqslant |I|$ mean that there exists a slowly decreasing function with exponential type at most $|I|/2$ that vanishes on $\Lambda$? Now we show that the answer is negative.

Consider the function

We aim to show that $D_{\mathrm{sd}} (\Lambda) =1$, but nonetheless $\Lambda$ is not a subset of zeros of a slowly decreasing function of exponential type $\pi$.

Recall some well-known facts on the asymptotic behaviour of the functions $\sin\pi z$ and ${\sin\pi\sqrt{z}}/{\sqrt{z}}$. For any sufficiently small fixed $\delta>0$ we have

On the other hand, for any $\varepsilon\in (0,1)$ set

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