N. F. Abuzyarova, “Representation of invariant subspaces of the Schwartz space”, Sb. Math., 213:8 (2022), 1020

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

This article is cited in 3 scientific papers (total in 3 papers)

Representation of invariant subspaces of the Schwartz space

N. F. Abuzyarova^ab

^a Bashkir State University, Ufa, Russia
^b Institute of Mathematics with Computing Centre, Ufa Federal Research Centre of the Russian Academy of Sciences, Ufa, Russia

English version PDF (575 kB) HTML full-text Citations (3) Russian version article

References:

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

Abstract: A subspace

$W$ of the Schwartz space

$C^{\infty} (a,b)$ such that the restriction of the operator of differentiation to

$W$ has a discrete spectrum is considered. Conditions for the representation of

$W$ as a direct algebraic and topological sum of two subspaces, namely, the residual subspace and the subspace spanned by the exponential monomials from

$W$ , are investigated. One condition ensuring this representation turns out to be the existence of a functional annihilating

$W$ such that the Fourier-Laplace transform of this functional is a slowly decreasing entire function. A new characteristic of complex sequences is introduced and investigated. Using this characteristic, the condition that an invariant subspace is equal to the direct sum of its residual and exponential subspaces can be put into a form that is similar to the previously discovered conditions for the possibility of weak spectral synthesis.
Bibliography: 19 titles.

Keywords: invariant subspace, spectral synthesis, Fourier-Laplace transform, slowly decreasing entire function, Schwartz spaces.

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).

Received: 04.11.2021 and 14.04.2022

Bibliographic databases:

Document Type: Article

MSC: Primary 30D15, 47A15; Secondary 42B10

Language: English

Original paper language: Russian

§ 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

$\begin{equation} W_I=\{ f\in\mathcal E(a,b)\colon f^{(k)}(t)=0,\ t\in I, \ k=0,1, 2, \dots\}. \end{equation} \tag{1.1}$

The existence of

$I_W$ was first shown in [1], Theorem 4.1; it also follows from the general dual scheme that we used in investigating the problem of spectral synthesis for the operator of differentiation in

$\mathcal E (a,b)$ (see [2] and [3], § 2).

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

$\begin{equation} W=\overline{\operatorname{span} \operatorname{Exp} (\Lambda) +W_{I_W}} \end{equation} \tag{1.2}$

holds, where

$\operatorname{span} X$ is the linear span of the set

$X\subset\mathcal E(a,b)$ . For a

$D$ -invariant subspace

$W$ such a representation is a generalization of the equality

$\begin{equation} W=\overline{\operatorname{span}\operatorname{Exp} (\Lambda)}, \end{equation} \tag{1.3}$

which means that

$W$ admits spectral synthesis in the classical sense.

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.

Theorem A (see [2], Corollaries 2 and 3, and [4], Theorems 1.1–1.3). Let $W$ be a $D$ -invariant subspace with discrete spectrum $(-\mathrm{i}\Lambda)$ and residual interval $I_W$ .

If $|I_W| >2\rho (\Lambda)$ , then $W$ admits weak spectral synthesis (1.2).

If $|I_W|<2 \rho (\Lambda)$ , then $W=\mathcal E(a,b)$ .

There exist both $D$ -invariant subspaces with discrete spectrum $(-\mathrm{i}\Lambda)$ and residual interval $I_W$ of length $2\rho (\Lambda)$ that admit weak spectral synthesis (1.2) and ones that do not admit it.

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 $W_{I_W}$ :

$\begin{equation} W=\operatorname{span}\operatorname{Exp} (\Lambda) \oplus W_{I_W} \end{equation} \tag{1.4}$

(see [1], Proposition 6.1).

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

$\begin{equation} W=\overline{\operatorname{span}\operatorname{Exp} (\Lambda)} \oplus W_{I_W} \end{equation} \tag{1.5}$

as a direct (algebraic and topological) sum holds for a

$D$ -invariant subspace

$W$ with infinite discrete spectrum

$(-\mathrm{i}\Lambda)$ and residual interval

$I_W$ . However, it was indicated in [1] that the authors did not know the answer.

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 $\mathcal F$ :

$\begin{equation*} \mathcal P=\mathcal F (\mathcal E'), \quad \text{where } \mathcal F (S)=S(e^{-\mathrm itz}), \quad S\in\mathcal E'. \end{equation*} \notag$

The topology and linear structure on

$\mathcal P$ are inherited from

$\mathcal E'$ .

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

$\begin{equation*} \| \varphi\|_k =\sup_{z\in\mathbb C} \frac{|\varphi (z)|}{(1+|z|)^k e^{k|{\operatorname{Im}z}|}}, \qquad k=1,2,\dots\,. \end{equation*} \notag$

In particular, this means that

$\mathcal P$ is a locally convex space of type

$(\mathrm{LN}^*)$ (see [6]). Moreover, the operation of multiplication of functions is continuous in

$\mathcal P$ , that is,

$\mathcal P$ is a topological algebra.

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

$\begin{equation*} \| \varphi\|_{I,k} =\sup_{z\in\mathbb C} \frac{|\varphi (z)|}{(1+|z|)^k\exp (dy^{+}-cy^{-})}, \qquad y^{\pm}=\max\{ 0,\pm y\}, \quad z=x+\mathrm{i} y, \end{equation*} \notag$

provided that

$I=[c,d]$ . On the other hand, if

$I=[c,b)$ , then

$\widetilde{P}_k$ is the space of entire functions with finite norm

$\begin{equation*} \| \varphi\|_{I,k} =\sup_{z\in\mathbb C} \frac{|\varphi (z)|}{(1+|z|)^k\exp (d_ky^{+}-cy^{-})}, \qquad y^{\pm}=\max\{ 0,\pm y\}, \quad z=x+\mathrm{i} y, \end{equation*} \notag$

where

$c<d_1<\dots < d_{k}\nearrow b$ as

$k\to\infty$ . For intervals

$I$ of another form the definitions must be modified in an obvious way.

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)

$\begin{equation*} \mathcal F(\mathcal E'(I)) =\mathcal P(I). \end{equation*} \notag$

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

$\begin{equation*} W^0=\{ S\in\mathcal E' (I)\colon S(f)=0\ \forall\, f\in W\}. \end{equation*} \notag$

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

$\begin{equation} E(\Lambda, I)=\overline{\operatorname{span}\operatorname{Exp} (\Lambda)} \end{equation} \tag{2.1}$

(the closure is considered in

$\mathcal E(I)$ ). Clearly,

$E(\Lambda,I)$ is a nontrivial

$D$ -invariant subspace of

$\mathcal E(I)$ which admits spectral synthesis.

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

$\begin{equation*} (E(\Lambda,I) )^0=\{ S\in\mathcal E' (I)\colon S(f)=0\ \forall\, f\in E(\Lambda,I) \}. \end{equation*} \notag$

The strong dual of the Fréchet space

$E(\Lambda, I)$ is the quotient space

$\mathcal E' (I)/(E(\Lambda,I) )^0$ . Using the Fourier-Laplace transform

$\mathcal F$ we can realize it at the quotient space of entire functions

$\mathcal P(I)/\mathcal J(\Lambda, I)$ .

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

$\begin{equation*} \rho (\Lambda )=\pi D_{\mathrm{BM}}(\Lambda). \end{equation*} \notag$

It follows from this and the Paley-Wiener-Schwartz theorem that

$D_{\mathrm{BM}}(\Lambda)=+\infty$ unless

$\Lambda$ is a subset of zeros of some function

$\varphi\in\mathcal P$ ; in the last case

$D_{\mathrm{BM}}(\Lambda)$ is the infimum of the positive

$c$ such that the algebra

$\mathcal P$ contains a function

$\varphi$ of exponential type

$\pi c$ that vanishes on

$\Lambda$ .

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

$\begin{equation} \forall\, x\in\mathbb R\ \exists\, x'\in\mathbb R\colon |x-x'|\leqslant a\ln(2+|x|), \quad |\varphi (x')|\geqslant (2+|x'|)^{-a}. \end{equation} \tag{2.2}$

For what follows note that (2.2) can be replaced by the following condition, which has a more general form:

$\begin{equation} \forall\, x\in\mathbb R \quad \exists\, z'\in\mathbb C\colon |x-z'|\leqslant a\ln(2+|x|), \quad |\psi (z')|\geqslant (a+|z'|)^{-a} \end{equation} \tag{2.3}$

(see [10], § 3).

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

$\begin{equation} L(\varphi, a_0) =\{ z\colon \ln|\varphi (z)|< -a_0 (|{\operatorname{Im}z}|+ \ln (2+|z|))\} \end{equation} \tag{2.4}$

is relatively compact;

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

$\begin{equation*} |{\operatorname{Im} \zeta}|+ \ln (2+|\zeta|)\leqslant a_0 (|{\operatorname{Im}z}|+ \ln (2+|z|)), \qquad \zeta, z\in L_{\alpha}, \end{equation*} \notag$

holds.

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.

Theorem 1. Let $W$ be a $D$ -invariant subspace of $\mathcal E(a,b)$ with discrete spectrum $(-\mathrm{i}\Lambda )$ and residual interval $I_W$ , where $|I_W|<+\infty$ .

If $|I_W|>2\pi D_{\mathrm{sd}}(\Lambda)$ and the following relations hold:

$\begin{equation} \varlimsup_{j\to\infty} \frac{\operatorname{Im} \lambda_j}{|\lambda_j|}<+\infty\quad\textit{and} \quad \varliminf_{j\to\infty} \frac{\operatorname{Im} \lambda_j}{|\lambda_j|}>-\infty, \end{equation} \tag{2.5}$

then

$W$ has the form (1.5).

Conversely, assume that $W$ has the form (1.5). Then $|I_W|\geqslant 2\pi D_{\mathrm{sd}}(\Lambda)$ .

Moreover, if $I_W\Subset (a,b)$ , then both inequalities in (2.5) hold. If the inclusion $I_W\subset (a,b)$ is not compact and $a$ (or $b$ ) is an endpoint of $I_W$ , then the first relation (second relation, respectively) in (2.5) holds.

There exist both $D$ -invariant subspaces with discrete spectrum $(-\mathrm{i}\Lambda )$ and residual interval $I_W$ of length $2\pi D_{\mathrm{sd}}(\Lambda)$ that admit the representation (1.5) and ones that do not.

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

Theorem 2. Let $W$ be a $D$ -invariant subspace of $\mathcal E(a,b)$ with discrete spectrum $(-\mathrm{i}\Lambda )$ and residual interval $I_W =(-\infty,d]$ (or $I_W=[c,+\infty )$ ). Then the representation (1.5) holds for $W$ if and only if $D_{\mathrm{sd}}(\Lambda)<+\infty$ and the first relation (second relation, respectively) in (2.5) is satisfied.

Theorems 1 and 2 imply a corollary.

Corollary 1. Let $W$ be a $D$ -invariant subspace of $\mathcal E(a,b)$ with discrete spectrum $(-\mathrm{i}\Lambda )$ and residual interval $I_W$ .

If $D_{\mathrm{sd}}(\Lambda)=+\infty$ , then there is no representation (1.5).

§ 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 $I_W$ .

Proposition 1. A $D$ -invariant subspace $W$ admitting weak spectral synthesis (1.2) can be represented as the direct sum (1.5) if and only if the restriction operator

$\begin{equation} U\colon E(\Lambda, (a,b))\to E(\Lambda, I_W) \end{equation} \tag{3.1}$

is a linear topological isomorphism.

Proof. Note that by the well-known result that a mean-periodic extension of a function is unique (see [12], § 1, [13], § 9) the subspace

$W_{I_W}\cap E(\Lambda, (a,b))$ is trivial. Hence the algebraic sum of

$W_{I_W}$ and

$E(\Lambda, (a,b))$ is direct. If it coincides with

$W$ , then it is also a topological direct sum: in this case the correspondence

$(f_1,f_2) \to f_1+f_2$ defines a continuous map of the Fréchet space

$W_{I_W}\times E(\Lambda, (a,b))$ onto

$W$ , so it is an (algebraic and topological) isomorphism between these spaces.

We turn to the proof of sufficiency. The function $f\in \mathcal E(a,b)$ belongs to $W$ if and only if its restriction $f|_{I_W}$ to $I_W$ is an element of $E(\Lambda, I_W)$ (see [1], Proposition 6.2). Therefore, since the restriction operator $U$ is surjective, for each $f\in W$ we can find a function $f_1\in E(\Lambda, (a,b))$ such that $f|_{I_W}=f_1|_{I_W}$ . Then, clearly,

$\begin{equation*} f_2=f-f_1\in W_{I_W}\quad\text{and}\quad f=f_1+f_2\in E(\Lambda, (a,b))\oplus W_{I_W}. \end{equation*} \notag$

Necessity. By the results on the uniqueness of a mean-periodic extension the restriction operator (3.1) is an algebraic and topological monomorphism. For each $f_0\in E(\Lambda, I_W)$ any smooth extension $f$ of it to $(a,b)$ belongs to $W$ by [1], Proposition 6.2. Hence $f=f_1+f_2$ , where $f_1\in E(\Lambda, (a,b))$ and $f_2\in W_{I_W}$ . Therefore, $f_0=U(f_1)$ . Thus, $U$ is a continuous linear map of the Fréchet space $E(\Lambda, (a,b))$ onto the Fréchet space $S(\lambda, I_W)$ . Hence this map is a linear topological isomorphism. The proof is complete.

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

$\begin{equation*} U^*\colon \mathcal E' (I)/(E(\Lambda,I) )^0 \to\mathcal E' (\widetilde{I})/(E(\Lambda,\widetilde{I}) )^0 \end{equation*} \notag$

of the restriction operator

$U\colon E(\Lambda,\widetilde{I})\to E(\Lambda,I)$ . The Fourier-Laplace transform defines the lift

$\begin{equation*} \widehat{U}\colon \mathcal P(I)/\mathcal J(\Lambda, I)\to \mathcal P(\widetilde{I})/\mathcal J(\Lambda, \widetilde{I}), \end{equation*} \notag$

of this adjoint operator. With each coset

$\begin{equation*} [\psi]\in \mathcal P(I)/\mathcal J(\Lambda, I), \qquad\psi\in \mathcal P(I), \end{equation*} \notag$

it associates the coset

$\begin{equation*} \widehat{U} ([\psi] )\in \mathcal P(\widetilde{I})/\mathcal J(\Lambda, \widetilde{I}), \qquad \widehat{U} ([\psi] )=\bigl\{\Psi=\psi+\Phi\colon \Phi\in \mathcal J(\Lambda, \widetilde{I}) \bigr\}. \end{equation*} \notag$

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

$\begin{equation} \widehat{U} (\mathcal P(I)/\mathcal J(\Lambda, I)) =\mathcal P(\widetilde{I})/\mathcal J(\Lambda, \widetilde{I}). \end{equation} \tag{3.2}$

In that case

$\widehat{U}$ is also a linear topological isomorphism. Equality (3.2) is equivalent to the property that for each function

$\Psi\in\mathcal P(\widetilde{I})$ there exists

$\psi\in\mathcal P(I)$ such that

$(\Psi -\psi)\in \mathcal J(\Lambda, \widetilde{I})$ , that is,

$\Psi (\Lambda) =\psi (\Lambda )$ .

We summarize the above reasonings in the following proposition.

Proposition 2. The restriction operator $U$ from $E(\Lambda,\widetilde{I})$ to $E(\Lambda, I)$ is a linear topological isomorphism if and only if the following interpolation problem is solvable: for each function $\Psi \in\mathcal P(\widetilde{I})$ there exists $\psi\in\mathcal P(I)$ such that the difference $(\Psi-\psi )$ vanishes on $\Lambda$ .

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 $\mathcal P(I)$ .

Proposition 3. A $D$ -invariant subspace $W$ defined by (1.2) can be represented as the direct sum (1.5) if and only if the interpolation problem on $\Lambda$ is solvable for the pair of spaces $\mathcal P(a,b)$ and $\mathcal P(I_W)$ .

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 ‘ [ ’.

Theorem 3. Let $\Lambda=\{\lambda_j\}$ and $I=\langle c,d\rangle$ be a sequence and an interval such that the exponential system $\operatorname{Exp} (\Lambda)$ is not complete in $\mathcal E(I)$ .

If $2\pi D_{\mathrm{sd}}(\Lambda)<|I|$ and both relations in (2.5) hold, then the interpolation problem on $\Lambda$ is solvable for the pair of spaces $\mathcal P$ and $\mathcal P(I)$ .

Assume that $|I|=+\infty$ , $D_{\mathrm{sd}}(\Lambda)<+\infty$ and the first or second relation in (2.5) holds, depending on whether $I=(-\infty, d\rangle$ or $I=\langle c,+\infty )$ , respectively.

Then the interpolation problem on $\Lambda$ for the pair of spaces $\mathcal P$ and $\mathcal P(I)$ is solvable.

Assume that there exists an interval $\widetilde{I}\supset I$ such that the interpolation problem on $\Lambda$ is solvable for the pair of spaces $\mathcal P(\widetilde{I})$ and $\mathcal P(I)$ .

Then $D_{\mathrm{sd}}(\Lambda)<+\infty$ and $2\pi D_{\mathrm{sd}}(\Lambda)\leqslant |I|$ . If, in addition,

$\begin{equation*} d\in (\widetilde{I}\setminus I)\cup (I\setminus\partial \widetilde{I}) \quad (\textit{or } c\in (\widetilde{I}\setminus I)\cup (I\setminus\partial \widetilde{I})), \end{equation*} \notag$

then the first relation (second relation, respectively) in (2.5) is satisfied.

Proof. I. 1) By assumption there exist a slowly decreasing function

$\varphi\in\mathcal P(I)$ which vanishes on

$\Lambda$ and a positive constant

$M_0$ such that

$\Lambda$ lies in the curvilinear strip

$\begin{equation*} S=\bigl\{z=x+\mathrm{i} y \colon |y|\leqslant M_0\ln (2+|x|), \ x\in\mathbb R\bigr\}. \end{equation*} \notag$

Hence, taking (2.2) into account, it is easy to conclude that there exists

$A_1>0$ such that

$\begin{equation*} \forall\, \lambda_j\in\Lambda \quad \exists\, x_j\in\mathbb R\colon |\lambda_j-x_j|\leqslant A_1\ln (2+|x_j|), \quad |\varphi (x_j)|\geqslant (2+|x_j|)^{-A_1}. \end{equation*} \notag$

Applying the theorem on a lower estimate for the modulus of an analytic function in a disc to

$f_j={\varphi}/{\varphi (x_j)}$ and the disc

$|z-x_j|\leqslant 2A_1\ln (2+|x_j|)$ we find a circle

$C_j$ with centre

$x_j$ and radius

$\begin{equation*} r_j \in \bigl(A_1\ln (2+|x_j|), 2A_1\ln (2+|x_j|) \bigr) \end{equation*} \notag$

such that

$\begin{equation*} |\varphi (z)|\geqslant (2+|z|)^{-A_0}, \qquad z\in C_j, \end{equation*} \notag$

where

$A_0\geqslant 0$ is independent of

$j$ (and

$\lambda_j$ lies obviously inside

$C_j$ ).

Let $K_j$ be the open disc with boundary $C_j$ , and let

$\begin{equation*} \mathcal K=\bigcup_j K_j, \qquad \mathcal C=\bigcup_j C_j\quad\text{and} \quad \widetilde{\mathcal U} =\mathcal K\setminus\mathcal C. \end{equation*} \notag$

The set

$\widetilde{\mathcal U}$ consists of a countable number of relatively compact connected components

$U_k$ . Generally speaking, not all the

$U_k\subset\widetilde{\mathcal U}$ have a nonempty intersection with

$\Lambda$ . Consider the set

$\begin{equation*} \mathcal U=\bigcup_{U_k\cap \Lambda\neq \varnothing} U_k; \end{equation*} \notag$

and for any

$A> A_0$ put

$\begin{equation*} S_A=\{ z\in\mathbb C\colon |\varphi (z)|< (2+|z|)^{-A}\}. \end{equation*} \notag$

Clearly,

$S_A\subset\mathcal U$ .

It follows from the membership relations $\varphi$ , $\varphi'={\mathrm{d}\varphi}/{\mathrm{d}z}\in\mathcal P(I)$ (where the second relation follows from Bernstein’s theorem ([15], Ch. 11) that there exists a positive constant $M_1$ such that

$\begin{equation} |\varphi (z)|\leqslant (2+|z|)^{M_1}\quad\text{and} \quad |\varphi ' (z)|\leqslant (2+|z|)^{M_1} \quad \forall\, z\in \overline{\mathcal U}. \end{equation} \tag{3.3}$

Taking account of estimate (3.3) for $|\varphi '|$ and the fact that each connected component $U_k\subset \mathcal U$ has diameter $O(\ln |z|)$ for any $z\in U_k$ as $k\to\infty$ , it is easy to see that there exists $A'>A_0$ such that for all $U_k$ the distance from $U_k\cap S_{A'}$ to the boundary of $U_k$ is at least $\sup_{z\in U_k} (2+|z|)^{-A'}$ .

Consider an infinitely differentiable function $\eta$ on the complex plane that vanishes outside $\mathcal U$ , is equal to one on $\mathcal U\cap S_{A'}$ and such that

$\begin{equation} \biggl|\frac{\partial\eta (z)}{\partial\overline z} \biggr|\leqslant (2+|z|)^{M_2}, \qquad z\in\mathbb C, \end{equation} \tag{3.4}$

where

$M_2>0$ is independent of

$z$ (see [16], Ch. I, § 1).

For an arbitrary $\Psi\in\mathcal P$ set $v=-\Psi \varphi^{-1}\,{\partial\eta}/{\partial\overline{z}}$ . Bearing in mind the intrinsic description of the algebra $\mathcal P$ and the properties of $\varphi$ and $\eta$ (including the bound (3.4)), we conclude that $v\in C^{\infty} (\mathbb C)$ and

$\begin{equation*} |v(z)|\leqslant (2+|z|)^{M_3}, \qquad z\in\mathbb C, \end{equation*} \notag$

where

$M_3$ is a positive constant. By a well-known result of Hörmander (Theorem 4.4.2 in [17]) there exists a infinitely differentiable function

$u$ in

$\mathbb C$ such that

$\begin{equation*} \frac{\partial u}{\partial \overline{ z}} =v, \end{equation*} \notag$

and moreover,

$\begin{equation} |u(z)|\leqslant (2+|z|)^{M_4}, \qquad z\in\mathbb C, \end{equation} \tag{3.5}$

where

$M_4$ is a positive constant.

The function $\psi=\varphi u+\Psi\eta$ is the required solution of the interpolation problem. In fact, $\Psi (\Lambda) =\psi (\Lambda)$ because $\varphi (\Lambda )=0$ . Now,

$\begin{equation*} \frac{\partial \psi}{\partial \overline{z}} =\varphi \frac{\partial u}{\partial \overline{ z}}+\Psi\frac{\partial\eta}{\partial\overline z }=0, \end{equation*} \notag$

so that

$\psi$ is an entire function.

Note that $\mathcal U\subset\widetilde{S}$ , where

$\begin{equation*} \widetilde{S}=\{z=x+\mathrm{i} y \colon |y|\leqslant \widetilde{M}_0\ln (2+|x|)\} \end{equation*} \notag$

and

$\widetilde{M}_0$ is a positive function. Now,

$\begin{equation*} \eta (z)=0, \qquad z\notin\mathcal U, \end{equation*} \notag$

and

$\Psi$ has polynomial growth in

$\widetilde{S}$ . Hence, taking (3.5) into account, we obtain

$\psi\in\mathcal P(I)$ .

2) Assume for definiteness that $I=(-\infty, d\rangle$ . Suppose that $\Lambda$ satisfies the first relation in (2.5) and let $\varphi\in\mathcal P(I)$ be a slowly decreasing function that vanishes on $\Lambda$ .

We need a special partition of the complex plane into two unbounded domains $G_+$ and $G_{-}$ , the first of which contains the open upper half-plane, while the second lies accordingly in the open lower half-plane.

To construct this partition we use the definition (2.2) of a slowly decreasing functions and note that the inequality $|x-x'|\leqslant a \ln(2+|x|)$ in it can be replaced by $|x-x'|\leqslant a \ln(2+|x'|)$ without loss of generality. For each point $x\in\mathbb R$ we find a point $x'\in\mathbb R$ as in (2.2) and apply the theorem on a lower bound for the modulus of an analytic function to the function $f_x={\varphi}/{\varphi (x')}$ in the disc $|z-x'|\leqslant 2a\ln (2+|x'|)$ . Then we obtain

$\begin{equation} |\varphi (z)|\geqslant (2+|z|)^{-a'}, \qquad z\in C_x, \end{equation} \tag{3.6}$

where the constant

$a'$ is independent of

$x$ and

$C_x$ is a circle with centre

$x'$ and radius

$\begin{equation*} r_x \in \bigl(a\ln (2+|x'|), 2a\ln (2+|x'|) \bigr). \end{equation*} \notag$

Let $K_x$ be the open disc with boundary $C_x$ , and let $I_x=K_x\cap \mathbb R$ . From the cover $\{I_x\}_{x\in\mathbb R}$ we extract a countable locally finite subcover $\{ I_{x_j}\}$ : $\bigcup_{j}I_{x_j}=\mathbb R$ . It is easy to see that, of arcs of the circles $C_{x_j}$ lying in the lower half-plane, we can make a continuous curve $\Gamma$ dividing the complex plane into two unbounded domains, $G_+$ and $G_{-}$ .

We use the version of the definition of a slowly decreasing function that consists of conditions (SD1) and (SD2) (see § 2.2). We can assume without loss of generality that the positive constant $a_0$ involved in (SD1) and (SD2) satisfies $a_0\geqslant a'$ , where $a'$ is the constant in (3.6).

It is clear that the set $L(\varphi,a_0)$ defined by (2.4) is centred about the sequence of zeros of $\varphi$ in the following sense: it contains all zeros of $\varphi$ , and each connected component $L_{\alpha}$ of it contains at least one zero of $\varphi$ .

Set $\Lambda_+=\Lambda\cap G_+$ and $\Lambda_{-}=\Lambda\setminus\Lambda_+$ , and let $\mathcal L_+$ denote the union of the set of those connected components $L_{\alpha}\subset L(\varphi,a_0)$ for which

$\begin{equation*} L_{\alpha}\cap \Lambda_+\neq\varnothing \end{equation*} \notag$

and

$\mathcal L_{-}$ denote the union of those

$L_{\alpha}\subset L(\varphi,a_0)$ for which

$\begin{equation*} L_{\alpha}\cap \Lambda_-\neq\varnothing. \end{equation*} \notag$

The estimate (3.6) holds on the whole of the common boundary

$\Gamma$ between

$G_+$ and

$G_{-}$ . Therefore,

$\begin{equation*} \mathcal L_+\subset G_+, \qquad \mathcal L_{-}\subset G_{-}. \end{equation*} \notag$

In view of the above and since the first relation in (2.5) holds for $\Lambda$ , we can argue similarly to the construction of $\eta$ in part 1) of this proof, except that we replace the connected components $U_k$ of $\mathcal U$ by connected components $L_{\alpha}$ of $\mathcal L_{+}$ . Then, as a result, we find a constant $a_1>a_0$ and an infinitely differentiable function $\eta_{+}$ on the complex plane that vanishes outside $\mathcal L_{+}$ , is equal to one on $\mathcal L_{+}\cap L (\varphi, a_1)$ and satisfies

$\begin{equation*} \biggl|\frac{\partial\eta_{+} (z)}{\partial\overline z}\biggr|\leqslant (2+|z|)^{b_1}, \qquad z\in\mathbb C, \end{equation*} \notag$

where

$b_1$ is a positive constant independent of

$z$ .

Now, taking condition (SD2) into account, since $\varphi'\in\mathcal P(I)$ , we conclude that there exists $a_2>a_0$ such that for each component $L_{\alpha}\subset \mathcal L_{-}$ the distance from $L_{\alpha}\cap L(\varphi, a_2)$ to the boundary of $L_{\alpha}$ is at least

$\begin{equation*} \sup_{z\in L_{\alpha}} \exp \bigl(-a_2(|{\operatorname{Im}z}|+\ln (2+|z|))\bigr), \end{equation*} \notag$

where

$a_2$ is a positive constant independent of

$z$ . Hence there exists an infinitely differentiable function

$\eta_{-}$ on the complex plane that vanishes outside

$\mathcal L_{-}$ , is equal to one on

$\mathcal L_{-}\cap L (\varphi, a_2)$ and such that

$\begin{equation*} \biggl|\frac{\partial\eta_{-}(z)}{\partial\overline z}\biggr|\leqslant \exp (b_2(|{\operatorname{Im}z}|+\ln (2+|z|))) \qquad z\in\mathbb C, \end{equation*} \notag$

where

$b_2$ is a positive constant independent of

$z$ .

Let $\Psi\in\mathcal P$ be an arbitrary function. Then

$\begin{equation*} |\Psi (z)|\leqslant (2+|z|)^M, \qquad z\in\mathcal L_{+}, \end{equation*} \notag$

and

$\begin{equation*} |\Psi (z)|\leqslant (2+|z|)^{M}e^{M|{\operatorname{Im}z}|}, \qquad z\in\mathcal L_{-}, \end{equation*} \notag$

where

$M$ is a positive constant independent of

$z$ . Set

$\begin{equation*} \eta=\eta_++\eta_-\quad\text{and} \quad v=-\Psi \varphi^{-1}\frac{\partial\eta}{\partial\overline{z}}. \end{equation*} \notag$

It follows from the above that

$v$ is infinitely differentiable in

$\mathbb C$ and there exists

$C_0>0$ such that

$\begin{equation*} ve^{-p}\in L^2 (\mathbb C), \end{equation*} \notag$

where

$p$ is the subharmonic function defined by

$\begin{equation*} p(z):=\begin{cases} C_0\ln (2+|z|),&\operatorname{Im} z\geqslant 0, \\ -C_0\operatorname{Im} z+C_0\ln (2+|z|),&\operatorname{Im} z <0. \end{cases} \end{equation*} \notag$

By a theorem of Hörmander ([17], Theorem 4.4.2) there exists a solution

$u$ of the equation

$\begin{equation*} \frac{\partial u}{\partial \overline{ z}} =v \end{equation*} \notag$

that is infinitely differentiable in

$\mathbb C$ and satisfies

$\begin{equation*} |u(z)|\leqslant \mathrm{const} \exp\{p(z)+\mathrm{const}\, \ln (2+|z|)\}. \end{equation*} \notag$

As in part 1) of this proof, we consider the function $\psi=\varphi u+\Psi\eta$ and verify that $\Psi (\Lambda) =\psi (\Lambda)$ and $\psi\in\mathcal P(I)$ .

II. Let $\varepsilon_0=\operatorname{dist} (I,\partial\widetilde{I})$ . Fix some $\varepsilon \in (0,\varepsilon_0]$ such that there exists a point $a_{\varepsilon}\in \widetilde{I}\setminus I$ such that $\operatorname{dist}(a_{\varepsilon}, I)<\varepsilon$ . For definiteness assume that $a_{\varepsilon}\geqslant d$ . This corresponds to $d\in (\widetilde{I}\setminus I)\cup (I\setminus\partial \widetilde{I})$ . By assumption, for $\Psi (z)=\exp\{-\mathrm{i} a_{\varepsilon}z\}$ we can find $\psi\in \mathcal P(I)$ such that $\Phi:=\Psi-\psi$ vanishes on $\Lambda$ .

We have the following estimate for $\psi$ :

$\begin{equation*} \ln |\psi (z)| \leqslant d_1 y + M_{\psi}\ln (2+|x|), \quad z=x+\mathrm{i}y\quad \text{for } y>0, \end{equation*} \notag$

where

$m_{\psi}>0$ and either

$d_1<d$ or we can find

$a_{\varepsilon}> d$ , so that

$a_{\varepsilon }-d_1>0$ in either case. Hence there exist a positive constant

$M_0$ , a positive

$r_0$ which depends only on

$M_0$ , and a positive

$y_0$ which depends only on

$M_0$ and

$r_0$ such that

$\begin{equation*} \ln |\Phi (z)|\geqslant \frac12, \end{equation*} \notag$

for all

$\begin{equation*} z\in \{z=x+\mathrm{i}y\colon |x|\geqslant r_0,\ y\geqslant M_0\ln |x|\} \cup\{z=x+\mathrm{i}y\colon |x|\leqslant r_0,\ y\geqslant y_0\}. \end{equation*} \notag$

Hence

$\Phi$ is a slowly decreasing function. Moreover, all of its zeros, possibly apart from a finite number of them, lie in the set

$\begin{equation*} \bigl\{z=x+\mathrm{i}y\colon y\leqslant \max(y_0,\, M_0 \ln (|x|+2)), \ x\in\mathbb R \bigr\}. \end{equation*} \notag$

Hence the first relation in (2.5) holds.

Multiplying $\Phi$ by an appropriate exponential $\exp(-\mathrm{i}c_0z)$ we obtain a slowly decreasing function of exponential type at most $(|I|+\varepsilon)/2$ . As $\varepsilon$ can be auxiliary, we conclude that $D_{\mathrm{sd}} (\Lambda)<+\infty$ and $2\pi D_{\mathrm{sd}} (\Lambda ) \leqslant |I|$ .

If $c\in (\widetilde{I}\setminus I)\cup (I\setminus\partial \widetilde{I})$ , then similar arguments involving $a_{\varepsilon}\leqslant c$ show that the second relation in (2.5) is satisfied.

Theorem 3 is proved.

Remark 1. We see from the proof of part II of Theorem 3 that if a sequence $\Lambda$ satisfies (2.5) and $D_{\mathrm{sd}}(\Lambda)<\infty$ , then for each $\varepsilon$ there exists a slowly decreasing function of exponential type at most $\pi (D_{\mathrm{sd}} (\Lambda)+\varepsilon)$ such that relations of the form (2.5) hold for its full set of zeros $\Lambda'\supset\Lambda$ .

Remark 2. We can also see from the proof of Theorem 3 that the interpolation problem on $\Lambda$ for the pair of spaces $\mathcal P$ and $\mathcal P(I)$ is solvable when the sequence $\Lambda$ satisfies (2.5) and a certain weaker condition than $2\pi D_{\mathrm{sd}}(\Lambda)<|I|$ , namely, that the submodule $\mathcal J(\Lambda,I)$ contains a slowly decreasing function.

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.

Corollary 2. Given an interval $I$ , assume that there exists an interval $\widetilde{I}\supset I$ such that the interpolation problem on $\Lambda$ is solvable for the pair of spaces $\mathcal P(\widetilde{I})$ and $\mathcal P(I)$ and, in addition, if $|I|<\infty$ , then both relations in (2.5) hold.

Then the interpolation problem on $\Lambda$ is solvable for the pair of spaces $\mathcal P$ and $\mathcal P(I)$ .

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.

Proposition 4. Assume that the embedding $I\subset\widetilde{I}$ is not compact and $|\widetilde{I}|<\infty$ .

If the interpolation problem on $\Lambda$ is solvable for the pair of spaces $\mathcal P(\widetilde{I})$ and $\mathcal P(I)$ , then this problem is solvable for the pair of spaces $\mathcal P (\widetilde I_{\infty})$ and $\mathcal P(I)$ , where $\widetilde I_{\infty}$ is the ray containing $\widetilde{I}$ that has a common endpoint with $I$ .

Proof. We can assume without loss of generality that

$I=[c,b)$ and

$\widetilde{I} = (a,b)$ . Then

$\widetilde{I}_{\infty} =(-\infty,b)$ .

Setting $\sigma_0=c-a$ we show that for each $\sigma\in (0,\sigma_0)$ the interpolation problem on $\Lambda$ is solvable for the spaces $\mathcal P (a',b)$ and $\mathcal P (I)$ , where $a'=a-\sigma$ .

Let $c'=c-\sigma$ , $b'=b-\sigma$ and let $\Phi\in \mathcal P (a',b)$ be an arbitrary function. Consider a representation

$\begin{equation} \Phi =\Phi_1 +\Phi_2, \qquad \Phi_1\in\mathcal P(a',b'), \quad \Phi_2\in\mathcal P(a,b). \end{equation} \tag{3.7}$

(We explain below that the representation (3.7) is possible.)

By assumption, for the functions $\Phi_1e^{-\mathrm{i}\sigma z}$ and $\Phi_2$ in $\mathcal P(a,b)$ there exist functions $\widetilde\varphi_1$ and $\varphi_2$ in $\mathcal P(I)$ that coincide with them on $\Lambda$ . Now, for $\psi_1 =\widetilde\varphi_1e^{\mathrm{i}\sigma z}\in\mathcal P(a,b)$ there exists $\varphi_1\in\mathcal P(I)$ such that

$\begin{equation*} \varphi_1 (\Lambda) =\psi_1 (\Lambda) =\Phi_1 (\Lambda). \end{equation*} \notag$

Set

$\varphi =\varphi_1+\varphi_2$ . We see that

$\varphi\in\mathcal P(I)$ и

$\varphi (\Lambda) =\Phi (\Lambda)$ . Thus the interpolation problem on

$\Lambda$ is solvable for the pair of spaces

$\mathcal P(a',b)$ and

$\mathcal P(I)$ .

We prove in a similar way that the interpolation problem on $\Lambda$ is solvable for the pair of spaces $\mathcal P(a'',b)$ and $\mathcal P(I)$ , where $a'' =a'-\sigma'$ and $\sigma'$ is an arbitrary positive number less than $(c - a')$ . Continuing in this way, we see that the interpolation problem on $\Lambda$ is solvable for the pair of spaces $\mathcal P(\widetilde{a},b)$ and $\mathcal P(I)$ for an arbitrary finite $\widetilde{a}$ , and therefore for the spaces $\mathcal P(-\infty,b)$ and $\mathcal P(I)$ .

To complete the proof it remains to verify that for $\Phi\in \mathcal P (a',b)$ a representation (3.7) is possible. By the Paley-Wiener-Schwartz theorem $\Phi = \mathcal F(S)$ for some $S\in\mathcal E' (a',b)$ . We denote the closure of the convex hull of the support of a distribution $S$ by $\operatorname{ch}\operatorname{supp} S$ . Then we have

$\begin{equation*} \operatorname{ch}\operatorname{supp} S=[t_1,t_2]\subset (a',b). \end{equation*} \notag$

$(a,b')\cap (t_1,t_2)=\varnothing$ , then we can set one of the terms on the right-hand side of (3.7) to be zero. A less trivial case is when

$\begin{equation*} (a,b')\cap (t_1,t_2)\neq\varnothing. \end{equation*} \notag$

Then dividing

$\Phi$ by a suitable polynomial

$p$ if necessary we obtain a function

$\Psi =\Phi p^{-1}\in \mathcal P (a',b)$ such that

$\begin{equation*} T=\mathcal F^{-1} (\Psi)\in(C [t_1,t_2])'. \end{equation*} \notag$

It is well known that the action of

$T$ on elements of

$C [t_1,t_2]$ (in particular, on

$f\in\mathcal E(a',b)$ ) is described by a Stieltjes integral

$\begin{equation*} T(f) =\int_{t_1}^{t_2} f(t)\,\mathrm{d} v(t), \end{equation*} \notag$

where

$v$ is a function of bounded variation on

$[t_1,t_2]$ which depends on

$T$ and can be determined up to an additive constant.

We fix some $t_0\in (a,b')\cap (t_1,t_2)$ such that $v$ is continuous near it, and set

$\begin{equation*} v_1(t)=\begin{cases} v(t)\quad &\text{for }t\in [t_1,t_0], \\ v(t_0)\quad &\text{for }t\in (t_0,t_2], \end{cases}\quad\text{and} \quad v_2=v-v_1. \end{equation*} \notag$

Then

$\begin{equation*} T=T_1+T_2, \qquad T_j(f) =\int_{t_1}^{t_2} f(t)\,\mathrm{d} v_j (t), \quad j=1,2, \end{equation*} \notag$

where

$\operatorname{ch}\operatorname{supp} T_1 \subset [t_1,t_0]\subset (a',b')$ and

$\operatorname{ch}\operatorname{supp} T_2 \subset [t_0,t_2]\subset (a,b)$ . Therefore,

$\begin{equation*} \Psi=\Psi_1+\Psi_2, \qquad \Psi_1=\mathcal F(T_1) \in\mathcal P (a',b'), \quad \Psi_2=\mathcal F(T_2) \in\mathcal P (a,b). \end{equation*} \notag$

Setting

$\Phi_j=\Psi_j p$ ,

$j=1,2$ , we obtain the required representation (3.7) for

$\Phi$ . The proof is complete.

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

Corollary 3. 1) Under the assumptions of Theorem 3, if

$\begin{equation*} 2\pi D_{\mathrm{sd}}(\Lambda)>|I| \quad\textit{or } D_{\mathrm{sd}}(\Lambda)=+\infty, \end{equation*} \notag$

then for no interval

$\widetilde{I}\supset I$ is the interpolation problem on

$\Lambda$ solvable for the pair of spaces

$\mathcal P(\widetilde{I})$ and

$\mathcal P(I)$ .

Let $\Lambda$ and $I=\langle c,d \rangle$ be as in Theorem 3. If $\widetilde{I}\supset I$ is an interval such that

$\begin{equation*} d\notin \partial \widetilde{I}, \qquad \varlimsup_{j\to\infty} \frac{\operatorname{Im} \lambda_j}{\ln |\lambda_j|}=+\infty \end{equation*} \notag$

$\begin{equation*} c\notin \partial \widetilde{I}, \qquad \varliminf_{j\to\infty} \frac{\operatorname{Im} \lambda_j}{\ln |\lambda_j|}=-\infty, \end{equation*} \notag$

then the interpolation problem on

$\Lambda$ is not solvable for the pair of spaces

$\mathcal P(\widetilde{I})$ and

$\mathcal P(I)$ .

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.

Corollary 4. Let $\Lambda\subset\mathbb C$ , $D_{\mathrm{BM}} (\Lambda) <+\infty$ and $I=( -\infty,d\rangle$ (or $I = \langle c,+\infty)$ ).

For the interpolation problem on $\Lambda$ to be solvable for the pair of spaces $\mathcal P$ and $\mathcal P(I)$ it is necessary and sufficient that ${\mathrm{sd}}(\Lambda)<+\infty$ and the first relation (second relation, respectively) in (2.5) holds.

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

$\begin{equation} W_S =\{ f\in\mathcal E (a,b)\colon S(f^{(k)}) =0,\ k=0,1,\dots\}. \end{equation} \tag{3.8}$

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

$\begin{equation*} \varphi (z)=\frac{\sin\pi z}{s_0(z)\omega (z)}, \quad\text{where }\ s_{0}(z) =\frac{\sin(\pi\sqrt{z})}{\pi\sqrt{z}} \ \ \text{and}\ \ \omega(z) =\prod_{k=1}^{\infty}\biggl(1-\frac{z}{2^{2n}+1}\biggr). \end{equation*} \notag$

The subspace

$W_T\subset \mathcal E (-\pi,\pi)$ defined by formula (3.8) for

$T$ in place of

$S$ has the discrete spectrum

$(-\mathrm{i}\Lambda)$ , where

$\Lambda$ is the set of zeros of

$\varphi$ . It is clear that

$\Lambda \subset\mathbb Z$ and

$D_{\mathrm{sd}}(\Lambda)=D_{BM}(\Lambda)=1$ . On the other hand it is known that

$W_T$ does not admit weak spectral synthesis (1.2) (see [1], Theorem 1.2). A fortiori,

$W_T$ has no representation (1.5).

§ 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

$\begin{equation*} \varphi (z) =\frac{s(-\mathrm{i} z)\, \sin\pi z }{s(z)}, \quad \text{where } s(z)= \prod_{k=1}^{\infty}\biggl( 1-\frac{z}{2^k}\biggr). \end{equation*} \notag$

Its set of zeros

$\begin{equation*} \mathcal M =\bigl(\mathbb Z\setminus \{2^k\}_{k=1}^{\infty}\bigr)\cup \{2^k\mathrm{i}\}_{k=1}^{\infty} \end{equation*} \notag$

fails the first relation (2.5) but satisfies the second. The counting function

$\nu$ of the sequence

$\mathcal M$ exhibits the asymptotic behaviour

$\begin{equation*} \nu (t) =(\log_2 t +O(1)), \qquad t\to\infty. \end{equation*} \notag$

Using well-known techniques from the theory of entire functions (for instance, see [15], III.3.5) it is easy to verify that for any sufficiently small positive

$\delta$ the inequalities

$\begin{equation} A\ln^2(2+ |z|)-B_{\delta}\ln (2+|z|)\leqslant\ln |s(z)|\leqslant A\ln^2(2+ |z|)+B_{\delta}\ln (2+|z|) \end{equation} \tag{4.1}$

hold outside the discs

$|z-2^{k}|< \delta$ ,

$k=1,2,\dots$ . Here

$A =(\ln 2)^{-1}$ and the positive constant

$B_{\delta}$ depends only on

$\delta$ .

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

$\begin{equation} \ln |{\sin\pi z}| =\pi|{\operatorname{Im}z}| + O(1)\quad\text{as } |z|\to\infty, \quad |z-k|\geqslant\delta, \quad k\in\mathbb Z. \end{equation} \tag{4.2}$

It follows from (4.1) and (4.2) that

$\begin{equation} \pi |{\operatorname{Im}z}|-C_{\delta}\ln (2+|z|)\leqslant\ln |\varphi (z)|\leqslant \pi|{\operatorname{Im}z}|+C_{\delta}\ln (2+|z|) \end{equation} \tag{4.3}$

for all

$z$ such that

$|z-\mu_j|\geqslant \delta$ ,

$\mu_j\in\mathcal M$ , where

$\delta>0$ is sufficiently small. Hence

$D_{\mathrm{sd}} (\Lambda)=1$ .

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:

$\begin{equation*} \eta (z)= \begin{cases} 0 &\text{for }z\notin{\displaystyle\bigcup_{j} K(\mu_j,2\delta)}, \\ 1 &\text{for }z\in{\displaystyle\bigcup_{j} K(\mu_j,\delta)}, \end{cases} \end{equation*} \notag$

and

$\begin{equation*} \biggl|\frac{\partial\eta}{\partial\overline z} \biggr|\leqslant a_0, \qquad z\in\mathbb C, \end{equation*} \notag$

where

$a_0>0$ is independent of

$z$ and

$K(\mu,r) =\{z\colon |z-\mu|\leqslant r\}$ .

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

$\begin{equation*} v=-\Psi \varphi^{-1}\frac{\partial\eta}{\partial\overline{z}}\quad\text{and} \quad p(z)= C_0\ln (2+|z|), \end{equation*} \notag$

where

$C_0$ is a positive constant. It follows from the above that if

$C_0$ is sufficiently large, then

$\begin{equation*} v\in C^{\infty} (\mathbb C) \quad\text{and}\quad ve^{-p}\in L^2 (\mathbb C). \end{equation*} \notag$

By a theorem of Hörmander ([17], Theorem 4.4.2) the equation

$\begin{equation*} \frac{\partial u}{\partial \overline{ z}} =v \end{equation*} \notag$

has a solution that is infinitely differentiable in

$\mathbb C$ and satisfies

$\begin{equation*} |u(z)|\leqslant \mathrm{const}\exp\bigl(p(z)+\mathrm{const}\,\ln (2+|z|)\bigr). \end{equation*} \notag$

Set

$\psi=\varphi u+\Psi\eta$ . It is easy to see that

$\Psi (\Lambda) =\psi (\Lambda)$ and

$\psi\in\mathcal P(I)$ . As

$\Psi\in\mathcal P(-\infty, \pi+\varepsilon)$ can be arbitrary, we conclude that the interpolation problem on

$\mathcal M$ is solvable for the pair of spaces

$\mathcal P(-\infty, \pi+\varepsilon)$ and

$\mathcal P(I)$ .

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).

Theorem 4. Assume that a $D$ -invariant subspace $W\subset\mathcal E(a,b)$ has the form (1.5) and at least one of the conditions $|I_W|=+\infty$ and (2.5) is satisfied.

Then the sequence $\Lambda$ can be decomposed into disjoint finite subsets $\Lambda_k$ , $k=1,2,\dots$ , so that each function $f\in W$ can uniquely be represented as a sum ${f=f_1+f_2}$ , where

$\begin{equation} f_1\in W_{I_W}, \quad f_2 (t)=\sum_{k=1}^{\infty}\biggl(\sum_{\lambda_j\in\Lambda_k} p_{j}(t)\exp(-\mathrm{i}\lambda_jt)\biggr) \end{equation} \tag{4.4}$

and the

$p_j$ are polynomials, so that the outer sum (with respect to

$k$ ) for

$f_2$ converges in the topology of

$\mathcal E (\mathbb R)$ .

Proof. By assumption, each

$f\in W$ has a unique representation

$\begin{equation} f=f_1+f_2, \qquad f_1\in W_{I_W}, \quad f_2 \in E(\Lambda, (a,b)). \end{equation} \tag{4.5}$

Using part I, 2) of Theorem 1, Theorem 2, part I, 2) of Theorem 3 and Proposition 3 we conclude that, first, the function

$f_2$ in (4.5) can uniquely be extended to a function

$F_2\in E(\Lambda,\mathbb R)$ , and second, there exists a slowly decreasing function

$\psi\in\mathcal P(a,b)$ that vanishes on

$\Lambda$ . Here we bear in mind that either

$|I_W|=+\infty$ or (2.5) holds.

Since $\mathcal P(a,b)$ contains a slowly decreasing function vanishing on $\Lambda$ , it follows from Theorem 3.1 in [10] (see also [11], Theorem 9) that $F_2$ can be represented as a series (4.4) convergent in $\mathcal E(\mathbb R)$ . The proof is complete.

Remark 3. Under the assumptions of Theorem 4, if $I_W$ and $(a,b)$ have a common endpoint (for instance, $I_W =[c,b)$ ), then the subspace $W$ consists of the restrictions to $(a,b)$ of functions from the $D$ -invariant subspace $\widetilde{W}\subset \mathcal E (a,+\infty)$ ; moreover,

$\begin{equation*} \widetilde{W} =W_{[c,+\infty)}\oplus E(\Lambda,\mathbb R), \end{equation*} \notag$

and Theorem 4 holds for

$\widetilde{W}$ .

Remark 4. If a $D$ -invariant subspace $W\subset\mathcal E(a,b)$ has the form (1.5), the embedding $I_W\subset (a,b)$ is not compact, and $|I_W|<+\infty$ , then the example in § 4.1 shows that one of the relations in (2.5) can fail in general. For example, let $I_W=[c,b)$ . Then by Propositions 3 and 4 we can only ensure that $f_2$ in (4.5) extends to a mean-periodic function on $(-\infty,b)$ , that is, an element of $E(\Lambda, (-\infty, b))$ . Hence Theorem 3.1 in [10] on the representation of mean-periodic functions on the whole line by a series of exponential polynomials cannot be applied in general. However, analysing the proof of that theorem we can see that, in the context of this remark, a mean-periodic extension of $f_2$ to the ray $(-\infty,b)$ expands in a series of exponential polynomials convergent with brackets in $\mathcal E (-\infty, b-d)$ , provided that $d>0$ is large enough. On the other hand, the analysis of that proof shows that in some cases $d$ can be taken equal to $0$ . In particular, this holds for functions in the $D$ -invariant subspace considered in § 4.1.

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

$\begin{equation*} D_{\mathrm{BM}}(\Lambda)=0\quad\text{and} \quad D_{\mathrm{sd}}(\Lambda) =\infty. \end{equation*} \notag$

This sequence consists of the points

$j^2$ which are taken with multiplicity

$[\ln^{3/2}j]$ each,

$j=2,3,\dots$ .

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

$\begin{equation*} 0<D_{\mathrm{BM}}(\Lambda)<\delta\quad\text{and} \quad D_{\mathrm{sd}}(\Lambda) =1. \end{equation*} \notag$

This sequence provides a nontrivial illustration to Remark 2.

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

$\begin{equation*} \Lambda''_0 =\{ k_0M_0+1, \ k_0M_0+2,\ \dots, \ k_0M_0+[\ln^3(k_0M_0)]\} \end{equation*} \notag$

and

$\begin{equation*} \begin{gathered} \, \Lambda''_j =\{ k_jM_0+1, \ k_jM_0+2,\ \dots, \ k_jM_0+[\ln^3(k_jM_0)]\}, \\ k_j =\biggl[\frac{2^{k_{j-1}M_0}}{M_0}\biggr]+1, \qquad j=1,2,\dots; \end{gathered} \end{equation*} \notag$

here we take a positive integer

$k_0$ sufficiently large so that

$\begin{equation*} \Lambda'\cap\Lambda''=\varnothing\quad\text{and} \quad\Lambda'_{j}\cap\Lambda'_{j+1}=\varnothing, \quad j=0,1,\dots\,. \end{equation*} \notag$

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

$\begin{equation} n_{Z_{\varphi}} (t)-\gamma t=O(\ln^2|t|), \qquad |t|\to\infty, \end{equation} \tag{4.6}$

must hold, where

$\gamma =\sigma/\pi \in (M_0^{-1},1)$ .

Setting

$\begin{equation*} \widetilde t_j=k_jM_0\quad\text{and} \quad t_j=k_jM_0+[\ln^3(k_jM_0)], \quad j=0, 1,\dots, \end{equation*} \notag$

we can write

$\begin{equation} n_{\Lambda} (\widetilde t_j)=\frac{1}{M_0}\widetilde t_j +O(\ln^2\widetilde t_j), \qquad j\to\infty, \end{equation} \tag{4.7}$

and

$\begin{equation} n_{\Lambda} ( t_j)=\frac{1}{M_0}\widetilde t_j +[\ln^3(\widetilde t_j)]+O(\ln^2\widetilde t_j), \qquad j\to\infty. \end{equation} \tag{4.8}$

It follows from (4.6) and (4.7) that

$\begin{equation*} n_{\mathcal M} (\widetilde t_j) = \biggl(\gamma-\frac{1}{M_0}\biggr)\widetilde t_j +O(\ln^2\widetilde t_j), \qquad j\to\infty. \end{equation*} \notag$

Hence we deduce from (4.8) that

$\begin{equation*} n_{Z_{\varphi}} (t_j)\geqslant \gamma \widetilde t_j +[\ln^3(\widetilde t_j)]+O(\ln^2\widetilde t_j), \qquad j\to\infty, \end{equation*} \notag$

because it is obvious that

$n_{Z_{\varphi}} (t_j)=n_{\Lambda} (t_j)+n_{\mathcal M} (t_j)$ and

$n_{\mathcal M} (t_j)\geqslant n_{\mathcal M} (\widetilde t_j)$ .

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

$\begin{equation*} n_{Z_{\varphi}} (t_j)\geqslant \gamma t_j + (1-\gamma )[\ln^3t_j]+O(\ln^2t_j), \qquad j\to\infty, \end{equation*} \notag$

which contradicts estimate (4.6).

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

$\begin{equation} \Phi (z)=\frac{\sqrt{z}\sin \pi z}{\sin\pi\sqrt{z}}+\frac{\sin\pi z}{s(z)}, \end{equation} \tag{4.9}$

where

$s (z)=\prod_{k=1}^{\infty}( 1-{z}/{2^{k}})$ . It is clear that

$\Phi\in\mathcal P$ , the exponential type of

$\Phi$ is equal to

$\pi$ , and the set of zeros

$\Lambda$ of this function satisfies

$D_{\mathrm{BM}} (\Lambda ) =1$ .

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

$\begin{equation} \ln |\sin\pi z|=\pi|{\operatorname{Im}z}| + O(1), \qquad |z|\to\infty, \quad |z-k|\geqslant\delta, \quad k\in\mathbb Z, \end{equation} \tag{4.10}$

$\begin{equation} \ln\biggl|\frac{\sin\pi\sqrt{x}}{\sqrt{x}}\biggr|= O(1), \qquad x\to+\infty, \quad |x-k^2|\geqslant \delta, \quad k=1,2,\dots, \end{equation} \tag{4.11}$

and

$\begin{equation} \ln\biggl|\frac{\sin\pi\sqrt{x}}{\sqrt{x}}\biggr|= \pi \sqrt{|x|}+ O(1), \qquad x\to -\infty. \end{equation} \tag{4.12}$

Taking estimate (4.1) for

$\ln |s|$ and relations (4.10)–(4.12) into account we can easily deduce that

$|\Phi |$ is bounded above and below by positive constants on

$[0,\infty)$ , but it decreases more rapidly than any function

$|x|^{-n}$ ,

$n=1,2,\dots$ , as

$x\to -\infty$ . Hence

$\Phi$ is not a slowly decreasing function. Assume that for some

$\Psi\in\mathcal P$ the ratio

$\Psi/\Phi$ is an entire function of order one and minimal type. Then by (4.1) and (4.10)–(4.12) the order of

$\Psi/\Phi$ in the whole plane is 0. Taking into account that this function has polynomial growth on the positive half-axis we conclude that

$\Psi/\Phi$ is a polynomial. Hence

$\Lambda$ cannot be a subset of zeros of a slowly decreasing function of exponential type

$\pi$ .

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

$\begin{equation*} \omega_{\varepsilon} (z)=s(z)\Phi (-\varepsilon z). \end{equation*} \notag$

Using (4.1) and (4.9)–(4.12) we can easily verify that

$\Phi\omega_{\varepsilon}$ is a slowly decreasing function of exponential type

$\pi (1+\varepsilon)$ . As

$\varepsilon>0$ can be arbitrary, we conclude that

$D_{\mathrm{sd}}(\Lambda)=1$ .



Bibliography

1.	A. Aleman and B. Korenblum, “Derivation-invariant subspaces of $C^{\infty}$ ”, Comput. Methods Funct. Theory, 8:1–2 (2008), 493–512
2.	N. F. Abuzyarova, “Spectral synthesis in the Schwartz space of infinitely differentiable functions”, Dokl. Ross. Akad. Nauk, 457:5 (2014), 510–513 ; English transl. in Dokl. Math., 90:1 (2014), 479–482
3.	N. F. Abuzyarova, “Spectral synthesis for the differentiation operator in the Schwartz space”, Mat. Zametki, 102:2 (2017), 163–177 ; English transl. in Math. Notes, 102:2 (2017), 137–148
4.	A. Aleman, A. Baranov and Yu. Belov, “Subspaces of $C^{\infty}$ invariant under the differentiation”, J. Funct. Anal., 268:8 (2015), 2421–2439
5.	N. F. Abuzyarova, “Representation of synthesizable differentiation-invariant subspaces of the Schwartz space”, Dokl. Ross. Akad. Nauk Mat. Inform. Protsessy Upr., 498 (2021), 5–9 ; English transl. in Dokl. Math., 103:3 (2021), 99–102
6.	J. Sebastião e Silva, “Su certe classi di spazi localmente convessi importanti per le applicazioni”, Rend. Mat. e Appl. (5), 14 (1955), 388–410
7.	L. Hörmander, The analysis of linear partial differential operators, v. I, Grundlehren Math. Wiss., 256, Distribution theory and Fourier analysis, Springer-Verlag, Berlin, 1983, ix+391 pp.
8.	N. F. Abuzyarova, “Closed submodules in the module of entire functions of exponential type and polynomial growth on the real axis”, Ufim. Mat. Zh., 6:4 (2014), 3–18 ; English transl. in Ufa Math. J., 6:4 (2014), 3–17
9.	P. Koosis, The logarithmic integral, v. II, Cambridge Stud. Adv. Math., 21, Cambridge Univ. Press, Cambridge, 1992, xxvi+574 pp.
10.	L. Ehrenpreis, “Solution of some problems of division. IV. Invertible and elliptic operators”, Amer. J. Math., 82:3 (1960), 522–588
11.	C. A. Berenstein and B. A. Taylor, “A new look at interpolation theory for entire functions of one variable”, Adv. Math., 33:2 (1979), 109–143
12.	A. M. Sedletskii, “On functions periodic in the mean”, Izv. Akad. Nauk SSSR Ser. Mat., 34:6 (1970), 1391–1415 ; English transl. in Izv. Math., 4:6 (1970), 1406–1428
13.	A. F. Leont'ev, “Properties of sequences of Dirichlet polynomials which are convergent on an interval of the imaginary axis”, Izv. Akad. Nauk SSSR Ser. Mat., 29:2 (1965), 269–328 (Russian)
14.	J. Diedonné and L. Schwartz, “La dualité dans les espaces $(\mathscr F)$ et $(\mathscr{LF})$ ”, Ann. Inst. Fourier (Grenoble), 1 (1950), 61–101
15.	R. P. Boas, Jr., Entire functions, Academic Press Inc., New York, 1954, x+276 pp.
16.	V. S. Vladimirov, Generalized functions in mathematical physics, 2nd ed., Nauka, Moscow, 1979, 319 pp. ; English transl., Mir, Moscow, 1979, xii+362 pp.
17.	L. Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, NJ–Toronto, ON–London, 1966, x+208 pp.
18.	N. F. Abuzyarova, “Some properties of principal submodules in the module of entire functions of exponential type and polynomial growth on the real axis”, Ufim. Mat. Zh., 8:1 (2016), 3–14 ; English transl. in Ufa Math. J., 8:1 (2016), 1–12
19.	N. F. Abuzyarova, “On conditions of invertibility in the sense of Ehrenpreis in the Schwartz algebra”, Lobachevskii J. Math., 42:6 (2021), 1141–1153

Citation: N. F. Abuzyarova, “Representation of invariant subspaces of the Schwartz space”, Sb. Math., 213:8 (2022), 1020–1040

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\paper Representation of invariant subspaces of the Schwartz space

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This publication is cited in the following 3 articles:

N. F. Abuzyarova, Z. Yu. Fazullin, “Invariant subspaces in non-quasianalytic spaces of $\Omega$ -ultradifferentiable functions on an interval”, Eurasian Math. J., 15:3 (2024), 9–24
N. F. Abuzyarova, “Invariantnye podprostranstva v nekvazianaliticheskikh prostranstvakh $\Omega$ -ultradifferentsiruemykh funktsii na intervale”, Izv. vuzov. Matem., 2023, no. 11, 86–91
N. F. Abuzyarova, “Invariant Subspaces in Nonquasianalytic Spaces of Ω-Ultradifferentiable Functions on an Interval”, Russ Math., 67:11 (2023), 75

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§ 1. Introduction

§ 2. Statements of the main results

2.1. Auxiliary facts

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

§ 3. Proof of Theorems 1 and 2

3.1. A reduction to the dual interpolation problem

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

3.3. End of the proof of Theorems 1 and 2

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

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

4.2. Expanding in a series of exponential polynomials with brackets

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

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

Bibliography

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

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

§ 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)

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

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