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This article is cited in 2 scientific papers (total in 2 papers) Representation of invariant subspaces of the Schwartz space N. F. Abuzyarovaab 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
Russian version:
Matematicheskii Sbornik, 2022, Volume 213, Number 8, DOI: https://doi.org/10.4213/sm9687 
Bibliographic databases:
    § 1. IntroductionLet $(a,b)$ 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 $I_W$. 1) If $|I_W| >2\rho (\Lambda)$, then $W$ admits weak spectral synthesis (1.2). 2) If $|I_W|<2 \rho (\Lambda)$, then $W=\mathcal E(a,b)$. 3) 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 results2.1. Auxiliary factsRecall 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) 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 resultsGiven 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 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. I. Let $W$ be a $D$-invariant subspace of $\mathcal E(a,b)$ with discrete spectrum $(-\mathrm{i}\Lambda )$ and residual interval $I_W$, where $|I_W|<+\infty$. 1) 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). 2) 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. II. 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 23.1. A reduction to the dual interpolation problemIn 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 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 $\mathcal E(I)$. I. 1) 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 $\mathcal P(I)$. 2) 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. II. 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 such that 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 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 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, 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
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 and $\mathcal L_{-}$ denote the union of those $L_{\alpha}\subset L(\varphi,a_0)$ for which 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 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 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 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
$$
\begin{equation*}
\operatorname{ch}\operatorname{supp} S=[t_1,t_2]\subset (a',b).
\end{equation*}
\notag
$$
If $(a,b')\cap (t_1,t_2)=\varnothing$, 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 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 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 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 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 $\mathcal P(I)$. 2) Let $\Lambda$ and $I=\langle c,d \rangle$ be as in Theorem 3. If $\widetilde{I}\supset I$ is an interval such that or 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 2Note 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:
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 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 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 bracketsFrom [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 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, 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 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 $D_{\mathrm{sd}}(\Lambda)=1$.
Citation in format AMSBIB
\Bibitem{Abu22} |
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