Sbornik: Mathematics, 2023, Volume 214, Issue 5, Pages 676–702 DOI: https://doi.org/10.4213/sm9802e (Mi sm9802)

DOI: https://doi.org/10.4213/sm9802e

Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 5, Pages 69–96
DOI: https://doi.org/10.4213/sm9802

Bibliographic databases:

§ 1. Introduction

In 1938 Levinson proved the following result (see [1], Ch. VIII, Theorem XLIII), which is a “far-reaching generalization of the principle of the maximum of the modulus for analytic functions” (see [2]).

Note that, independently of Levinson and apparently at the same time, this result, in a slightly different form, was established by Sjöberg [3]. On the other hand, long before this, Carleman [4] proved the following.

Just this result of Carleman’s was subsequently developed by Levinson and Sjöberg, who extended it to the most general case. Note, however, that Carleman’s theorem holds without additional assumptions about the majorant $M(\varphi)$. Subsequently, Wolf [5] transferred the Levinson-Sjöberg theorem to a wide class of functions. Another, simpler proof of Theorem 1 was proposed in [2].

We present a version of this theorem (see [6] and [7]).

As Koosis showed, condition (1.1), under which Levinson’s theorem hold, is also necessary (see [6]): if the integral in (1.1) is divergent, then there exists a sequence of polynomials $P_n(z)$ such that

1) $|P_n(z)|\leqslant K M(|y|)$, $K=\mathrm{const}$, for $n\geqslant1$ and all $z$ in the rectangle

2) as $n\to\infty$,

Note that under some additional assumptions about the behaviour of $M(y)$ a similar result was proved by Levinson in [1]. On the other hand it was shown in [7] that in Levinson’s theorem one can replace the monotonicity of $M(y)$ by its Lebesgue measurability. The following version of a Levinson-type result was presented in [8] (also see [9]).

The sufficiency part of this result follows form Levinson’s theorem. In fact, setting $a=b=1$ in Theorem 1 it is sufficient to consider the family of functions $\{f_{n}(z)\}$ such that $f_{n}(z)=f(z+n)$, $z\in Q$, $n\in\mathbb{Z}$.

In [10] Levinson’s theorem was generalized to the case when, in place of the real interval $[-a,a]$, we have some rectifiable arc $\gamma$ or, more precisely, an arc with bounded slope.

Recall the definition (also see [10]): an arc $\gamma$ with equation $y=g(x)$, $|x|<a$, that satisfies the Lipschitz condition

Now we state the result of [10].

Let $M=M(y)$ be the function from Theorem 1 and $F_M$ be the family of analytic functions $f$ in the curvilinear quadrilateral

It was shown in [10] that if the integral (1.1) is convergent, then for each $\delta>0$ there exists a constant $C_M(\delta)$, which only depends on $\delta$ and $M$, such that for all $f\in F_M$, in the domain

The main step of the prof of this result is constructing a so-called cutoff function, that is, an analytic function $F$ in a neighbourhood $G$ of $\gamma$ such that for each $f\in F_M$ the ratio $f/F$ is analytic in $G$ and continuous in $\overline{G}$ (here $G$ is a curvilinear rectangle with pointed corners). The construction of this function is based on Ahlfors’s theorem on distortion under conformal mappings. In fact, estimates (1.5) for the distance $\rho(z)$ are also used.

In this paper we discuss some questions closely connected with the Levinson-Sjöberg-Wolff theorems and their applications to approximation theory,^1[x]1The statements of these problems are due to the first-named author. and, in particular, the Dyn’kin problems, which he stated in the 1970s. The above overview of results will perhaps allow one take a broader view on these questions in the future and discover other versions of these problems of Dyn’kin’s.

§ 2. Dyn’kin problem of an effective estimate for the growth majorant

Let $E$ be a compact set in $\mathbb{R}$ and $M$ be a majorant from Levinson’s theorem satisfying the bi-logarithmic condition (1.1). In [11] Dyn’kin introduced the system $F_{E}^{0}(M)$ of functions $f$ defined and analytic away from $E$ and such that

By Theorem 1 the set $F_{E}^{0}(M)$ is normal, that is, for each $\delta>0$

Thus, $M^*$ is the least function such that

Let $M$ be a function such that $\log M(e^{-\sigma})$ is a convex function of $\sigma$.

Set

As claimed in [12] (see p. 61, § 2.1, the remark), the problem of an effective estimate for the majorant “in the form $M^*\simeq P^{-1}$ with unknown $P$ was established in [11]”. Here and throughout, $M^*\simeq P^{-1}$ means that

Now we look at the results in [11] more closely. In that paper the author considered only regular sequences $\{M_n\}$, that is, sequences such that the numbers $m_n={M_n}/{n!}$ have the following properties:

1) $m_n^{1/n}\to\infty$ as $n\to\infty$;

2) $\displaystyle \sup_{n\geqslant0}\biggl(\dfrac{m_{n+1}}{m_n}\biggr)^{1/n}<\infty$;

3) $m_n^2\leqslant m_{n-1} m_{n+1}$, $n\geqslant1$.

As is well known, for $\alpha>0$ the Carleman class $C_{I}((n!)^{1+\alpha})$ is called a Gevrey class. It is regular because the numbers $M_n=(n!)^{1+\alpha}$ satisfy conditions 1)–3).

The associated weight is the function $H^*(r)=[h^*(r)]^{-1}$ (see [11]), where

The class $C_{I}(M_n)$ is quasianalytic if and only if any of the following equivalent conditions is satisfied (see [11]):

1) $\displaystyle\sum_{n=0}^{\infty} \frac{M_n}{M_{n+1}}=\infty$;

2) $\displaystyle\int_{0}^{1} \log^{+}\log H^*(t)\,dt=\infty$.

Let us present the results from [13], where the first Dyn’kin problem on estimates for the majorant $M^*$ was solved. Consider the square

In [11] and [12] Dyn’kin asked about the precise behaviour of the majorant

An upper bound for $M^*$ can be obtained using one method due to Domar [7], [9] (see [6]).

Using duality, Matsaev showed that the Levinson-Sjöberg theorem is equivalent to the Denjoy-Carleman theorem on the quasianalytic classes $C_{I}(M_n)$ (see [14]). Subsequently, this fact was re-discovered by Dyn’kin [15], while in [12] he claimed two-sided bounds for $M^*$ in terms of the quantity

Let

• 1) $\varphi(t)\geqslant0$, $t\in\mathbb{R}_{+}=[0,+\infty)$;
• 2) $\varphi(t)\uparrow\infty$ as $t\to +\infty$ and $\displaystyle\lim_{t\to\infty} \frac{\varphi(t)}{\log t}=\infty$;
• 3) $\displaystyle\int_{\mathbb{R}} \frac{\varphi(t)}{t^2+1}\,dt<\infty$;
• 4) $\varphi(e^x)$ is convex in $x$ on $\mathbb{R}_{+}$.

Occasionally, authors also impose a further condition on $\varphi$:

• 5) $\varphi(t)$ is concave (upward convex) on $\mathbb{R}_{+}$, and
  $$ \begin{equation*} \lim_{t\to\infty} t \varphi'(t+0)=\infty. \end{equation*} \notag $$

For the logarithm of the majorant $M$ in (2.2) let

The following result was proved in [13].

§ 3. Second Dyn’kin problem of estimates for the function $J_M(s)$

The problem discussed in this section goes back historically to Bang [17].

Let $\{M_n\}_{n=0}^{\infty}$ be an arbitrary positive sequence such that $M_n^{1/n}\to \infty$ (but not necessarily regular). Then it has the greatest logarithmically convex minorant $\{M_n^{c}\}_{n=0}^{\infty}$, which is a sequence satisfying $M_n^{c}\leqslant M_n$ for $n\geqslant0$, and $(M_n^{c})^2\leqslant M_{n-1}^{c} M_{n+1}^{c}$ for $n\geqslant1$. The sequence $\{M_n^{c}\}$ is called the convex regularization of $\{M_n\}$ by logarithms (see [18]).

Let $P=\{n_i\}$ be the sequence of principal indices, so that $M_{n_i}=M_{n_i}^{c}$ for $i\geqslant1$. In [17], for each function $f\in C^{\infty}(I)$ Bang considered the quantity

The central result in [17] is as follows.

Note that in this statement $q$ does not necessarily belong to the set $P$ of principal indices. The parameter $h$ is chosen so that the shift $x+h$ belongs to $I$.

Bang used Theorem 6 to prove a criterion for the class $C_{I}(M_n)$ to be quasianalytic. We are only interested in the sufficiency part of this criterion, because its proof implies a simple estimate for each function $f$ in the class $C_{I}^{0}(M_n)=\{f\colon f\in C_{I}^{N}(M_n),\ f^{(n)}(0)=f^{(n)}(1)=0,\ n\geqslant0\}$ in a neighbourhood of $x=0$. Some authors extend this estimate groundlessly to the extremal function $J_M(M_n)$ (see [12] and [13]).

Making no claim to originality, we give a short proof of the following result due to Bang: if a class $C_{I}^{0}(M_n)$ is not quasianalytic, then

By assumption there exists a function $f$ in $C_{I}^{0}(M_n)$ such that $f(x)\not\equiv0$. Hence $B_{f}(x)\not\equiv0$ too. Therefore, there exist $p_1\in P$ and $x_1\in I$ such that $B_{f}(x_1)=e^{-p_1}$. Next we construct recursively a sequence $\{x_n\}_{n=1}^{\infty}$: such that $x_n\downarrow0$, $B_{f}(x_j)=e^{-p_j}$ for $p_j\in P$, $p_1<p_2<\dots<p_n<\dotsb$. If $x=x_j$ and $x+h=x_{j-1}$, then $h>0$. By Theorem 6

However, the left-hand side here is

The proof is complete. However, we are interested in inequality (3.4) itself, because Bang obtained an important estimate for $f$ on its basis: if $x\in I$ and

It should be noted that here $p_1$ depends on the particular function $f$: the smaller $\|f\|$, the greater $p_1=p_1(f)$.

Using Taylor’s formula Bang also obtained another inequality, which yields the bound

To see the difference between (3.5) and (3.6) we look at an example.

Consider the sequence of numbers

Let $f$ be the function from the above proof of the sufficiency part of Theorem 6; it satisfies (3.5). From (3.6) we also obtain

In view of the rapid growth of $H_1(x)$ as $x\to0$, we can write (3.7) as follows:

The fact that $C_{I}^{N}(M_n)$ is not quasianalytic follows from the condition

However, using Bang’s estimate (3.5) we can deduce a sharper estimate, albeit for a fixed function $f$ (see [17]): there exists $x_0=x_0(f)$ such that for all $x$, ${0<x<x_0(f)}$, and some $c=c(f)>0$ we have

A natural question is as follows: which of inequalities (3.8) and (3.10) reflects faithfully the behaviour of the extremal function $J_M(x)$?

An attempt to answer was made in [12] (also see [19] in this connection).

Let $\{M_n\}$ be a regular sequence and $H_{0}$ be the corrected associated weight function, that is,

In view of the proof of Bang’s theorem and the above comments to (3.5), we can conclude that, in place of $J_M(x)$, estimates (3.16) must involve the particular function $f$ constructed in [10], and the constants $q_1$ and $q_2$ must depend on this function $f$, that is, $q_1=q_1(f)$ and $q_2=q_2(f)$. This means that, contrary to the author’s claim (see [12]), the second Dyn’kin problem was in fact not solved in [12].

It is easy to see that, for the sequence $M_n=n!\, [\log (n+e)]^{(1+\beta)n}$, where $\beta>0$ and $n\geqslant0$, we have

In [13] the corresponding inequality, in place of ${1}/{|f(x)|}$, involved incorrectly the quantity $\delta_{\{M_n\}}(s)=\sup\{|g(s)|,\ g\in C_{I}^{0}(M_n)\}$.

Thus, although Bang’s asymptotic estimate (3.10), is better than (3.8) for each fixed $f$ of the above type, it does not describe adequately the behaviour of $J_M(x)$.

The following result was established in [19].

Now we compare estimates (3.17) for $J_M(x)$ with Dyn’kin’s estimates (3.16) for the function $f\in C_{I}^{0}(M_n)$ constructed in [10] and [12] using Gurarii’s method of cutoff function. In doing this it is natural to limit ourselves to the case when^2[x]2This condition fails for the Gevrey class $C_{I}([n!]^{1+\alpha})$, where $\alpha>0$, because $t |\psi'(t)|\sim 1/\alpha$. However, then $\theta(y)\sim \alpha y$ as $y\to0$, so estimates (3.16) do hold formally and coincide with (3.17). $t |\psi'(t)|\to\infty$ as $t\to0$. Then, using Dyn’kin’s estimate (3.15) we obtain

Set

Estimates of the form (3.17) for $J_M(x)$ are important for applications, for example, in problems concerning the asymptotic behaviour of entire Dirichlet series on the real axis (see [20]).

We go over to the results in [13] related to estimates for the extremal function $J_M(y)$. The authors of [13] claimed that they also solved the second Dyn’kin problem in that paper.

Let $\varphi(r)=\log T(r)$, where $T(r)=\sup_{n\geqslant0}(r^n/M_n)$ is the trace function of the sequence $\{M_n\}$ which satisfies the nonquasianalyticity condition

The following result was presented in [13], Theorem 2.1: if

We see that $\delta_{\{M_n\}}(x)\equiv J_M(x)$; however, neither the regularity of the sequence $\{M_n\}$ nor the convergence of (3.12) were explicitly assumed in [13]. We use this notation for the extremal function in what follows, when we discuss the results of [13].

If $\varphi(r)=\log T(r)$ is a concave function on $\mathbb{R}_{+}$ such that $\varphi(r) \log^{-3}(r)\uparrow\infty$ as $r\to\infty$, then it was shown in [13] that condition (3.20) holds for it, but the authors knew of no weaker condition that would be easier to verify and could replace (3.20) (see [13]). However, the assumption that $\log T(r)$ itself is concave restrict excessively the class of sequences $\{M_n\}$. Usually, in such problem a natural object of consideration is the least concave majorant $\omega_{T}(r)$ of the function $\log T(r)$, about which one assumes that it belongs to the convergence class, that is, satisfies (3.19). In fact,

We see from the proof of the result of [13] presented above that it is reasonable to consider separately two cases: when only the integral (3.19) converges and when the analogous integral for the function $\omega_{T}(r)$ is convergent. In fact, the verification of the asymptotic equality (3.21) relies essentially on Lemma 2.1 in [13]: let $W$ be the outer function in the upper half-plane $\mathbb{C}_{+}$ with logarithmic weight $\varphi(t)=\log T(|t|)$. Then

Note that $W(z)\neq0$ in $\mathbb{C}_{+}$, and we have $W(z)=e^{u(z)+i v(z)}$, where

It was proved in Lemma 4.1 in [13] that if $\varphi(r)=\log T(r)$ belongs to the convergence class, then

Thus, taking account of the upper estimate in (3.23) and (3.26) one arrives at the following assertion (see [13], Theorem 2.1): if the integral in (3.19) converges, then, as $s\to0$,

We obtain a lower bound for $\log \delta_{\{M_n\}}(s)$ from the one in (3.23) and an appropriate lower bound for $\rho_{1,W}(s)$, which however is obtained under the additional assumption (3.20). Since this case reduces in fact to the convergence of the bi-logarithmic integral (3.22) for $H(y)=\exp(m(y))$ or, equivalently, for the associated weight $H_0$, we consider just it in our paper. Here we obtain a relation, quite different from the one in (3.21), which provides a solution of the second Dyn’kin problem.

As regards inequality (3.27), it is false (see § 5): its proof in [13] is inaccurate and contains a significant gap. In fact, the asymptotic inequality (3.27) holds for each fixed function $g\in C_{I}^{0}(M_n)$, but in its own neighbourhood of zero, that is, for ${0<s\leqslant s_{0}(g)}$, rather than for the extremal function $\delta_{\{M_n\}}(s)$. That is, there was no solution of the Dyn’kin problem in question in [13].

§ 4. The main result: solving the second Dyn’kin problem

Let $\{M_n\}$ be a regular sequence and $H_0$ be the associated weight introduced above.

If the integral

We have obtained an upper bound for $J_M(x)$. For a lower one we consider the normed space $F_{I}(H_0)$ of analytic functions in $\mathbb{C}\setminus I$ satisfying the estimate

In place of $I$ we could consider any closed set $E\subset\mathbb{R}$ (see [11]). So taking $E=\{0\}$ consider the linear functional $G$ on the space $F_{\{0\}}(H_{0})$ such that $\langle G,f\rangle=f(\delta)$ for some fixed $\delta\in (0,1]$. Then we obviously have $|\langle G,f\rangle|\leqslant C_{f} H_{0}(\delta)$. Because the integral (4.1) is convergent, by Levinson’s theorem the set of functions $F_{\{0\}}^{0}$ is normal. Hence setting $C_{f}^{0}=\inf C_{f}$ we obtain $\sup_{f\in F_{\{0\}}^{0}(H_{0})} C_{f}^{0}=C<\infty$. Therefore, $\|G\|\leqslant C H_{0}(\delta)$ (the positive constant $C$ is independent of $\delta$). Now, since $F_{\{0\}}(H_{0})\subset F_{I}(H_{0})$, by the Hahn-Banach theorem the functional $G$ can be extended to the whole of $F_{I}(H_{0})$ with the same norm. We keep the notation $G$ for this functional and consider the function

Now consider the function $g$ such that $g(t)=1+\eta(t)(t-\delta)$. Since

1) $g^{(n)}(0)=0$, $n\geqslant0$;

2) $\|g^{(n)}\|\leqslant K H_{0}(\delta) M_n$, $n\geqslant0$, where $K=(1+L)C$;

3) $g(\delta)=1$.

Hence the function

We state the result obtained as the following theorem.

Thus, we see from (4.3) that, as $x\to0$,

Estimates (4.3), by contrast with (3.17), describe the asymptotic behaviour of the extremal function $J_M(x)$ as accurately as possible. The meaning of this theorem is that $J_M(x)$ tends to zero as $x\to0$ much slower than any function $f$ in the class $C_{I}^{0}(M_n)$ (see above). Note that in Theorem 9 we need not rely on Theorem 5 from [13], which solves the first Dyn’kin problem, or, more precisely, on the relation

As we show in § 5, the function $\log H_{0}(x)$ exhibits a considerably slower growth as $x\to0$ than $P_{\varphi}(x)$: $\log H_{0}(x)=o(P_{\varphi}(x))$ as $x\to0$. This means that the upper bound in (2.1), on which the authors of some papers mentioned above relied, is incorrect. In the context of the proof of Theorem 9, $M^*(x)=C H_{0}(x)$.

§ 5. On an upper bound for $\delta_{\{M_n\}}(s)$

For a function $g\in C_{I}^{0}(M_n)$ its Fourier transform

Using (5.1) and the maximum modulus principle, we have

Thus,

Let $\tau_s$ be such that

Setting $\tau=1/s$ we obviously obtain $m(s)=\log T(\tau_s)-s \tau_s\geqslant \log T(1/s)-1$, so that for $0<s\leqslant s_0$

Now we claim that

On the other hand, since $\sqrt{n}<2^{n+1}$ for $n\geqslant0$, in a similar way we obtain

Thus, using estimates for Fourier transforms we have obtained an estimate for $J_M(s)$ analogous to the right-hand estimate in (3.17) (although in terms of the associated weight $H_{0}$, which is not very important). Note that in [13] the authors attempted to establish the ostensibly finer asymptotic estimate

We can also obtain an estimate of type (5.10) via Taylor’s formula (see above). On the other hand the main difficulty in Theorem 9 consists in estimating $J_M(s)$ from below in terms of $H_{0}(s)$. We have obtained such an estimate under assumption (4.1), but in fact condition (3.20), on which we commented above, has the same meaning as the convergence of the bilogarithmic integral (4.1).

It follows from (5.9) and estimates (4.3) in Theorem 9 that

Note that condition (5.14) in this lemma holds, for instance, for the regular sequence $M_n=n!\,[\log (n+e)]^{(1+\beta)n}$, $\beta>0$, for $n\geqslant0$. In this case

§ 6. Applying the main result: an estimate for the distance between the algebraic polynomials and the imaginary exponentials in a weighted space

Following [13], let $C_{T}^{0}$ denote the weighted space of continuous functions $f$ on $\mathbb{R}$ such that

Which functions in $C_{T}^{0}$ can actually be approximated by polynomials in this weighted space? It is known that the limit function must be the restriction of an entire function of minimal exponential type to $\mathbb{R}$ (see [24], Supplements and problems, §§ 12 and 13).

The following problem was discussed in [13]: what is the asymptotic behaviour as $s\to0$ of the quantity

However, this result was deduced from (3.21) with the help of Lemma 2.2 in [13]: let $W$ be an outer function in $\mathbb{C}_{+}$ with logarithmic weight $\varphi(t)=\log T(|t|)$ (in this case $|W(t)|=T(|t|)$, $t\in\mathbb{R}$, where the function $W$ was defined above). Then

The general form of a continuous linear functional on $C_{T}^{0}$ is as follows:

Now we find a lower estimate for $d_{T}(s)$.

For each functional $\mu^*\in \mathscr{P}^{\perp}$ and any algebraic polynomial $\mathscr{P}$ we have

Let $g$ be an arbitrary function in $C_{I}^{0}(M_{n-2})$, let $M_{-2}=M_{-1}=M_{0}$, and let $f(x)=(F^{-1}g)(x)$. Then $f\in C^{\infty}(\mathbb{R})$ and $f(x)\equiv0$ for $x\leqslant0$. We also have

Set $d \nu(t)=c f(t) \, dt$, where $c>0$ is a normalizing coefficient (to be specified in what follows). Clearly, $\mu^*\in \mathscr{P}^{\perp}$, where

Therefore, for each function $g\in C_{I}^{0}(M_{n-2})$

Thus, in view of (6.5), (6.7) and (6.9) we have the following result.

Thus, under the assumptions of Theorem 10, as $s\to0$,

Acknowledgements

The authors are grateful to the participants of the seminar Complex and harmonic analysis (at the Institute of Mathematics with Computer Center of the Ufa Federal Research Center of Russian Academy of Sciences) for discussions of the main results obtained in this paper. The authors are also obliged to referees for useful comments.

Bibliography
1.	N. Levinson, Gap and density theorems, Amer. Math. Soc. Colloq. Publ., 26, Amer. Math. Soc., New York, 1940, viii+246 pp.
2.	V. P. Gurarii, “On Levinson's theorem concerning normal families of analytic functions”, Investigations in linear operators and function theory, Pt. 1, Semin. Math., Springer, Boston, MA, 1972, 124–127
3.	N. Sjöberg, “Sur les minorantes subharmoniques d'une function donée”, Comptes rendus du IX congres des mathématiciens scandinaves (Helsinki 1938), Helsingfors, 1939, 309–319
4.	T. Carleman, “Extension d'un théorème de Liouville”, Acta Math., 48:3–4 (1926), 363–366
5.	F. Wolf, “On majorants of subharmonic and analytic functions”, Bull. Amer. Math. Soc., 48:12 (1942), 925–932
6.	P. Koosis, The logarithmic integral, v. I, Cambridge Stud. Adv. Math., 12, Cambridge Univ. Press, Cambridge, 1988, xvi+606 pp.    ; Corr. reprint of the 1988 original, 1998, xviii+606 pp.
7.	Y. Domar, “On the existence of a largest subharmonic minorant of a given function”, Ark. Mat., 3:5 (1958), 429–440
8.	A. Borichev and H. Hedenmalm, “Completeness of translates in weighted spaces on the half-plane”, Acta Math., 174:1 (1995), 1–84
9.	Y. Domar, “Uniform boundedness in families related to subharmonic functions”, J. London Math. Soc. (2), 38:3 (1988), 485–491
10.	A. M. Gaĭsin and I. G. Kinzyabulatov, “A Levinson-Sjöberg type theorem. Applications”, Sb. Math., 199:7 (2008), 985–1007
11.	E. M. Dyn'kin, “Growth of an analytic function near its set of singular points”, J. Soviet Math., 4:4 (1975), 438–440
12.	E. M. Dyn'kin, “The pseudoanalytic extension”, J. Anal. Math., 60 (1993), 45–70
13.	V. Matsaev and M. Sodin, “Asymptotics of Fourier and Laplace transforms in weighted spaces of analytic functions”, St. Petersburg Math. J., 14:4 (2003), 615–640
14.	V. Matsaev, Uniqueness, completeness and compactness theorems related to classical quasianaliticity, Kandidat dissertation, Physics and Technology Institute of Low Temperatures of the Academy of Sciences of Ukr.SSR, Khar'kov, 1964 (Russian)
15.	E. M. Dyn'kin, “Functions with given estimate for $\partial f/\partial\overline z$, and N. Levinson's theorem”, Math. USSR-Sb., 18:2 (1972), 181–189
16.	N. Nikolski, “Yngve Domar's forty years in harmonic analysis”, Festschrift in honour of Lennart Carleson and Yngve Domar (Uppsala 1993), Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., 58, Uppsala Univ., Uppsala, 1995, 45–78
17.	T. Bang, “The theory of metric spaces applied to infinitely differentiable functions”, Math. Scand., 1 (1953), 137–152
18.	S. Mandelbrojt, Séries adhérentes. Régularisation des suites. Applications, Gauthier-Villars, Paris, 1952, xiv+277 pp.
19.	A. M. Gaisin, “Extremal problems in nonquasianalytic Carleman classes. Applications”, Sb. Math., 209:7 (2018), 958–984
20.	A. M. Gaisin, “Dirichlet series with real coefficients that are unbounded on the positive half-axis”, Sb. Math., 198:6 (2007), 793–815
21.	A. M. Gaisin, “Levinson's condition in the theory of entire functions: equivalent statements”, Math. Notes, 83:3 (2008), 317–326
22.	P. Koosis, Introduction to $H^p$ spaces, With an appendix on Wolff's proof of the corona theorem, London Math. Soc. Lecture Note Ser., 40, Cambridge Univ. Press, Cambridge–New York, 1980, xv+376 pp.
23.	G. M. Fichtenholz, A course of calculus, v. II, 8th ed., Fizmatlit, Moscow, 2006, 864 pp. (Russian); German transl., G. M. Fichtenholz, Differential- und Integralrechnung, v. II, Hochschulbücher fur Math., 62, 10. Aufl., VEB Deutscher Verlag der Wissenschaften, Berlin, 1990, 732 pp.
24.	N. I. Akhiezer, Lectures on approximation theory, 2nd ed., Nauka, Moscow, 1965, 407 pp. (Russian)    ; German transl., N. I. Achieser, Vorlesungen über Approximationstheorie, Math. Lehrbücher und Monogr., II, 2. verbesserte Aufl., Akademie-Verlag, Berlin, 1967, xiii+412 pp.

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