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B. I. Korenblyum
Quasianalytic Classes of Functions in the Disk
(Presented by Academician A. N. Kolmogorov, February 8, 1965)
Mathematics
1°. In the present paper, for functions analytic in the unit disk \(|z| < 1\) and infinitely differentiable on the circle \(|z| = 1\), a problem analogous to the classical Hadamard—Denjoy—Carleman problem of quasianalyticity is solved.
2°. Let \(\mathfrak{D}\) be the class of functions \(f(z)\) infinitely differentiable in the closed unit disk \(K\) \((|z| \leqslant 1)\). This means that at each point \(z_0 \in K\) and for each function \(f(z) \in \mathfrak{D}\) there is an asymptotic expansion
\[ f(z) \sim \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!}(z-z_0)^n \quad (x \to z_0,\ z \in K), \]
and, for \(|z_0|<1\), this series obviously has a positive radius of convergence. The boundary values \(\widetilde f(\theta)=f(e^{i\theta})\) \((f\in\mathfrak{D})\) form a class of periodic functions of the real variable \(\theta\), which we shall denote by \(\widetilde{\mathfrak{D}}\).
Let \(\{A_n\}_0^\infty\) be a prescribed nondecreasing sequence of positive numbers. Introduce the following subclasses of \(\mathfrak{D}\) and \(\widetilde{\mathfrak{D}}\):
1) \(\mathfrak{D}\{A_n\}\) is the class of functions \(f(z)\in\mathfrak{D}\) for which
\[ \max_{z\in K} |f^n(z)| \leqslant C A_n \quad (n=0,1,2,\ldots), \tag{1} \]
where the constant \(C\) depends on \(f(z)\).
2) \(\widetilde{\mathfrak{D}}\{A_n\}\) is the class of functions \(\widetilde f(\theta)\in\widetilde{\mathfrak{D}}\) for which
\[ \max_{-\infty<\theta<\infty} |\widetilde f^{\,n}(\theta)| \leqslant C A_n \quad (n=0,1,2,\ldots). \tag{2} \]
Definition 1. The class \(\mathfrak{D}\{A_n\}\) is called quasianalytic if from \(f(z)\in\mathfrak{D}\{A_n\}\) and
\[ f^{(n)}(z_0)=0 \quad (n=0,1,2,\ldots), \tag{3} \]
where \(z_0\) is some point of \(K\), it follows that \(f(z)\equiv 0\).
Obviously, (3) can hold for a function \(f(z)\in\mathfrak{D}\) not identically equal to zero only when \(|z_0|=1\).
Definition 2. The class \(\widetilde{\mathfrak{D}}\{A_n\}\) is called quasianalytic if from \(\widetilde f(\theta)\in\widetilde{\mathfrak{D}}\{A_n\}\) and
\[ \widetilde f^{(n)}(\theta_0)=0 \quad (n=0,1,2,\ldots), \tag{4} \]
where \(\theta_0\) is some point \((-\infty<\theta_0<\infty)\), it follows that \(\widetilde f(\theta)\equiv 0\).
Remark. The class \(\widetilde{\mathfrak{D}}\{A_n\}\), generally speaking, does not coincide with the class of functions \(f(e^{i\theta})\) that are boundary values of functions from \(\mathfrak{D}\{A_n\}\).
Definition 3. We shall say that the sequence \(\{A_n\}_0^\infty\) satisfies the Carleman–Ostrowski–Mandelbrojt conditions \((K-O-M)\) if any one of the following equivalent (see (1), p. 29) conditions is fulfilled:
a) If we put \(\beta_n=\inf_{k\ge n} A_k^{1/k}\), then
\[ \sum^\infty \frac{1}{\beta_n}=\infty . \tag{5} \]
b) If we put \(T(r)=\sup_{n\ge 1} r^n/A_n\), then
\[ \int^\infty \frac{\log T(r)}{r^2}\,dr=\infty . \tag{6} \]
c) either \(\lim_{n\to\infty} A_n^{1/n}<\infty\), or \(\lim_{n\to\infty} A_n^{1/n}=\infty\) and
\[ \sum^\infty \frac{A_n^c}{A_{n+1}^c}=\infty, \tag{7} \]
where \(\{A_n^c\}\) is the convex regularization by means of logarithms (see (1), p. 24) of the sequence \(\{A_n\}\).
3°. The following theorem gives the solution of the problem posed.
Theorem 1. In order that the class \(\mathfrak D\{A_n\}\) be quasianalytic, it is necessary and sufficient that the sequence \(\{\sqrt{A_n}\}\) satisfy the \(K-O-M\) conditions.
The same conditions are necessary and sufficient for the quasianalyticity of the class \(\widetilde{\mathfrak D}\{A_n\}\).
Remark 1. In the theory of quasianalytic functions one usually considers the broader classes
\[ \mathbf D\{A_n\}=\bigcup_{k>0}\mathfrak D\{k^n A_n\},\qquad \widetilde{\mathbf D}\{A_n\}=\bigcup_{k>0}\widetilde{\mathfrak D}\{k^n A_n\}. \tag{8} \]
It can be shown that Theorem 1 is also valid for the classes \(\mathbf D,\widetilde{\mathbf D}\).
Remark 2. The class \(\mathfrak D\{A_n\}\) is a proper part of the class \(\widetilde D\{A_n\}\) of all periodic (with period \(2\pi\)) functions satisfying inequalities (2). From Carleman’s classical result \((^2)\) it follows that condition (5) is necessary and sufficient for the quasianalyticity of \(\widetilde D\{A_n\}\).
The condition of Theorem 1 is substantially broader. Thus, for example, the class \(\mathfrak D\{(n!)^2\}\) is quasianalytic, although \(\widetilde D\{(n!)^2\}\) does not possess this property.
4°. To prove Theorem 1 it is convenient to map the disk \(K\) conformally onto the half-plane \(\operatorname{Re} z\ge 0\) in such a way that the point \(z_0\), at which (3) or (4) holds, goes to zero. Under this mapping the sequence \(\{A_n\}\) becomes another one, but the fulfillment of the \(K-O-M\) conditions for \(\{\sqrt{A_n}\}\) is not disturbed. Moreover, instead of inequalities of type (1), (2) it is convenient to consider analogous inequalities for mean-square norms on the straight lines \(\operatorname{Re} z=c\ge 0\). Thus we arrive at a theorem which is essentially equivalent to Theorem 1.
Theorem 2. In order that there not exist a function \(f(z)\not\equiv 0\), infinitely differentiable in the half-plane \(\operatorname{Re} z\ge 0\) and such that
\[ \max_{0\le x<\infty}\int_{-\infty}^{\infty}\left|f^n(x+iy)\right|^2\,dy\le A_n^2 \qquad (n=0,1,2,\ldots), \tag{9} \]
\[ f^{(n)}(0)=0 \qquad (n=0,1,2,\ldots), \tag{10} \]
* It is known that the maximum in (9) is attained at \(x=0\).
necessary and sufficient that the sequence \(\{\sqrt{A_n}\}\) satisfy the \(K—O—M\) conditions.
Proof. Suppose such a function \(f(z)\) exists. As is known \((^3)\), \(f(z)\) can be represented in the form
\[ f(z)=\frac{1}{\sqrt{2\pi}}\int_0^\infty g(t)e^{-tz}\,dt \qquad (\operatorname{Re} z \geqslant 0), \]
where from (9) and (10) it follows that
\[ \int_0^\infty t^{2n}|g(t)|^2\,dt \leqslant A_n^2,\qquad \int_0^\infty t^n g(t)\,dt=0 \quad (n=0,1,2,\ldots). \tag{11} \]
We have arrived at the problem of uniqueness of the solution of a certain Stieltjes-type moment problem, but for measures from \(L^2\). We shall show that problem (11) reduces to the classical Watson problem.*
Consider the function
\[ G(z)=\frac{1}{\sqrt{2\pi}}\int_0^\infty t^{z-\frac12}g(t)\,dt =\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} e^{(z+\frac12)\xi}g(e^\xi)\,d\xi \qquad (\operatorname{Re} z>0). \]
From (11) it follows that \(G(z)\) has the following properties:
a) \(M(x)=\displaystyle\int_{-\infty}^{\infty}|G(x+iy)|^2\,dy\) is bounded in every finite interval \(0<x<a\);
b) \(M(n)\leqslant A_n^2\) \((n=0,1,2,\ldots)\);
c) \(G\left(\frac12+n\right)=0\) \((n=0,1,2,\ldots)\).
Hence it is easy to conclude that the function \(\Phi(z)=G(z)\sec \pi z\) is analytic in the half-plane \(\operatorname{Re} z>0\), and
\[ M_1(x)=\int_{-\infty}^{\infty}\operatorname{ch}^2\pi y\,|\Phi(x+iy)|^2\,dy \]
is bounded in every finite interval \((0,a)\), and
\[ M_1(n)\leqslant A_n^2 \quad (n=0,1,2,\ldots). \tag{12} \]
Finally, consider the Fourier transform of \(\Phi(z)\)
\[ \varphi(\xi)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\Phi(iy)e^{-iy\xi}\,dy. \tag{13} \]
From (12) it follows that \(\varphi(\xi)\) is analytic in the strip \(|\operatorname{Im}\xi|<\pi\), and its boundary values on the lines \(\operatorname{Im}\xi=\pm\pi\) are square-summable. Using the analyticity of \(\Phi(z)\), we may shift the path of integration in (13):
\[ \varphi(\xi)=\frac{1}{\sqrt{2\pi}}e^{-n\xi}\int_{-\infty}^{\infty}\Phi(n+iy)e^{-iy\xi}\,dy. \]
By virtue of (12),
\[ \int_{-\infty}^{\infty}|\varphi(\xi+i\eta)|^2e^{2n\xi}\,d\xi = \int_{-\infty}^{\infty}|\Phi(n+iy)|^2e^{2\eta y}\,dy \leqslant \]
\[ \leqslant 4\int_{-\infty}^{\infty}|\Phi(n+iy)|^2\operatorname{ch}^2\pi y\,dy \leqslant 4A_n^2 \quad (-\pi<\eta<\pi;\ n=0,1,2,\ldots). \]
* The connection between the uniqueness of the Stieltjes moment problem and the Watson problem is known: see, for example, \((^4)\).
Thus, for every function \(f(z)\ne 0\), analytic in the half-plane \(\operatorname{Re} z>0\) and satisfying (9), (10), one can construct \(\psi(\zeta)=1/\varphi(\zeta)\ne 0\), analytic in the strip \(|\operatorname{Im}\zeta|<\pi\) and satisfying the conditions
\[ \int_{-\infty}^{\infty} |\psi(\xi+i\eta)|^2 e^{2n\xi}\,d\xi \leq A_n^2 \quad (-\pi<\eta<\pi;\ n=0,1,2,\ldots), \tag{14} \]
and conversely. Problem (14) is easily reduced to the classical Watson problem for the strip \(|\eta|<\pi\), if one considers the function
\[ \psi_\delta(\xi+i\eta)=\int_{\xi}^{\xi+\delta}\psi(\xi+i\eta)\,d\xi \]
(\(\delta>0\)), which, by virtue of (14), satisfies the inequalities
\[ |\psi_\delta(\xi+i\eta)|\leq C A_n e^{-n\xi} \quad (|\eta|<\pi,\ \xi>0;\ n=0,1,2,\ldots). \tag{15} \]
As is known ([1], p. 55), for the existence of a nonzero solution of problem (15) it is necessary and sufficient that the sequence \(\{\sqrt[n]{A_n}\}\) not satisfy the C–O–M conditions. The same is necessary and sufficient for the existence of a nonzero solution of problem (14). The theorem is proved.
Kyiv Civil Engineering Institute
Received
8 II 1965
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