UDC 517.531+517.51

MATHEMATICS

S. P. GEISBERG

ASYMPTOTIC BEHAVIOR OF QUASIANALYTIC FUNCTIONS

(Presented by Academician V. I. Smirnov on 13 VII 1970)

1°. Let \(C\{M_n\}\) be a quasianalytic class of functions \(f(x)\) defined on the entire real axis,
\(\|f\|=\sup_n M_n^{-1}\max_x |f^{(n)}(x)|<\infty\). Denote by \(I_a(F)\) the subset of \(C\{M_n\}\) consisting of functions \(f(x)\) for which \(|f(x)|=O(F(x-a))\). B. I. Korenblum \((^1)\) established the existence of such a function \(F\) that \(I_a(F)\) is a nonempty closed subspace of \(C\{M_n\}\) and \(\bigcap_a I_a(F)=\varnothing\). This function determines the limiting rate of decrease in the class \(*\,C\{M_n\}\).

In the present note it is shown that every function from the class \(C\{M_n\}\) is representable as the sum of a function from \(I_a(F)\) and a function holomorphic and belonging to the class \(**\,A\) in a certain narrowing infinite strip \(D_a(F)\). The strip \(D_a(F)\) is characterized by the following property: if \(\Psi(z)\) maps \(D_a(F)\) conformally onto the right half-plane, then \(\Psi(x)=O(-\ln F(x))\). This result makes it possible to establish an analogue of the well-known Ahlfors–Heins theorem \((^4)\) for quasianalytic functions and thereby to give a description of their asymptotic behavior along the real axis.

2°. Without loss of generality one may assume that the sequence \(\{M_n\}\) is logarithmically convex,
\(\lim_{n\to\infty}(n^{-1}M_n)^{1/n}=\infty\), \(M_0=1\) \((^5)\).

Simple computations show that, under these conditions, the function

\[ \omega(x)=-\ln \left[x^4\min_{n\ge0} x^{-n}M_n\right],\quad x\ge1, \]

has the following properties:
\(0\le \omega(x_1+x_2)\le \omega(x_1)+\omega(x_2)\),

\[ \frac{x\omega'(x)}{\omega(x)}\nearrow 1\quad\text{as }x\to\infty,\qquad \int_1^\infty x^{-2}\omega(x)\,dx<\infty. \]

Put \(\omega(\varphi_n)=n\),

\[ \alpha(x)=\prod_{n=1}^{\infty}\left(1+x^2\varphi_n^{-2}\right)^{1/2}. \]

The following lemma establishes the connection between functions of the class \(C\{M_n\}\) and Fourier transforms of functions integrable with weight \(\alpha(x)\).

Lemma 1. Let \(f(x)\in C\{M_n\}\),

\[ g(x)=\int_{-\infty}^{\infty} f(\tau)(1+\tau^2)^{-1}e^{ix\tau}\,d\tau. \]

* An earlier weaker result was obtained by Hirschman \((^2)\).

** A function \(\varphi(z)\) belongs to the class \(A\) in the strip \(D_a(F)\) if the image \(\varphi\) under a conformal mapping of \(D_a(F)\) onto the right half-plane belongs to the class \(A\) \(((^3),\) p. 289).

Then

\[ \|g\|_\alpha=\int_{-\infty}^{\infty}|g(x)|\alpha(x)\,dx \leq K\|f\|, \]

where \(K\) does not depend on \(f\).

Put

\[ \xi(x)=\sum_{n=1}^{\infty}\frac{x^{2}}{\varphi_n(x^{2}+\varphi_n^{2})},\qquad \eta(\xi(x))=x, \]

\[ F(x)= \begin{cases} \exp\left[-\frac{1}{10}\exp\left(\dfrac{\pi}{2}\int_{1}^{x}\dfrac{\eta(t)}{\omega(\eta(t))}\,dt\right)\right], & \text{for } x\geq 2,\\[1.2em] F(2), & \text{for } x<2, \end{cases} \]

and, for real \(a\), denote by \(D_a(F)\) the domain in the complex plane \(z=\sigma+i\tau\) defined by the inequalities

\[ |\tau|\leq \frac{\omega(\eta(\sigma-a))}{\eta(\sigma-a)} \]

for \(\sigma\geq a+1\), and \(|\tau|\leq \omega(\eta(1))/\eta(1)\) for \(\sigma<a+1\).

Lemma 2. If \(\Psi(z)\) maps \(D_1(F)\) conformally onto the right half-plane, then \(\Psi(x)=O(-\ln F(x))\).

For functions integrable with weight \(\alpha(x)\), the following holds.

Theorem 1. Let

\[ \int_{-\infty}^{\infty}|g(x)|\alpha(x)\,dx<\infty,\qquad \tilde g(x)=\int_{-\infty}^{\infty}g(t)e^{ixt}\,dt. \]

Then for every \(a\in(-\infty,\infty)\) there is a representation

\[ \tilde g(x)=\tilde g_a^{-}(x)+\tilde g_a^{+}(x), \]

where \(\tilde g_a^{+}(x)\) is the restriction to the real axis of an entire function \(\tilde g_a^{+}(z)\) belonging to class \(A\) in the domain \(D_a(F)\), \(\tilde g_a^{-}(x)\in I_a(F)\), and

\[ \sup_x|\tilde g_a^{-}(x)|\leq c_g(a)\|g\|_\alpha,\qquad \lim_{a\to-\infty}c_g(a)=0. \]

Remark 1. From the results of Warschawski \((^6)\) it follows that, if a function \(\psi(z)\) belongs to class \(A\) in the strip \(D_a(F)\) and \(\psi(\sigma)\in I_{a-\varepsilon}(F)\), \(\varepsilon>0\), then \(\psi(z)\equiv0\). Therefore \(\tilde g_a^{+}(x)\) is the principal part of the function \(\tilde g(x)\).

Remark 2. The strip \(D_a(F)\) in Theorem 1 cannot be substantially enlarged. Namely, it cannot be replaced by any domain containing every strip \(D_a(F)\) for sufficiently large \(\sigma\).

The main role in the proof of Theorem 1 is played by

Lemma 3. Let

\[ \Phi(x)=\prod_{n=1}^{\infty}\left(1-\frac{ix}{\varphi_n}\right)^{-1}e^{ix/\varphi_n},\qquad \tilde\Phi(z)=\int_{-\infty}^{\infty}\Phi(x)e^{izx}\,dx,\qquad z=\sigma+i\tau. \]

Then:

a) \(\tilde\Phi(z)\) is an entire function, bounded in every half-plane \(\sigma\leq\sigma_0\);

b) \(\tilde\Phi(z)\) belongs to class \(A\) in the strip \(D_0(F)\);

c) \(\tilde\Phi(z)\in I_0(F)\).

Proof of Theorem 1. Since \(\|g\|_\alpha<\infty\), we have \(g(x)=\Phi(x)r(x)\), where \(r(x)\in L^1(-\infty,\infty)\). Therefore

\[ \tilde g(x)=\int_{-\infty}^{\infty}\tilde\Phi(x-t)\tilde r(t)\,dt = \int_{-\infty}^{a}\tilde\Phi(x-t)\tilde r(t)\,dt + \int_{a}^{\infty}\tilde\Phi(x-t)\tilde r(t)\,dt = \tilde g_a^{-}(x)+\tilde g_a^{+}(x). \]

Using assertion b) of Lemma 3 and the inclusion \(D_p(F)\subset D_q(F)\) for \(p<q\), we obtain that \(\tilde g_a^{+}(x)\) is the restriction to the real axis of a function \(\tilde g_a^{+}(z)\) belonging to class \(A\) in the strip \(D_a(F)\). From assertion c) of Lem—

we and the inequality \(2F(x) \geq \displaystyle\int_x^\infty F(t)\,dt\), we obtain that \(\tilde g_a^{-}(x) \in I_a(F)\). The theorem is proved.

From Lemma 1 and Theorem 1 there follows the main

Theorem 2. Let \(f(x) \in C\{M_n\}\). Then, for every \(a\), \(f(x)\) admits the representation

\[ f(x)=f_a^{+}(x)+f_a^{-}(x), \]

where \(f_a^{+}(x)\) is the contraction to the real axis of a function \(f_a^{+}(z)\) belonging to the class \(A\) in the strip \(D_a(F)\), \(f_a^{-}(x) \in I_a(F)\), and

\[ \sup_x |f_a^{-}(x)| \leq c_f(a)\|f\|,\qquad c_f(a)\leq 1,\qquad \lim_{a\to-\infty} c_f(a)=0. \]

3°. Let us apply the results obtained to the study of the asymptotic behavior of functions of the class \(C\{M_n\}\) as \(|x|\to\infty\). To this end, we note that the asymptotic behavior of functions belonging to the class \(A\) in the domain \(D_a(F)\) is known. Namely, from the Ahlfors–Heins theorem (\(^4\)) and the results of Warschawski (\(^6\)) it follows that, for a function \(g(z)\ne 0\) belonging to the class \(A\) in \(D_a(F)\), we have

\[ \lim_{\sigma\to\infty}^{*}\frac{\ln |g(\sigma)|}{\ln F(\sigma-a)}=\gamma_g>-\infty, \]

where \(\lim^{*}\) means that, in tending to infinity, \(\sigma\) does not take values from a certain set \(E\), for which

\[ \int_E \frac{\eta(x)}{\omega(\eta(x))}\,dx<\infty . \]

Combining this result with Theorem 2, we obtain the following theorem.

Theorem 3. Let \(f(x)\in C\{M_n\}\), \(f(x)\ne 0\). Then there exists an \(a\) such that*

\[ \lim_{x\to\infty}^{*}\frac{\ln |f(x)|}{\ln F(x-a)}=0. \]

An analogous result can be formulated for the case when \(x\to-\infty\). Theorem 3 should naturally be regarded as an analogue of the Ahlfors–Heins theorem for quasianalytic functions. It shows that the asymptotics of functions of the class \(C\{M_n\}\) is identical with the asymptotics along the real axis of functions of the class \(A\) in the strip \(D_a(F)\).

Leningrad State Pedagogical Institute
named after A. I. Herzen

Received
2 VII 1969

CITED LITERATURE

\(^1\) B. I. Korenblyum, Studies on contemporary problems of the theory of functions of a complex variable, Moscow, 1961.
\(^2\) B. Ya. Levin, Distribution of zeros of entire functions, Moscow, 1956.
\(^3\) I. I. Hirschman, Am. J. Math., 72, 1 (1950).
\(^4\) H. Ahlfors, M. Heins, Ann. Math., 50, No. 2 (1949).
\(^5\) C. Mandelbrojt, Contiguous Series. Regularization of Sequences. Applications, Moscow, 1955.
\(^6\) C. E. Warschawski, Mathematics, Collected Translations, 2–4 (1958).
\(^7\) I. I. Hirschman, Am. J. Math., 72, 2 (1950).

* Theorem 3 sharpens a result obtained by Hirschman ((\(^7\)), Theorem 3a).