UDC 517.535.4

MATHEMATICS

I. F. KRASICHKOV

ON CLOSED IDEALS IN A LOCALLY CONVEX ALGEBRA OF ENTIRE FUNCTIONS WITH AN ARBITRARY GROWTH MAJORANT

(Presented by Academician I. M. Vinogradov on 11 I 1966)

Let \(M(t)\) be a monotonically increasing positive function of \(t \ge 0\), satisfying the condition: for sufficiently large \(t\), \(\ln \ln M(t)\) is convex in \(\ln t\). From this condition, in particular, there follows the following property of the function \(M(t)\): for any positive numbers \(a, b\) there exist numbers \(C=C(a,b)\) and \(c=c(a,b)\) such that for all \(t \ge 0\) the inequality

\[ M(at)M(bt) \le C M(ct). \tag{*} \]

holds.

Define the class \(P\) of entire functions as follows: \(\varphi \in P\) if and only if there exist numbers \(a=a(\varphi)\), \(A=A(\varphi)\) such that, for all \(z\),

\[ |\varphi(z)| \le A M(a|z|). \]

If \(\varphi_1, \varphi_2 \in P\), then, by virtue of \((*)\), also \(\varphi_1\varphi_2 \in P\). Therefore, with the usual operations of addition and multiplication, \(P\) is an algebra over the field of complex numbers. By the letter \(m\) we shall denote any monotonically increasing positive function \(m(t)\) of the argument \(t \ge 0\). Let

\[ \mathfrak{B}=\left\{m:\ \lim_{t\to\infty}\frac{m(t)}{M(at)}=\infty \quad \text{for any } a>0\right\}. \]

Introduce in \(P\) a locally convex topology by taking, as a fundamental system of neighborhoods of zero (see \((^1)\), p. 367), the system of all absolutely convex and absorbing sets of the form

\[ V_m=\{\varphi \in P:\ |\varphi(z)| \le m(|z|) \text{ for all } z\}, \quad m \in \mathfrak{B}. \]

With this topology \(P\) is a semi-reflexive space. Convergence of a countable sequence \(\varphi_n \to \varphi\) in the topology of \(P\) is equivalent to the following:

1) \(\varphi_n(z)\to \varphi(z)\) uniformly on each compact set in the complex plane;

2) there exist numbers \(A=A(\{\varphi_n\})\), \(a=a(\{\varphi_n\})\) such that, for all \(z\) and \(n\), the inequalities

\[ |\varphi_n(z)-\varphi(z)| \le A M(a|z|) \]

hold.

These features of the space \(P\) indicate that the topology under consideration is natural for the class \(P\).

The operation of pointwise multiplication of functions turns out to be continuous in this topology. Hence \(P\) is a locally convex algebra. This algebra, however, is not countably normed. Using the terminology of article \((^2)\), we shall say that an ideal \(I \subset P\) is fixed if there exists at least one point \(\lambda\) such that \(f(\lambda)=0\) for every \(f \in I\). The point \(\lambda\) will be called a zero of the ideal \(I\). The multiplicity of a zero of an ideal is defined in the natural way.

For the algebra \(P\) the following is valid.

Theorem on closed ideals. Every nontrivial closed ideal \(I \subset P\) is fixed. If \(\lambda_i\) \((i=1,\ldots)\) is the sequence of all its zeros, then for some \(a>0\)

\[ \overline{\lim_{t\to\infty}} \frac{n(t)}{\ln M(at)} < \infty, \tag{**} \]

where \(n(t)\) is the number of zeros \(\lambda_i\), counted with their multiplicities, in the disk \(\{z:\ |z|\le t\}\). Conversely, if \(\lambda_i\) \((i=1,\ldots)\) is an arbitrary sequence of points satisfying, for some \(a>0\), condition \((**)\), then there exists a nontrivial closed ideal \(I \subset P\) whose zeros are the points \(\lambda_i\) and only these points. The ideal \(I\) is uniquely determined by its zeros.

Thus, the closed ideals in \(P\) are completely characterized by their zeros. The theorem formulated above is a generalization of the result of Ehrenpreis \((^3)\), which pertains only to the algebra of entire functions of exponential type: \(M(t)=e^t\). It is also useful to compare it with the results of F. Schilling \((^4)\) and P. K. Rashevskii \((^5)\). The proof uses the notions of a generalized shift of entire functions and a generalized convolution of functionals in the space of all entire functions. In this setting the algebra \(P\) turns out to be isomorphic to an algebra of linear functionals with convolution. At the final stage of the proof an infinite product is constructed representing a function from \(P\). The method for constructing such a product was indicated by the author earlier (see \((^6)\), p. 175). Here an essential role is played by the condition that \(\ln\ln M(t)\) is convex in \(\ln t\). In conclusion, we note that the notion of generalized shift is closely connected with the theory of analytic operators of A. F. Leont’ev (see \((^7)\), p. 19).

Received
27 XII 1965

CITED LITERATURE

\(^1\) L. V. Kantorovich, G. P. Akilov, Functional Analysis in Normed Spaces, Moscow, 1957.
\(^2\) M. Henriksen, Pacif. J. Math., 2, No. 2 (1952).
\(^3\) L. Ehrenpreis, Am. J. Math., 77, No. 2 (1955).
\(^4\) F. Schilling, Bull. Am. Math. Soc., 52, No. 11 (1946).
\(^5\) P. K. Rashevskii, DAN, 162, No. 3 (1965).
\(^6\) I. F. Krasichkov, Matem. sborn., 56, No. 2 (1962).
\(^7\) A. F. Leont’ev, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 39 (1951).