Mathematics

B. A. RYMARENKO

ON REAL ENTIRE FUNCTIONS BELONGING TO THE CLASS \(W_{\sigma}^{(1)}\)

(Presented by Academician S. N. Bernstein on 31 X 1964)

Denote by \(W_{\sigma}^{(1)}\) the class of real entire functions \(\varphi_{\sigma}(z)=\sum_{k=0}^{\infty} c_k z^k\) of finite degree \(\sigma\), nonnegative on the real axis and such that

\[ A_{\varphi_\sigma}\equiv \|\varphi_\sigma\|=\int_{-\infty}^{\infty}\varphi_\sigma(x)\,dx<\infty . \tag{1} \]

Let \(F_j(\varphi_\sigma)\) \((j=1,\ldots,s)\) be linear functionals defined on the set \(W_\sigma\). We shall call a function \(\varphi_\sigma^*(x)\in W_\sigma^{(1)}\) an extremal function of the problem if its norm \(A_{\varphi_\sigma}\) assumes the smallest value among the norms of all functions \(\varphi_\sigma(x)\in W_\sigma^{(1)}\) subject to the constraints

\[ F_j(\varphi_\sigma)=\gamma_j\qquad (j=1,\ldots,s), \tag{2} \]

where \(\gamma_j\) are prescribed real numbers (we shall consider only admissible constraints, i.e., those compatible and not contradicting the membership of the functions \(\varphi_\sigma(x)\) in the class \(W_\sigma^{(1)}\)). Then the following holds.

Theorem. There exists an extremal function satisfying the constraints (2), for \(s\le 2\), of the form

\[ \varphi_\sigma^*(x)=\left[\psi_{\sigma/2}(x)\right]^2, \tag{3} \]

where \(\psi_{\sigma/2}(x)\) is some real entire function of degree \(\sigma/2\).

Remark. 1) The uniqueness of the extremal function \(\varphi_\sigma^*(x)\) is not asserted here; 2) if \(s>2\), i.e., if more than two constraints of type (2) are prescribed, then the theorem may fail to hold.

By virtue of conditions (1) and (3), the function \(\psi_{\sigma/2}(x)\in W_\sigma^{(2)}\), and therefore, by the Wiener—Paley theorem \((^{1})\), it can be represented in the form

\[ \psi_{\sigma/2}(x)=\int_{-\sigma/2}^{\sigma/2} f(t)e^{ixt}\,dt, \tag{4} \]

where \(f(t)\in L_2[-\sigma/2,\sigma/2]\), and

\[ A_{\varphi_\sigma}\equiv \|\varphi_\sigma\| =\int_{-\infty}^{\infty}\left|\psi_{\sigma/2}(x)\right|^2\,dx =\pi\int_{-\sigma/2}^{\sigma/2}|f(t)|^2\,dt . \tag{5} \]

Thus, the problem of finding an extremal function \(\varphi_\sigma^*(x)\in W_\sigma^{(1)}\) subject to the constraints (2), and its norm, reduces to the minimization of the integral

\[ \int_{-\sigma/2}^{\sigma/2} f^2(x)\,dx \]

under certain constraints imposed on the function \(f(x)\), which follow from conditions (2) and relations (3) and (4).

To solve this problem, let us write for the function \(f(x)\) the Fourier series in the system \(\{\hat P_n(x)\}_0^\infty\) of Legendre polynomials normalized on \([-\sigma/2,\sigma/2]\):

\[ f(x)\sim \sum_{k=0}^{\infty} a_k \hat P_k(x). \tag{6} \]

Taking into account that the system \(\{\hat P_n(x)\}_0^\infty\) is closed and that \(f(x)\in L_2[-\sigma/2,\sigma/2]\), we obtain:

\[ A_{\varphi_\sigma}=\pi\sum_{k=0}^{\infty} a_k^2, \tag{7} \]

and conditions (2) will be rewritten in the corresponding way in terms of the coefficients \(a_k\).

Consider the following examples:

1. Let \(F(\varphi_\sigma)\equiv c_0=1\) \((s=1)\). Then, by virtue of (3), (4), and (6),

\[ \psi_{\sigma/2}(0)= \int_{-\sigma/2}^{\sigma/2}\sum_{k=0}^{\infty} a_k\hat P_k(x)\,dx=1, \]

and, consequently,

\[ a_0=\sigma^{-1/2}. \tag{8} \]

Minimizing \(A_{\varphi_\sigma}\) (expression (7)) under condition (8), we find

\[ a_k=0\ (k=1,2,\ldots);\qquad f(x)=\frac{1}{\sigma};\qquad \varphi_\sigma^*(x)=\left(\frac{\sin \sigma x/2}{\sigma x/2}\right)^2;\qquad A_{\varphi_\sigma^*}=\frac{\pi}{\sigma}. \]

The analogous problem for functions \(\varphi_\sigma(x)\in B_\sigma\) that are monotone on the entire real axis was solved by S. N. Bernstein \((^2)\).

1. Let \(F_1(\varphi_\sigma)\equiv c_0=1\) and \(F_2(\varphi_\sigma)\equiv c_2=\gamma\) \((s=2)\). Performing calculations similar to the preceding ones, we find

\[ a_0=\sigma^{-1/2};\qquad a_2=-\frac{6\sqrt5}{\sigma^{5/2}}\left(\gamma+\frac{\sigma^2}{12}\right);\qquad a_1=0;\qquad a_k=0\quad (k=3,4,\ldots); \]

\[ f(x)=\frac{1}{\sigma} -\frac{6\sqrt5}{\sigma^{5/2}}\left(\gamma+\frac{\sigma^2}{12}\right)\hat P_2(x); \qquad A_{\varphi_\sigma^*} =\pi\left[\frac{1}{\sigma} +\frac{180}{\sigma^5}\left(\gamma+\frac{\sigma^2}{12}\right)^2\right] \]

(the expression for \(\varphi_\sigma^*(x)\) is not given here because of its cumbersomeness, but it is easy to obtain it by taking (4) into account). For functions \(\varphi_\sigma\in B_\sigma\) that are monotone on the real axis, the analogous problem was solved by us earlier \((^3)\).

It is easy to see that, putting in this example \(c_2=-\sigma^2/12\), we obtain for \(A_{\varphi_\sigma^*}\) the minimal value coinciding with \(A_{\varphi_\sigma^*}\) in the first example (i.e., corresponding to the prescription of only one coefficient \(c_0=1\)).

For real polynomials of finite degree \(\varphi_\sigma(x)\in W_\sigma^{(p)}\), nonnegative on the real axis, the analogous theorem indicated here is also valid.

Leningrad Mechanical
Institute

Received
22 X 1964

REFERENCES

\(^1\) R. Paley, N. Wiener, Fourier Transforms in the Complex Domain, N. Y., 1934.
\(^2\) S. N. Bernstein, Collected Works, 2, No. 85, 1954.
\(^3\) B. A. Rymarenko, DAN, 155, No. 1 (1964).