Abstract
Full Text
UDC 519.21
MATHEMATICS
O. A. PRESNYAKOVA
ON THE ANALYTIC STRUCTURE OF SUBSPACES GENERATED BY RANDOM HOMOGENEOUS FIELDS
(Presented by Academician Yu. V. Linnik on 22 X 1969)
- Consider a homogeneous random field \(\xi(\mathbf{t})\), \(\mathbf{t}\in R^n\), with spectral density \(f(\mathbf{x})\), \(\mathbf{x}\in R^n\). Let \(M\subset R^n\) be some closed set of values of the time parameter \(\mathbf{t}\). With the field under consideration one naturally associates the space \(H(M)\)—the closed linear span, in the mean-square sense, of the random variables \(\xi(\mathbf{t})\) such that \(\mathbf{t}\in M\). The linear space \(H(R^n)\) is a Hilbert space with inner product \(\langle \xi,\eta\rangle=\mathbf{E}\xi\eta\). The spaces \(H(M)\), \(M\subset R^n\), are its subspaces. In the case \(n=1\), the question of the structure of the spaces \(H(M)\), when \(M\) is an interval, was completely solved by M. G. Krein. His results are formulated in the paper \(\left({}^{1}\right)\). Related questions are also investigated in the paper of Levinson and McKean \(\left({}^{2}\right)\) (the work of M. G. Krein \(\left({}^{1}\right)\), apparently, remained unknown to the authors of \(\left({}^{2}\right)\)).
We shall study the structure of subspaces in the case where \(M\) is a rectangle:
\[ M=K^\rho=\{\mathbf{t}=(t_1,\ldots,t_n):\ -\rho_j\leq t_j\leq \rho_j,\ j=\overline{1,n};\ \rho=(\rho_1,\ldots,\rho_n)\}. \]
As is known \(\left({}^{3}\right)\), there exists an isometric correspondence between the space \(H(R^n)\) and the space \(L^2(R^n,f)\) of functions square-summable with weight \(f(\mathbf{x})\). Under this correspondence, the subspace \(H(M)\) corresponds to the subspace \(L^2(M,f)\), which is the closure in \(L^2(R^n,f)\) of the linear span of the set of functions \(\exp i(\mathbf{x},\mathbf{t})\) such that \(\mathbf{t}\in M\). Here \((\mathbf{x},\mathbf{t})=x_1t_1+\cdots+x_nt_n\). Thus, the study of the structure of the subspace \(H(K_\rho)\) is reduced to the study of \(L^2(K_\rho,f)\).
- Introduce the notation:
\[ f_{t_0}^{*}(t)=\inf_{t_0^2\leq |\mathbf{x}|^2\leq t^2} f(\mathbf{x}),\qquad |\mathbf{x}|^2=x_1^2+\cdots+x_n^2, \]
\[ g^{*}(t)=\inf_{|\mathbf{x}|^2=t^2} f(\mathbf{x}), \]
\(S(M,\varepsilon)\) is the closure of the \(\varepsilon\)-neighborhood of the set \(M\).
\[ L^2(M^{+},f)=\bigcap_{n=1}^{\infty} L^2(S(M,\varepsilon_n),f),\qquad \text{where }\varepsilon_n\downarrow 0. \]
Theorem 1. If for some \(t_0\geq 0\) the inequality
\[ \int_{t_0}^{\infty}\frac{\ln f_{t_0}^{*}(t)}{1+t^2}\,dt>-\infty, \tag{1} \]
holds, then the set of functions \(L^2(K_\rho^{+},f)\) coincides with the set of entire analytic functions \(\varphi(z)\), \(z\in C^n\), satisfying the conditions:
\[ \int_{R^n}|\varphi(\mathbf{x})|^2 f(\mathbf{x})\,d\mathbf{x}<\infty \tag{2} \]
and
\[ \varphi(z)\text{ is an entire function of type }\rho=(\rho_1,\ldots,\rho_n) \tag{3} \]
(the definition of the type of an entire function here is the same as in \((^4)\)).
We denote by \(L^\circ_\rho\) the set of entire functions satisfying (2) and (3).
In the proof of the theorem Bernstein’s function is used, which is defined (see \((^5)\), p. 376) as follows. Let a sequence of numbers \(t_k>0,\ k=1,2,\ldots\), be given such that
\[ p=\pi\sum_{k=1}^{\infty}\frac{1}{t_k}<\infty. \]
Let \(a_k^2=\dfrac{1}{p^2t_k^2}\). From the sequence \(\{a_k\}\) we construct the Bernstein function
\[ B(z)=\prod_1^\infty \sin^2\frac{\pi}{2}\sqrt{1+a_k^2z^2/(1+a_k^2z^2)}. \]
From the same sequence \(\{a_k\}\) one can construct a function of \(n\) variables
\[ B(z)=B(z_1,\ldots,z_n)= \]
\[ =\prod_1^\infty \sin^2\frac{\pi}{2}\sqrt{1+a_k^2(z_1^2+\ldots+z_n^2/[1+a_k^2(z_1^2+\ldots+z_n^2)])}. \]
\(B(z)\) and \(B(z_1,\ldots,z_n)\) are entire analytic functions of finite degree. The type of each of them is 1. The proof of the theorem is based on the following lemmas.
Lemma 1. Let \(u(t)>0,\ t\in(0,\infty)\), be a monotonically increasing function satisfying the conditions:
\[ \int_0^\infty \frac{\ln u(t)}{1+t^2}\,dt<+\infty \qquad u(t)\xrightarrow[t\to+\infty]{}+\infty. \]
Then there exists a sequence \(t_k>0,\ k=1,2,\ldots\), such that
\[ \sum_{k=1}^{\infty}\frac{1}{t_k}<\infty, \]
and the Bernstein function \(B(z)\) constructed from it \((z=t+is)\) has the property: for every \(\lambda\ne0\),
\[ u(t)B(\lambda t)\le c(\lambda), \]
where \(c(\lambda)\) is a constant depending on \(\lambda\).
This lemma is a consequence of a theorem proved by O. I. Inozemtsev and V. A. Marchenko in \((^6)\). Put \(f^*_{t_0=0}(t)=f^*(t)\). For \(t_0=0\) the function \(1/\sqrt{f^*(t)}\) satisfies the conditions of Lemma 1; therefore the following lemma makes sense.
Lemma 2. Let \(B(z)\) be such a Bernstein function that for every \(\lambda\ne0\)
\[ B(\lambda t)/\sqrt{f^*(t)}\le c(\lambda), \]
where \(c(\lambda)\) is a constant depending on \(\lambda\). Then one can construct a Bernstein function \(B(z_1,\ldots,z_n)\) such that, whatever \(\lambda_1,\ldots,\lambda_n\ne0\) may be,
\[ B^2(\lambda_1x_1,\ldots,\lambda_nx_n)/f(x_1,\ldots,x_n)\le d(\lambda_1,\ldots,\lambda_n), \]
where \(d(\lambda_1,\ldots,\lambda_n)\) is a constant depending on \(\lambda_1,\ldots,\lambda_n\).
Lemma 3. Let \(\varphi(z_1,\ldots,z_n)\) be an entire function of finite degree, and let
\[ \varphi(z_1,\ldots,z_n)B(\lambda_1z_1,\ldots,\lambda_nz_n) \]
have type
\[ \sigma=(\rho_1+\max(\lambda_j,\ j=1,n),\ldots,\rho_n+\max(\lambda_j,\ j=1,n)). \]
Then the type of the function \(\varphi(z_1,\ldots,z_n)\) is equal to \(\rho=(\rho_1,\ldots,\rho_n)\).
3. Theorem 2. Let the function \(g^*(t)\) satisfy the conditions:
a) \(g^*(t)\) is continuous,
b) \(g^*(t_1+t_2)\ge c\,g^*(t_1)g^*(t_2)\);
c)
\[ \int_0^\infty \frac{\ln g^*(t)}{1+t^2}\,dt>-\infty; \tag{4} \]
then the set of functions \(L^2(K_\rho^+,f)\) coincides with \(L^\circ_\rho\).
The proof of the theorem is analogous to the proof of Theorem 1.
- In the paper of V. N. Tutubalin and M. I. Freidlin \({}^{7}\) it is proved for a random stationary process that if its spectral density \(f(x) \geqslant 1/|x|^{p}\) for some \(p>0\) and all \(x\) such that \(|x|>x_{0}>0\) (\(x_{0}\) is any positive number), then the functions \(1, ix, \ldots, (ix)^{k}\) form a basis in the space \(L^{2}(K_{0}^{+}, f)\) (\(k\) is the number of derivatives possessed by the trajectories of the process). The following is connected with their result.
Theorem 3. Let the spectral density \(f(x)\) satisfy condition (1) or conditions (4). Let \(\{M_{n}\}_{n=1}^{\infty}\) be a sequence of bounded closed sets for which the point \(\{O\}=\{t_{1}=0,\ldots,\ldots,t_{n}=0\}\) is an interior point, and
\[ \bigcap_{n=1}^{\infty} M_{n}=\{O\}. \]
Then
\[ \bigcap_{n=1}^{\infty} L^{2}(M_{n}^{+}, f)=L^{0}. \]
The proof of the theorem uses Lemma 2; its central part is the proof of the following lemma:
Lemma 4. Let \(M_{1}\) and \(M_{2}\) be two bounded closed sets for which the point \(O\) is an interior point, and let \(f(x)\) satisfy condition (1) for \(t_{0}=O\) or conditions (4); then
\[ L^{2}(M_{1}^{+}, f)\cap L^{2}(M_{2}^{+}, f)=L^{2}(M_{1}^{+}\cap M_{2}^{+}, f). \]
Corollary. If the conditions of the last lemma are satisfied, then there exists no closed bounded set \(M\) such that
\[ L^{2}(M^{+}, f)=L^{2}(R^{n}, f). \]
The author takes this opportunity to express deep gratitude to I. A. Ibragimov for posing the problem and for valuable advice.
Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
9 X 1969
CITED LITERATURE
\({}^{1}\) M. G. Krein, DAN, 94, No. 1 (1954).
\({}^{2}\) M. Levinson, H. P. McKean, Acta Math., 112, 1–2 (1964).
\({}^{3}\) I. M. Gel'fand, N. Ya. Vilenkin, Some Applications of Harmonic Analysis. Rigged Hilbert Spaces, Moscow, 1961.
\({}^{4}\) S. M. Nikol'skii, Approximation of Functions of Several Variables and Embedding Theorems, Moscow, 1969.
\({}^{5}\) N. I. Akhiezer, Lectures on Approximation Theory, Moscow, 1965.
\({}^{6}\) I. I. Ibragimov, V. A. Marchenko, UMN, 11, issue 2 (68) (1956).
\({}^{7}\) V. N. Tutubalin, M. I. Freidlin, Theory of Probability and Its Applications, 7, 196 (1962).