Abstract
Full Text
Mathematics
Yu. A. ROZANOV
ON LINEAR INTERPOLATION OF STATIONARY PROCESSES WITH DISCRETE TIME
(Presented by Academician A. N. Kolmogorov on 16 IV 1957)
In this note, the condition of interpolability of a stationary sequence due to A. N. Kolmogorov,
[
\int_{-\pi}^{\pi} \frac{d\lambda}{f_\lambda}=\infty
\tag{*}
]
and the results of work ({}^{(2)}) are generalized to multidimensional and infinite-dimensional processes. In § 1 a coordinate-free method is set forth for defining the very concept of a stationary process, which in § 2 leads to a more elegant formulation of the results.
§ 1. A stationary process and its spectral functional. In this paragraph we present jointly the cases of continuous time ((t) ranges over all real values) and discrete time ((t) ranges over all integer values).
In the coordinate approach, an (n)-dimensional stationary process is a collection of (n) random complex functions
[
x_1(t),\quad x_2(t),\ldots,x_n(t)
\tag{1}
]
such that
[
Mx_i(t)=m_i,\qquad Mx_i(t+\tau)\overline{x_j(t)}=B_{ij}(\tau);
\tag{2}
]
here, without loss of generality, one takes (m_i=0).
Consider (H), the collection of random complex variables (x) with (Mx=0) and (M|x|^2<\infty). If all (x\in H) that differ from one another only with probability equal to 0 are identified, then (H) becomes a Hilbert space with scalar product ((x,y)=Mx\overline{y}). To each vector (a=(a^1,a^2,\ldots,a^n)) of the affine (n)-dimensional space (A) we associate the element of the space (H)
[
x_t(a)=a^1x_1(t)+a^2x_2(t)+\cdots+a^nx_n(t).
]
It is easy to see that the scalar product
[
B_\tau(a,b)=\bigl(x_{t+\tau}(a),x_t(b)\bigr)
\tag{3}
]
does not depend on (t) for any (a,b\in A).
Obviously, the study of the system of functions (1) is equivalent to the study of the function (x_t(a)) of (t) and (a).
In the space (A) it is natural to introduce the scalar product ((a,b)=B_0(a,b)) and the norm (|a|=(a,a)^{1/2}), and to identify the elements (a) and (b) in the case (|a-b|=0). Then (A) becomes a unitary space, and the operators (x_t) turn out to be isometric.
The coordinate-free conception of stationary processes, proposed by A. N. Kolmogorov and used below, consists in the following. A stationary process ({A, x_t}) is a collection consisting of a unitary space (A) of “simultaneously observable quantities” and linear isometric operators (x_t) from (A) into (H), satisfying the condition that the scalar products (3) are independent of (t). It is natural to impose on (x_t) the continuity condition
[
\lim_{t-s\to 0}|x_t(a)-x_s(a)|=0
]
for all (a\in A).
Let (H_x) be the linear closure of the elements (x_t(a)) in (H). The equalities (U^\tau x_t(a)=x_{t+\tau}(a)) define on (H_x) ({}^{(1)}) unitary operators (U^\tau),
[
U^tU^\tau=U^{t+\tau},\quad |U^{t+\tau}x-U^\tau x|\to 0 \quad \text{as } t\to 0
]
and for every (x\in H).
For (U^\tau) the representation
[
U^\tau=\int e^{i\lambda\tau}\,dE_\lambda,
\tag{4}
]
holds, where (E_\lambda) is a spectral family (here and below the integration is over (-\pi\le \lambda\le \pi) in the discrete case, and over (-\infty<\lambda<+\infty) in the continuous case).
We shall call the bilinear functional
[
F_\lambda(a,b)=(E_\lambda x_0(a),\,x_0(b)).
\tag{5}
]
the spectral functional of the process. Obviously,
[
B_\tau(a,b)=\int e^{i\lambda\tau}\,dF_\lambda(a,b).
]
For fixed (a) and (b), the spectral functional (F_\lambda(a,b)) almost everywhere has a derivative
[
f_\lambda(a,b)=F'_\lambda(a,b),
\tag{6}
]
which we shall call the spectral density. Let (M_\lambda) be the set of (a\in A) for which (f_\lambda(a,a)) exists; every (a\in A) belongs to (M_\lambda) for almost all (\lambda); (f_\lambda(a,b)) is a bilinear functional on (M_\lambda), which, in the case of separability of (A), is everywhere dense in (A) for almost all (\lambda).
§ 2. On linear interpolation. We shall consider the discrete-time case. Let (T) be a finite set of integers, and let (H(T)) and (\hat H(T)) be, respectively, the linear closures of (x_t(a)) for (t\in T,\ a\in A), and of (x_t(a)) for (t\bar{\in}T,\ a\in A). The question is: when can (x_t(a)), (t\in T), be linearly interpolated, knowing (x_t(a)) for (t\bar{\in}T), i.e. when is (H(T)\subseteq \hat H(T))?
With respect to the process ({A,x_t}) we shall assume that (A) is separable and (f_\lambda) is bounded for almost all (\lambda), i.e. (|f_\lambda(a,b)|\le C_\lambda|a|\,|b|), (a,b\in M_\lambda).
Then (f_\lambda(a,b)=(f_\lambda a,b)), where (f_\lambda) is some positive bounded linear operator on (A).
Let (\theta(\lambda)) be the subspace of elements (a\in A) for which (f_\lambda a=0); (A(\lambda)=A\ominus\theta(\lambda)); on (A(\lambda)) there exists (f_\lambda^{-1}).
Consider the Hilbert space (\mathfrak B(T)) of functions (b_\lambda), defined almost everywhere on ([-\pi,\pi]) and such that:
[
1^\circ.\quad b_\lambda=\sum_{t\in T} e^{i\lambda t}b_t,\quad b_t\in A.
]
[
2^\circ.\quad b_\lambda\in f_\lambda A.
]
[
3^\circ.\quad \int_{-\pi}^{\pi}(f_\lambda^{-1}b_\lambda,b_\lambda)\,d\lambda<\infty.
]
The scalar product in (\mathfrak B(T)) is
[
\int_{-\pi}^{\pi} (f_\lambda^{-1} b_\lambda,\ b'_\lambda)\,d\lambda .
]
Main theorem. (\Delta(T)=H(T)\ominus \widehat H(T)) is isometric to (\mathfrak B(T)).
Let, for example, (A) be an (n)-dimensional unitary space; (a_1,a_2,\ldots,a_n) an orthonormal basis in it; (x_i(t)=x_t(a_i)). Then (f_\lambda) is given by the matrix (|f_{ij}(\lambda)|) of ordinary spectral densities:
[
f_{ij}(\lambda)=\frac{d}{d\lambda}(E_\lambda x_i(0),\,x_j(0)).
]
The function (b_\lambda) is the vector ((b_\lambda^1,\ldots,b_\lambda^n)), where (b_\lambda^k=\sum_{t\in T}e^{i\lambda t}\alpha_t) is a trigonometric polynomial; conditions (2^\circ) and (3^\circ) for (b_\lambda) look quite simple; for example, if (f_\lambda=|f_{ij}(\lambda)|) is non-singular for almost all (\lambda), then (2^\circ) is satisfied, and (3^\circ) is
[
\int_{-\pi}^{\pi}\sum_{i,j=1}^{n}p_{ij}(\lambda)b_\lambda^i\overline{b_\lambda^j}\,d\lambda<\infty,
\quad \text{where } |p_{ij}(\lambda)|=f_\lambda^{-1}.
]
For (n=1) one easily obtains the result of A. M. Yaglom ((^2)): (\Delta(T)\ne0) if and only if there exists a set of numbers (\alpha_t), (\sum_{t\in T}|\alpha_t|^2\ne0), and
[
\int_{-\pi}^{\pi}
\frac{\left|\sum_{t\in T} e^{i\lambda t}\alpha_t\right|^2}{f_\lambda}\,d\lambda<\infty .
\tag{7}
]
Let us further note that if the number of elements of (T) is (s), then the dimension (\Delta(T)\le ns); in order that the equality sign hold, it is necessary and sufficient that
[
\int_{-\pi}^{\pi} \operatorname{sp} f_\lambda^{-1}\,d\lambda<\infty,
\qquad
\operatorname{sp} f_\lambda^{-1}=\sum_{i=1}^{n}p_{ii}(\lambda).
\tag{8}
]
For (n=1,\ s=1) one obtains the result of A. N. Kolmogorov (*), contained in ((^1)).
For the proof of the main theorem we shall need the following lemma.
Lemma. Let (F_\lambda(a,b)), for arbitrary (a,b\in A), be an absolutely continuous function. Then (H_x) is isometric to the space (\mathfrak A) of functions (a_\lambda\in A(\lambda)), defined almost everywhere on ([-\pi,\pi]), ((a_\lambda,a)) measurable* for every (a\in A) and
[
\int_{-\pi}^{\pi}(a_\lambda,a_\lambda)\,d\lambda<\infty;
]
the scalar product in (\mathfrak A) is
[
\int_{-\pi}^{\pi}(a_\lambda,a'_\lambda)\,d\lambda;
]
moreover all (a_\lambda) equal to one another almost everywhere are identified.
The assertion of the lemma follows from the fact that the (\mathfrak A)-complete space, and the linear combinations of the functions (e^{i\lambda t}f_\lambda^{1/2}a) ((f_\lambda^{1/2}) is the positive square root of (f_\lambda), (a\in A), (-\infty<t<+\infty)) are everywhere dense in (\mathfrak A). Namely,
[
x_t(a)\leftrightarrow e^{i\lambda t}f_\lambda^{1/2}a
\tag{9}
]
generates an isometric correspondence between (H_x) and (\mathfrak A).
Proof of the main theorem. It is not difficult to show that, without loss of generality, one may assume the spectral functional (F_\lambda(a,b)) to be absolutely continuous for arbitrary (a) and (b). Then, by the lemma, (H_x) is isometric to (\mathfrak A). (\Delta(T)) is a subspace of (H_x), and it corresponds to some subspace (\mathfrak A(T)\subset \mathfrak A).
* Concerning the definition of measurability of (a_\lambda), see, for example, ((^3)), p. 54.
Take (\delta \in \Delta(T)) and the corresponding (a_\lambda \in \mathfrak A); ((\delta, x_t(a))=0) for (t \notin T), (a \in A). We have (see (9))
[
(\delta, x_t(a))=
\int_{-\pi}^{\pi} (a_\lambda, e^{i\lambda t} f_\lambda^{1/2} a)\,d\lambda
=
\int_{-\pi}^{\pi} e^{-i\lambda t}(f_\lambda^{1/2}a_\lambda,a)\,d\lambda=0,
]
[
(f_\lambda^{1/2}a_\lambda,a)=\sum_{t\in T} e^{i\lambda t}\alpha_t(a),
]
where
[
\alpha_t(a)=\frac{1}{2\pi}\int_{-\pi}^{\pi} e^{i\lambda t}(f_\lambda^{1/2}a_\lambda,a)\,d\lambda
]
is a bounded linear functional, since
[
|\alpha_t(a)|\le
\frac{1}{2\pi}\int_{-\pi}^{\pi} (a_\lambda,a_\lambda)^{1/2}(f_\lambda a,a)^{1/2}\,d\lambda
\le
]
[
\le
\frac{1}{2\pi}
\left(\int_{-\pi}^{\pi} (a_\lambda,a_\lambda)\,d\lambda\right)^{1/2}
\left(\int_{-\pi}^{\pi} (f_\lambda a,a)\,d\lambda\right)^{1/2}
=
\frac{1}{2\pi}
\left(\int_{-\pi}^{\pi} (a_\lambda,a_\lambda)\,d\lambda\right)^{1/2}|a|.
]
Hence
[
\alpha_t(a)=(b_t,a),\qquad b_t\in A,
]
[
(f_\lambda^{1/2}a_\lambda-\sum_{t\in T} e^{i\lambda t}b_t,a)=0,\qquad
f_\lambda^{1/2}a_\lambda=\sum_{t\in T} e^{i\lambda t}b_t.
]
Put (f_\lambda^{1/2}a_\lambda=b_\lambda). We have
[
\int_{-\pi}^{\pi} (a_\lambda,a_\lambda)\,d\lambda
=
\int_{-\pi}^{\pi} (f_\lambda^{-1}b_\lambda,b_\lambda)\,d\lambda<\infty.
]
Conversely, if
[
b_\lambda\in f_\lambda A,\qquad
\int_{-\pi}^{\pi} (f_\lambda^{-1}b_\lambda,b_\lambda)\,d\lambda<\infty,\qquad
b_\lambda=\sum_{t\in T} e^{-i\lambda T}b_t,
]
then
[
a_\lambda=f_\lambda^{-1/2}b_\lambda\in \mathfrak A(T)
]
and determines some element (\delta\in\Delta(T)).
Thus, (\Delta(T)) is isometric to (\mathfrak B(T)), as was required to prove.
Moscow State University
named after M. V. Lomonosov
Received
13 IV 1957
REFERENCES
- A. N. Kolmogorov, Bull. Moscow State Univ., No. 6, 1 (1941).
- A. M. Yaglom, Uspekhi Mat. Nauk, 4, No. 4, 171 (1949).
- E. Hille, Functional Analysis and Semigroups, IL, 1951.