Full Text
Yu. A. ROZANOV
ON THE INTERPOLATION OF STATIONARY PROCESSES WITH DISCRETE TIME
(Presented by Academician A. N. Kolmogorov, 25 IX 1959)
Let \(x(t)\) be a stationary, in the broad sense, random process (the parameter \(t\) assumes integer values), whose realization is known at all times \(t\), except for a finite number \(\{t_1, t_2, \ldots, t_s\}=T\); the note discusses the problem of linear interpolation of the quantities \(x(t)\), \(t\in T\), by means of the known part of the realization of the process \(x(t)\), \(t\notin T\).
As usual, we identify all random variables that differ from one another with probability equal to zero; then the totality of all random variables \(h\) with \(M|h|^2<\infty\) forms a Hilbert space with scalar product \((h_1,h_2)=Mh_1\overline{h_2}\). The quantity of the best linear interpolation of the unknown value \(x(t)\), \(t\in T\), is the projection \(\hat{x}_T(t)\) of the quantity \(x(t)\) onto the linear closure \(\hat{H}(T)\) of the quantities \(x(s)\), \(s\notin T\), and the least error \(\sigma_T(t)\) is equal to the “length” \(\|h_T(t)\|\) of the perpendicular \(h_T(t)\), dropped from \(x(t)\) to \(\hat{H}(T)\).
We shall say that the process \(x(t)\) is linearly interpolable if \(\sigma_T(t)\equiv 0\) for all \(T,t\). As was shown in \((^{2,3})\), for the interpolability of a process \(x(t)\) with spectral density \(f(\lambda)\), it is necessary and sufficient that
\[ \int_{-\pi}^{\pi}\frac{|P(e^{i\lambda})|^2}{f(\lambda)}\,d\lambda=\infty \tag{1} \]
for any trigonometric polynomial \(P(e^{i\lambda})\).
If condition (1) is not satisfied, then, as is not difficult to understand, there exists a minimal polynomial \(P_0(e^{i\lambda})\), having the form
\[ P_0(e^{i\lambda})=\prod_{k=1}^{n}(e^{i\lambda}-e^{i\alpha_k})^{m_k},\qquad \alpha_k\in[-\pi,\pi], \tag{2} \]
such that
\[ \int_{-\pi}^{\pi}\frac{|P_0(e^{i\lambda})|^2}{f(\lambda)}\,d\lambda<\infty \tag{3} \]
and every polynomial \(P(e^{i\lambda})\) for which
\[ \int_{-\pi}^{\pi}\frac{|P(e^{i\lambda})|^2}{f(\lambda)}\,d\lambda<\infty \]
is divisible by \(P_0(e^{i\lambda})\). We shall call the numbers \(\alpha_k\) the zeros of the spectral density \(f(\lambda)\) (in the case of a smooth spectral density \(f(\lambda)\), the numbers \(\alpha_k\) are its genuine zeros of multiplicity \(2m_k\)).
Denote by \(H(T)\) the linear closure of the quantities \(x(t)\), \(t\in T\); set
\[ \Delta(T)=H(T)\ominus \hat H(T). \tag{4} \]
As was shown in \((^3)\), the space \(\Delta(T)\) is isometric to the space \(\mathfrak B(T)\) of trigonometric polynomials \(b(\lambda)=\sum_{t\in T} b_t e^{i\lambda t}\) for which
\[ \int_{-\pi}^{\pi}\frac{|b(\lambda)|^2}{f(\lambda)}\,d\lambda<\infty, \]
with scalar product
\[ (b_1,b_2)=\int_{-\pi}^{\pi} b_1(\lambda)\overline{b_2(\lambda)}\frac{d\lambda}{f(\lambda)}. \]
For simplicity, let the set \(T\) be an “interval” \(\{t_0,t_0+1,\ldots,t_0+s-1\}\) of length \(s\).
From what was set forth above it follows:
Theorem 1. All unknown values \(x(t)\), \(t\in T\), are interpolated without error \((|\Delta(T)|=0)\) if and only if the length \(s\) of the “interval” \(T\) does not exceed the number \(m=\sum_{k=1}^{n} m_k\) of zeros of the spectral density \(f(\lambda)\).
In the case \(s>m\), the dimension of the “error space” \(\Delta(T)\) is equal to \(s-m\).
Next, let
\[ x(t)=\int_{-\pi}^{\pi} e^{i\lambda t}\Phi(d\lambda) \tag{5} \]
be the spectral representation of the process \(x(t)\). The best interpolation quantity \(\hat x_T(t)\) can be represented in the form
\[ \hat x_T(t)=\int_{-\pi}^{\pi} \hat\varphi_T(\lambda,t)\Phi(d\lambda), \tag{6} \]
where the function \(\hat\varphi_T(\lambda,t)\) is such that
\[ \int_{-\varphi}^{\pi} |\hat\varphi_T(\lambda,t)|^2 f(\lambda)\,d\lambda<\infty, \tag{7} \]
and, moreover, \(\hat\varphi_T(\lambda,t)\) can be approximated arbitrarily accurately in mean square with weight \(f(\lambda)\) by trigonometric polynomials \(\sum_{t_k\notin T} c_k e^{i\lambda t_k}\) \((^4)\); finding \(\hat x_T(t)\) essentially amounts to finding the function \(\hat\varphi_T(\lambda,t)\).
Consider the function
\[ A_T(\lambda,t)=e^{i\lambda t}-\hat\varphi_T(\lambda,t). \tag{8} \]
Theorem 2. The function \(A_T(\lambda,t)\) has the form
\[ A_T(\lambda,t)=\frac{P_0(e^{i\lambda})}{f(\lambda)} \sum_{k=t_0-m}^{t_0-s+1} c_k e^{i\lambda k}, \tag{9} \]
where \(s\) is the length of the “interval” \(T\); \(m\) is the degree of the minimal polynomial \(P_0(e^{i\lambda})\); the coefficients \(c_k\) can be found from the following system of linear equations:
\[ \sum_{k=t_0-m}^{t_0-s+1} A_{lk}c_k=B_l,\qquad t_0-m\le l\le t_0-s+1, \tag{10} \]
\[ A_{lk}=\int_{-\pi}^{\pi} e^{i\lambda(k-l)}\frac{|P_0(e^{i\lambda})|^2}{f(\lambda)}\,d\lambda, \]
\[ B_l=\int_{-\pi}^{\pi} e^{i\lambda(t-l)}\overline{P}_0(e^{i\lambda})\,d\lambda. \]
For example, in the case when only one value \(x(t)\) is unknown,
\[ A_T(\lambda,t)=2\pi\frac{e^{i\lambda t}}{f(\lambda)} \left(\int_{-\pi}^{\pi}\frac{d\mu}{f(\mu)}\right)^{-1} \tag{11} \]
and the error of the best interpolation is
\[ \sigma_T(t)=2\pi\left(\int_{-\pi}^{\pi}\frac{d\mu}{f(\mu)}\right)^{-1/2}. \tag{12} \]
Interpolability of the stationary process \(x(t)\) means that
\[ \hat S=\bigcap_T \hat H(T)=H, \tag{13} \]
where \(H\) is the linear closure of all quantities \(x(t)\), \(-\infty<t<\infty\).
Let us note that if the spectral function \(F(\lambda)\) of the process \(x(t)\) is not absolutely continuous, \(F(\lambda)=\int_{-\pi}^{\lambda} f(\lambda)+\Sigma(\lambda)\), then the totality \(\hat S=\bigcap_T \hat H(T)\) contains quantities \(s\) of the form
\[ s=\int_{\Delta_0} s(\lambda)\Phi(d\lambda), \tag{14} \]
where \(\Delta_0\) is a set of Lebesgue measure zero on which the measure \(d\Sigma(\lambda)\) is concentrated, and \(\int_{\Delta_0}|s(\lambda)|^2\,d\Sigma(\lambda)<\infty\) (see, for example, \((^4)\)).
Theorem 3. If the stationary process \(x(t)\) is not interpolable, then the totality \(\hat S\) consists of quantities of the form (14). In particular, if the spectral function \(F(\lambda)\) of the process \(x(t)\) is absolutely continuous, then \(\hat S=0\).
Proof. Without loss of generality, one may assume that the spectral function \(F(\lambda)\) of the process \(x(t)\) is absolutely continuous (see, for example, \((^4)\)). The space \(H\) in this case is isometric to the space \(\mathfrak A\) of functions \(a(\lambda)\),
\[ \int_{-\pi}^{\pi}|a(\lambda)|^2 f(\lambda)\,d\lambda<\infty. \]
The quantities \(\delta\in\Delta(T)\) will correspond to functions \(\delta(\lambda)=b(\lambda)/f(\lambda)\), where
\[ b(\lambda)=P_0(e^{i\lambda})\sum_k e^{i\lambda k}\in\mathfrak B(T). \]
For any function \(a(\lambda)\in\mathfrak A\) we have
\[ \int_{-\pi}^{\pi}|a(\lambda)-\delta(\lambda)|^2 f(\lambda)\,d\lambda = \int_{-\pi}^{\pi}|a(\lambda)f(\lambda)-b(\lambda)|^2\frac{d\lambda}{f(\lambda)} = \]
\[ =\int_{-\pi}^{\pi}\left|\frac{a(\lambda)f(\lambda)}{P_0(e^{i\lambda})}-\sum_k c_k e^{i\lambda k}\right|^2 \frac{|P_0(e^{i\lambda})|^2}{f(\lambda)}\,d\lambda. \tag{15} \]
Since the function \(g(\lambda)=|P_0(e^{i\lambda})|^2 f(\lambda)\) is integrable, and the function \(\psi(\lambda)=a(\lambda)f(\lambda)/P_0(e^{i\lambda})\) is square-integrable with weight \(g(\lambda)\), \(\psi(\lambda)\) can be approximated arbitrarily accurately in the mean-square sense with weight \(g(\lambda)\) by trigonometric polynomials \(\sum_k c_k e^{i\lambda k}\). Therefore it follows from relation (15) that
\[ \inf_{\delta}\int_{-\pi}^{\pi}|a(\lambda)-\delta(\lambda)|^2 f(\lambda)\,d\lambda=0, \tag{16} \]
where the infimum is taken over all \(\delta(\lambda)\) corresponding to quantities \(\delta\in\Delta(T)\) for all possible \(T\). Equality (16) means that \(H=\bigcup_T\Delta(T)\), or \(\bigcap_T \hat H(T)=0\), as was required to prove.
Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
1 VIII 1959
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