A. N. SHERSTNEV

ON PROBLEMS OF BEST APPROXIMATION IN RANDOM NORMED SPACES

(Presented by Academician A. N. Kolmogorov on 6 X 1962)

The present work is devoted to problems of best approximation in random normed spaces \((^1)\). Let us recall the corresponding definition. Let \(\mathfrak{B}\) be the set of all nonincreasing functions \(\xi(x)\), continuous from the left and defined on \(R=(-\infty,\infty)\), such that \(\xi(x)=1\) if \(x\leqslant 0\), and \(\xi(\infty)=0\). Let \(B\) be some subset of the set \(\mathfrak{B}\). For a given function \(\alpha\), defined on \(\mathfrak{B}\times\mathfrak{B}\) with values in \(\mathfrak{B}\), form the set
\[ B[\alpha]=\bigcup_{n=0}^{\infty}B_n[\alpha], \]
where
\[ B_0[\alpha]=B,\qquad B_n[\alpha]=B_{n-1}[\alpha]\cup \alpha\bigl(B_{n-1}[\alpha]\times B_{n-1}[\alpha]\bigr). \]
In the set \(\mathfrak{B}\) introduce an order relation, putting \(\xi\leqslant\eta\) if \(\xi(x)\leqslant\eta(x)\) for every \(x\in R\). We shall write \(\xi<\eta\) if \(\xi\leqslant\eta\) and there exists an \(x\in R\) such that \(\xi(x)<\eta(x)\). The function \(\alpha\) will be called a \(B\)-function if for all \(\xi,\eta,\zeta\in B[\alpha]\): 1) \(\alpha(\xi,\eta)=\alpha(\eta,\xi)\); 2) \(\alpha(\Delta,\xi)=\xi\), where \(\Delta\) is the function from \(\mathfrak{B}\) such that \(\Delta(x)=0\) for \(x>0\); 3) \(\alpha(\xi,\eta)\geqslant \alpha(\xi_1,\eta_1)\), if \(\xi\geqslant\xi_1,\eta\geqslant\eta_1\); 4) \(\alpha(\alpha(\xi,\eta),\zeta)=\alpha(\xi,\alpha(\eta,\zeta))\); 5)
\[ \alpha(\xi,\eta\mid x)\leqslant \inf_{t\in[0,1]}\min\{\xi(tx)+\eta((1-t)x),1\} \]
(here \(\alpha(\xi,\eta\mid x)\) is the value of the function \(\alpha(\xi,\eta)\) at the point \(x\in R\)).

Definition. A random normed space (r.n.s.) is a triple \((\Omega,f,\mu)\), where \(\Omega\) is some linear space over the field \(\Lambda\) of complex or real numbers; \(f\) is a mapping of \(\Omega\) into \(\mathfrak{B}\):
\[ \varphi\in\Omega \to f(\varphi)=\|\varphi\|=\|\varphi;\cdot\|\in\mathfrak{B}; \]
\(\mu\) is some \(f(\Omega)\)-function. At the same time the following axioms are satisfied:

I. \(\|\varphi\|=\Delta\) if and only if \(\varphi=\theta\) (\(\theta\) is the zero element of \(\Omega\)).

II. \(\|a\varphi;x\|=\left\|\varphi;\dfrac{x}{|a|}\right\|\) for every \(x\in R\) and every \(a\in\Lambda\).

III. \(\|\varphi+\psi\|\leqslant \mu(\|\varphi\|,\|\psi\|)\), \(\varphi\in\Omega;\ \psi\in\Omega\).

We shall call the function \(f\) a random norm. In what follows we shall speak of an r.n.s. \((\Omega,f)\) or \(\Omega\), allowing such abbreviations when the remaining elements of the triple are not important.

Let \((\varphi(t),\,t\in[0,\infty))\) be some process with continuous trajectories, the observation of which ceases at a random time \(\tau\). Let \(\varphi_0(t)\) be some realization of this process. It is required to approximate the function \(\varphi_0(t)\) by means of a linear combination of functions from a given system \(\{\varphi_1(t),\ldots,\varphi_n(t)\}\) from the space \(C_{[0,\infty)}\) of continuous functions on the half-line. The meaning of approximation may be understood in different ways. For example, one may seek constants \(c_1,\ldots,c_n\), \(c_k\in R\), from the condition that for each \(x>0\) the probability be minimal that
\[ \max_{0\leqslant t\leqslant \tau}\left|\varphi_0(t)-\sum_{k=1}^{n}c_k\varphi_k(t)\right|\geqslant x. \]
According to \((^2)\), in the space \(C_{[0,\infty)}\) one can introduce a random norm \(f_\tau\), putting
\[ \|\varphi;x\|=P\left(\max_{0\leqslant t\leqslant \tau}|\varphi(t)|\geqslant x\right). \]
Consequently, the problem just formulated is the problem of minimizing the random norm
\[ \left\|\varphi_0-\sum_{k=1}^{n}c_k\varphi_k\right\| \]
in the space \((C_{[0,\infty)},f_\tau)\). This problem can be formulated in the general case.

Let \((\Omega, f)\) be some r.n.s., \(\varphi_0 \in \Omega\), and let \(\{\varphi_1,\ldots,\varphi_n\}\) be a linearly independent system from \(\Omega\). We shall say that the polynomial
\[ \sum_{k=1}^{n} c_k^0 \varphi_k \]
realizes the \(f\)-best approximation for \(\varphi_0\), if there is no polynomial \(\sum_{k=1}^{n} c_k' \varphi_k\) for which
\[ \left\| \varphi_0-\sum_{k=1}^{n} c_k' \varphi_k \right\| < \left\| \varphi_0-\sum_{k=1}^{n} c_k^0 \varphi_k \right\|. \]
The problem of finding \(f\)-best approximations will be called the problem of \(f\)-best approximation.

We indicate one more variant of the approximation problem. We shall call the variance of the norm of an element \(\varphi\) of an r.n.s. the quantity
\[ \sigma^2(\|\varphi\|)=2\int_{0}^{\infty} z\|\varphi;z\|\,dz + \left[\int_{0}^{\infty} \|\varphi;z\|\,dz\right]^2, \]
provided the corresponding integrals exist. Let \(\varphi_0\in\Omega\), and let \(\{\varphi_1,\ldots,\varphi_n\}\subset\Omega\) be a linearly independent system. Suppose that for every \(\varphi\in\{\varphi_0,\varphi_1,\ldots,\varphi_n\}\) the integral
\[ \int_{0}^{\infty} z\|\varphi;z\|\,dz \]
exists. It is required, among all random norms of the form
\[ \left\|\varphi_0-\sum_{k=1}^{n} c_k\varphi_k\right\|, \]
to find those that minimize the quantity
\[ \sigma^2\left(\left\|\varphi_0-\sum_{k=1}^{n} c_k\varphi_k\right\|\right). \]
We shall call this problem the problem of \(\sigma^2\)-best approximation. Polynomials realizing the \(\sigma^2\)-best approximation will be called polynomials of \(\sigma^2\)-best approximation.

Theorem 1. Every polynomial of \(\sigma^2\)-best approximation is a polynomial of some \(f\)-best approximation.

Let us note the connection between the problems posed and problems of best approximation in particular spaces.

Theorem 2. Let \(N\) be a normed space with norm \(p(\cdot)\). Then the problem of best approximation with respect to the norm in this space is equivalent to the problem of \(f\)-best \((\sigma^2\)-best) approximation in the space \(N\) as an r.n.s. with random norm \(f\), given in the form
\[ \|\varphi;x\|= \begin{cases} 1, & x \leq p(\varphi),\\ 0, & x > p(\varphi). \end{cases} \]

In [1] it was shown that every countably normed space can be considered as an r.n.s. It is of interest to determine to what problem for a countably normed space the problem of best approximation in an r.n.s. reduces. Let \((\Phi,q)\) be an arbitrary countably normed space. Here
\[ q=q(\cdot)=\{|\cdot|_1,|\cdot|_2,\ldots\} \]
is a multinorm given on \(\Phi\). On the set \(\{q(\varphi)\}\) of values of \(q\) on \(\Phi\) we introduce an order relation, taking \(q(\varphi)\leq q(\psi)\) if and only if \(|\varphi|_k\leq|\psi|_k,\ k=1,2,\ldots\). Correspondingly, \(q(\varphi)<q(\psi)\) if \(q(\varphi)\leq q(\psi)\) and there exists a natural \(k\) for which \(|\varphi|_k<|\psi|_k\). Let \(\varphi_0\in\Phi\) and let \(\{\varphi_1,\ldots,\varphi_n\}\subset\Phi\) be some linearly independent system. We shall say that the polynomial
\[ \sum_{k=1}^{n} c_k^0\varphi_k \]
realizes the \(q\)-best approximation for

element \(\varphi_0\) among all polynomials of the form \(\sum_{k=1}^{n} c_k \varphi_k\), if there is no polynomial \(\sum_{k=1}^{n} c'_k \varphi_k\) for which
\[ q\left(\varphi_0-\sum_{k=1}^{n} c'_k\varphi_k\right) < q\left(\varphi_0-\sum_{k=1}^{n} c_k^0\varphi_k\right). \]
Now one may formulate the following result.

Theorem 3. Let \((\Phi,q)\) be a countably normed space and let the multinorm \(q\) be such that \(|\cdot|_1 \leq |\cdot|_2 \leq \cdots\). Let \(\tau\) be an arbitrary random variable taking nonnegative integer values, with \(P(\tau=m)>0,\ m=1,2,\ldots\). Then the problem of \(q\)-best approximation for the space \((\Phi,q)\) is equivalent to the problem of \(f\)-best approximation in the r.n.s. \((\Phi,f_\tau)\) with the random norm \(f_\tau\), given by the formula \(\|\varphi;x\|=P(|\varphi|_\tau \geq x)\).

Next we shall consider questions of existence and uniqueness for problems of best approximation in an r.n.s.

Theorem 4. Let \((\Omega,f)\) be an arbitrary r.n.s. and let \(\{\varphi_1,\ldots,\varphi_n\}\subset\Omega\) be an arbitrary linearly independent system. Then, whatever \(\varphi_0\in\Phi\) may be, there always exists some polynomial \(\sum_{k=1}^{n} c_k^0\varphi_k\) which is a polynomial of \(f\)-best approximation for the element \(\varphi_0\).

We note that the proof of the theorem is based on the axiom of choice.

Theorem 5. Let \((\Omega,f)\) be an arbitrary r.n.s., \(\varphi_0\in\Omega\), and let \(\{\varphi_1,\ldots,\varphi_n\}\subset\Omega\) be an arbitrary linearly independent system. Suppose the random norm \(f\) is such that
\[ \int_{0}^{\infty} z\,\|\varphi;z\|\,dz<\infty \]
for any \(\varphi\in\{\varphi_0,\varphi_1,\ldots,\varphi_n\}\). Then there always exists some polynomial \(\sum_{k=1}^{n} c_k^0\varphi_k\) which realizes the \(\sigma^2\)-best approximation to \(\varphi_0\) by means of the system \(\{\varphi_1,\ldots,\varphi_n\}\).

Theorems 4 and 5 solve the existence problem for problems of best approximation. Another important aspect is the problem of uniqueness. We shall dwell on the question of uniqueness for the problem of \(f\)-best approximation. We note that in solving this problem one has to “minimize” a quantity whose values are not numbers but functions (random norms). It may turn out that, as \(f\)-best approximations, there are several incomparable random norms, and if there are no additional requirements on the choice of a solution, we have no grounds for preferring any one solution. It may also happen that to each \(f\)-best approximation there corresponds only one polynomial realizing it. Conversely, the case is possible when the \(f\)-best approximation is unique, but there exist several realizations of this approximation. Thus, in the present case the concept of uniqueness has two aspects. We shall say that the problem of \(f\)-best approximation has an \(f\)-unique solution if to each \(f\)-best approximation there corresponds only one polynomial realizing this approximation. We shall say that the problem of \(f\)-best approximation has a unique solution if it has an \(f\)-unique solution and there exists only one \(f\)-best approximation. We note that the typical notion here is precisely that of \(f\)-uniqueness. Accordingly, we shall call a system \(\{\varphi_1,\ldots,\varphi_n\}\subset\Omega\) \(f\)-Chebyshev if, for any \(\varphi_0\in\Omega\), the problem of \(f\)-best approximation of \(\varphi_0\) by means of the system \(\{\varphi_1,\ldots,\varphi_n\}\) has an \(f\)-unique solution.

Theorem 6. Let $(\Omega, f, \mu_0)$ be a random normed space, where

\[ \mu_0(\xi,\eta \mid x)=\inf_{t\in[0,1]}\max\{\xi(tx),\eta((1-t)x)\}. \]

If $\Omega$ is such that $\|\varphi+\psi\|=\mu_0(\|\varphi\|,\|\psi\|)$ only when $\varphi=\lambda\psi$ $(\lambda\geqslant 0)$, then every linearly independent system in $\Omega$ is $f$-Chebyshev.

Theorem 7. In order that the system $\{\varphi_1(t),\ldots,\varphi_n(t)\}$ be $f$-Chebyshev in the space $(C_{[0,\infty)}, f_\tau, \mu_0)$, where $\tau$ is an arbitrary random variable taking nonnegative values with strictly decreasing function $\xi(x)=P(\tau\geqslant x)$, $x>0$, it is sufficient that every nontrivial polynomial

\[ \sum_{k=1}^{n} c_k \varphi_k(t) \]

have no more than $n-1$ zeros on the half-line $[0,\infty)$.

Kazan State University
named after V. I. Ulyanov-Lenin

Received
5 X 1962

REFERENCES

1. A. N. Sherstnev, DAN, 149, No. 2 (1963).