MATHEMATICS

S. I. ZUKHOVITSKII and G. I. ESKIN

ON CHEBYSHEV APPROXIMATION IN A HILBERT RING

(Presented by Academician N. N. Bogolyubov, 6 IX 1957)

1. Let (\varphi(q)) be a function continuous on the compact set (Q), with values in an arbitrary Hilbert ring (H) (see, for example, ((^1))). The problem of Chebyshev approximation of a function (f(q)), continuous on (Q) with values in (H), by means of functions (a\varphi(q)), where (a\in H), consists in finding such an (a^{(0)}\in H) that the function (a^{(0)}\varphi(q)) deviates least on (Q) from (f(q)), i.e., such that

[
\max_{q\in Q}\left|a^{(0)}\varphi(q)-f(q)\right|
=
\inf_{a\in H}\max_{q\in Q}\left|a\varphi(q)-f(q)\right|.
]

The case of a finite-dimensional ring was considered in ((^2)).

In particular, if (H) is the ring of all complex matrices (a={a_{\alpha\beta}}) for which

[
\sum_{\alpha,\beta}|a_{\alpha\beta}|^2<\infty
\quad\text{and}\quad
|a|^2=\sum_{\alpha,\beta}|a_{\alpha\beta}|^2,
]

then the preceding problem is the problem of Chebyshev approximation of a matrix-function (f(q)={f_{\alpha\beta}(q)}), continuous on (Q), by means of the product of matrices

[
a\varphi(q)={a_{\alpha\beta}}\cdot{\varphi_{\gamma\beta}(q)},
]

where (\varphi(q)={\varphi_{\gamma\beta}(q)}) is a fixed matrix-function continuous on (Q), and (a={a_{\alpha\beta}}) is an arbitrary numerical matrix from (H).

The classical problem of Chebyshev approximation of a numerical function (f_0(q)), continuous on (Q), by a polynomial

[
\sum_{k=1}^{n}\xi_k\varphi_k(q),
]

where (\varphi_1(q),\ldots,\varphi_n(q)) are fixed numerical functions continuous on (Q), is covered by the preceding problem when the matrix (\varphi(q)) is a matrix of the form

[
\varphi(q)=
\begin{pmatrix}
\varphi_1(q) & 0 & \ldots & 0\
\varphi_2(q) & 0 & \ldots & 0\
\cdot & \cdot & \cdot & \cdot\
\varphi_n(q) & 0 & \ldots & 0
\end{pmatrix},
]

and the matrix (f(q)) is a matrix of the form

[
f(q)=
\begin{pmatrix}
f_0(q) & 0 & \ldots & 0\
0 & 0 & \ldots & 0\
\cdot & \cdot & \cdot & \cdot\
0 & 0 & \ldots & 0
\end{pmatrix},
]

since

[
\inf_{a\in H}\max_{q\in Q}\left|a\varphi(q)-f(q)\right|
=
\inf_{\xi_k}\max_{q\in Q}
\left|
\sum_{k=1}^{n}\xi_k\varphi_k(q)-f(q)
\right|.
]

We note that the infimum on the left-hand side is attained on a matrix all of whose rows, beginning with the second, are zero.

The problem of Chebyshev approximation in an arbitrary Hilbert ring (H), in turn, is a special case of the following general problem considered in ((^{3,4})).

An operator-function (F(q)) is given, which for each (q\in Q) is a linear operator acting in the Hilbert space (H), and for each fixed (x\in H) the function (F(q)x) is continuous on (Q). It is required to find a vector (x_0\in H) such that

[
\max_{q\in Q}|F(q)x_0-f(q)|
=
\inf_{x\in H}\max_{q\in H}|F(q)x-f(q)|.
]

In the case under consideration, (F(q)) for each (q\in Q) is the operator of multiplication on the right by (\varphi(q)): (F(q)a=a\varphi(q)).

1. As follows from ((^3)) (see ((^4))), in order that for every function (f(q)), continuous on (Q) with values in (H), there exist a least-deviating function, it is necessary and sufficient that the condition

[
\max_{q\in Q}|a\varphi(q)|\geq m|a| \quad \text{for all } a\in S,
\tag{a}
]

be satisfied, where (S) is the orthogonal complement in (H) to the subspace (T) of vectors (a) for which (a\varphi(q)\equiv \theta) on (\theta), and (m) is a positive constant.

Obviously, (T) is a closed left ideal in (H), and (S), as the orthogonal complement to (T), is also a closed left ideal.

Choose in (S) and (T), respectively, a maximal system of mutually orthogonal irreducible Hermitian idempotents. The union of these two systems gives a maximal system of mutually orthogonal irreducible Hermitian idempotents ({p_\alpha}) for all of (H). Let (\alpha') denote the indices of those idempotents of the system ({p_\alpha}) which are contained in (S). Then, obviously,

[
\varphi(q)=\sum_{\alpha'} p_\alpha\varphi(q).
]

The orthogonal sum

[
\sum_{\alpha'}^{\oplus} p_\alpha H
]

is the least closed right ideal containing all values (\varphi(q)), when (q) ranges over the compact set (Q).

The following theorem establishes the structure which the function (\varphi(q)) must have in order that the preceding existence condition (a) be satisfied.

Theorem 1. In order that for every function (f(q)), continuous on (Q) with values in (H), there exist a function (a^{(0)}\varphi(q)) least-deviating from it, it is necessary and sufficient that the least closed right ideal containing the set of all values of the function (\varphi(q)) be the orthogonal sum of only a finite number of certain minimal right ideals (p_1H,\ldots,p_kH) of the ring (H):

[
\varphi(q)\in p_1H\oplus\cdots\oplus p_kH
\quad \text{for all } q\in Q,
]

or, what is the same,

[
S=H\ominus T=Hp_1\oplus\cdots\oplus Hp_k,
]

where (p_1,\ldots,p_k) are irreducible Hermitian idempotents.

1. The following theorem gives a necessary and sufficient condition for uniqueness of the function of least deviation. It need only be noted that, since (a\varphi(q)\equiv\theta) when (a\in T), the question of uniqueness of the function of least deviation reduces to the question of uniqueness of the vector (a^{(0)}\in S) for which (a^{(0)}\varphi(q)) is the function of least deviation.

Theorem 2. Suppose the function (\varphi(q)) satisfies the condition of Theorem 1. Then, in order that for every function (f(q)), continuous on (Q) with values in (H), there exist a unique function least-deviating from it, it is necessary and sufficient that, for every (a\in S,\ a\ne\theta), the equation (a\varphi(q)=\theta) have no roots on (Q).

Remark. The characteristic property of the function (a^{(0)}\varphi(q)) least deviating from (f(q)), as is easily seen from (3), consists in the fact that, for every (b\in H), the inequality

[
\min_{q\in M(a^{(0)},\, f)}
\operatorname{Re}\bigl(a^{(0)}\varphi(q)-f(q),\, b\varphi(p)\bigr)\leq 0
]

must hold, where (M(a^{(0)}, f)) is the set of those points of (Q) at which

[
\max_{q\in Q}|a^{(0)}\varphi(q)-f(q)|
]

is attained.

1. Let us illustrate the preceding two theorems by an example in which (H) is a simple Hilbert ring.

Theorem 1 asserts in this case that, in an arbitrary orthonormal basis ({p_{\alpha\beta}}), constructed from the above-mentioned maximal system of idempotents ({p_\alpha}), a function (\varphi(q)) satisfying condition (a) must have the form

[
\varphi(q)=\sum_{\beta}\bigl(\varphi_{1\beta}(q)p_{1\beta}+\cdots+\varphi_{k\beta}(q)p_{k\beta}\bigr),
]

where (\varphi_{1\beta}(q),\ldots,\varphi_{k\beta}(q)) are numerical continuous functions on (Q), while each vector (a\in S) has the form

[
a=\sum_{\alpha}\bigl(a_{\alpha 1}p_{\alpha 1}+\cdots+a_{\alpha k}p_{\alpha k}\bigr).
]

By virtue of the isomorphism of (H) to the ring of all complex matrices (a={a_{\alpha\beta}}) for which

[
\sum_{\alpha,\beta}|a_{\alpha\beta}|^2<\infty,
]

the function (\varphi(q)) corresponds to a matrix in which only the first (k) rows are nonzero, while an element (a\in S) corresponds to a matrix in which the elements different from zero may occur only in the first (k) columns.

Theorem 2, in turn, asserts that for uniqueness of the function of least deviation (for every approximated function continuous on (Q) with values in (H)) it is necessary and sufficient that, for every (q\in Q), the rank of the matrix corresponding to the function (\varphi(q)) be equal to (k).

Lutsk Pedagogical Institute
named after Lesya Ukrainka

Received
5 IX 1957

REFERENCES

1. M. A. Naimark, Normed Rings, Moscow, 1956.
2. S. I. Zukhovitskii, M. G. Krein, Uspekhi Mat. Nauk, 5, no. 1 (35), 217 (1950).
3. S. I. Zukhovitskii, Mat. Sbornik, 37 (79), no. 1, 3 (1955).
4. S. I. Zukhovitskii, G. I. Eskin, Dokl. Akad. Nauk SSSR, 116, no. 5 (1957).