Abstract
Full Text
MATHEMATICS
S. I. Zukhovitskii and G. I. Eskin
THE PROBLEM OF CHEBYSHEV APPROXIMATION IN A COMMUTATIVE HILBERT RING
(Presented by Academician N. N. Bogolyubov, 4 XII 1957)
- Let \(\varphi_1(q),\ldots,\varphi_n(q)\) be functions continuous on some compact set \(Q\), with values in an infinite-dimensional commutative Hilbert ring \(H\). Any element \(a \in H\) has the form \(a=\sum_\alpha a_\alpha e_\alpha\), where the coefficients \(a_\alpha\) are complex numbers, and \(\{e_\alpha\}\) is a basis of mutually orthogonal irreducible Hermitian idempotents \((^1)\). The functions \(\varphi_k(q)\) \((k=1,2,\ldots,n)\) have the form
\[ \varphi_k(q)=\sum_\alpha \varphi_{\alpha k}(q)e_\alpha, \]
where \(\varphi_{\alpha k}(q)\) are complex-valued functions continuous on \(Q\).
We shall approximate, in the best possible way, a function \(f(q)\), continuous on \(Q\) and with values in \(H\), by means of polynomials of the form \(\sum_{k=1}^n a_k\varphi_k(q)\), where \(a_1,\ldots,a_n\) are elements of \(H\), i.e., we shall seek among them such a polynomial \(\sum_{k=1}^n a_k^{(0)}\varphi_k(q)\) for which
\[ \max_{q\in Q}\left\|\sum_{k=1}^n a_k^{(0)}\varphi_k(q)-f(q)\right\| = \inf_{a_k\in H}\max_{q\in Q}\left\|\sum_{k=1}^n a_k\varphi_k(q)-f(q)\right\|. \]
The case of a finite-dimensional ring was considered in \((^2)\).
- Denote by \(T\) the set of all complexes \(a=(a_1,\ldots,a_n)\), where \(a_k\in H\) \((k=1,2,\ldots,n)\), for which
\[ \sum_{k=1}^n a_k\varphi_k(q)\equiv \theta \]
on \(Q\).
It is clear that \(T\) is a subspace of the Hilbert space \(H^n\) with norm
\[ \|a\|_{H^n}=\left(\sum_{k=1}^n \|a_k\|^2\right)^{1/2}. \]
Let \(S\) denote the orthogonal complement of \(T\) in \(H^n\).
We note that approximation by means of the polynomial \(\sum_{k=1}^n a_k\varphi_k(q)\) may be regarded as approximation by means of the operator-function \(A(q)\), which for each \(q\in Q\) is a linear bounded operator acting from \(H^n\) into \(H\) according to the formula
\[ A(q)a=\sum_{k=1}^n a_k\varphi_k(q), \]
and from \((^{3,4})\) we obtain that, for the existence for every continuous function of a polynomial
for least deviation it is necessary and sufficient that the condition
\[ \max_{q\in Q}\left\|\sum_{k=1}^{n} a_k\varphi_k(q)\right\|\geq m\left(\sum_{k=1}^{n}\|a_k\|^2\right)^{1/2} \quad \text{for all } (a_1,\ldots,a_n)\in S, \tag{a} \]
be satisfied, where \(m>0\) is a constant. From this condition there follows the following theorem.
Theorem 1. In order that for every function \(f(q)\), continuous on \(Q\) and with values in \(H\), there exist a polynomial of least deviation, it is necessary and sufficient that the subspace \(S\) be finite-dimensional, or, what is the same, that each of the functions
\[ \varphi_k(q)=\sum_{\alpha}\varphi_{\alpha k}(q)e_\alpha \quad (k=1,2,\ldots,n) \]
have only a finite number of coefficients \(\varphi_{\alpha k}(q)\) that are not identically equal to zero on \(Q\).
- Denote by \(L\) the set of indices \(\alpha\) such that \(\varphi_{\alpha k}(q)\not\equiv 0\) on \(Q\) for at least one \(k=1,2,\ldots,n\). The number \(l\) of such indices, by the preceding theorem, is finite, and \(\dim S\leq nl\).
Theorem 2. If the dimension of the subspace \(S\) is a multiple of \(l\): \(\dim S=tl\), then, in order that for every continuous function \(f(q)\) there exist a unique polynomial of least deviation from it
\[ \sum_{k=1}^{n} a_k^{(0)}\varphi_k(q), \]
where \((a_1^{(0)},\ldots,a_n^{(0)})\in S\), it is necessary and sufficient that every polynomial
\[ \sum_{k=1}^{n} a_k\varphi_k(q), \]
for which \(\sum_{k=1}^{n}\|a_k\|>0\) and \((a_1,\ldots,a_n)\in S\), vanish at no more than \(t-1\) points of the compact set \(Q\).
The proof of this theorem can be obtained by considering the polynomial
\[ \sum_{k=1}^{n} a_k\varphi_k(q)\quad ((a_1,\ldots,a_n)\in S) \]
as an operator-function acting from a finite-dimensional space into a finite-dimensional one, as in \((4)\).
The case in which \(H\) is finite-dimensional and \(\dim S=n\dim H\) was considered in \((2)\).
Remark. If \(\dim S\) is not a multiple of \(l\), then the question of uniqueness becomes more complicated, and, in addition to restrictions on the number of zeros of the polynomials
\[ \sum_{k=1}^{n} a_k\varphi_k(q) \]
\[ \left(\sum_{k=1}^{n}\|a_k\|>0\right), \]
additional restrictions in the spirit of \((5)\) are also needed.
- Let us now suppose that the functions \(\varphi_1(q),\ldots,\varphi_n(q)\), continuous on \(Q\) and with values in \(H\), are such that the corresponding subspace \(S\) is infinite-dimensional; in particular, \(S=H^n\) (in this latter case the functions \(\varphi_1(q),\ldots,\varphi_n(q)\) are “linearly independent” on \(Q\) in the sense that
\[ \sum_{k=1}^{n} a_k\varphi_k(q)\equiv 0 \]
only when \(a_1=\cdots=a_n=0\)).
Noting that \(S\) is always separable (even when \(H\) is nonseparable), we shall, for convenience, assume \(H\) separable and \(S=H^n\).
By Theorem 1, now not for every function \(f(q)\), continuous on \(Q\) and with values in \(H\), does there exist a polynomial of least deviation. Denote by \(F_\varphi\) the set of those functions, continuous on \(Q\) and with values in \(H\), for which such polynomials exist. We note that \(F_\varphi\) is dense in \(C(H,Q)\)—the Banach space of all functions continuous on \(Q\) and with values in \(H\).
Theorem 3. Let the continuous functions \(\varphi_1(q),\ldots,\varphi_n(q)\) be such that every polynomial
\[ \sum_{k=1}^{n} a_k\varphi_k(q) \left(\sum_{k=1}^{n}\|a_k\|>0\right) \]
vanishes at no more than \(n-1\) points of the compact set \(Q\) (which contains more than \(n\) points).
Then, in order that the polynomial \(\sum_{k=1}^{n} a_k^{(0)}\varphi_k(q)\) deviate least on \(Q\) from the function \(f(q)\in F_\varphi\), it is necessary that the deviation
\[ \max_{q\in Q}\left\|\sum_{k=1}^{n} a_k^{(0)}\varphi_k(q)-f(q)\right\| \]
be attained at no fewer than \(n+1\) points of the compact set \(Q\).
Theorem 4. In order that for every function \(f(q)\in F_\varphi\) there exist a unique polynomial of least deviation, it is necessary and sufficient that every polynomial \(\sum_{k=1}^{n} a_k\varphi_k(q)\) \(\left(\sum_{k=1}^{n}\|a_k\|>0\right)\) vanish at no more than \(n-1\) points of the compact set \(Q\). This condition is equivalent to requiring that, for each \(\alpha=1,2,\ldots\), the numerical functions \(\varphi_{\alpha 1}(q), \varphi_{\alpha 2}(q), \ldots, \varphi_{\alpha n}(q)\), where \(\varphi_k(q)=\sum_{\alpha=1}^{\infty}\varphi_{\alpha k}(q)e_\alpha\) \((k=1,2,\ldots,n)\), form a Chebyshev system on \(Q\).
- As follows from Theorem 1, in order that for every function \(f(q)\), continuous on \(Q\) and with values in \(H\), when approximating it by \(H\)-functions \(a\varphi(q)\), there exist a function \(a^{(0)}\varphi(q)\) of least deviation, one has to impose a very restrictive condition on the function \(\varphi(q)=\sum_{\alpha}\varphi_\alpha(q)e_\alpha\) (namely, that only a finite number of the coefficients \(\varphi_\alpha(q)\) be not identically equal to zero on \(Q\)).
If, however, as the approximating function one takes not the function \(a\varphi(q)\), but a function of the form \(\lambda a-a\varphi(q)\), where \(\lambda\ne0\) is some complex number, then, as established in (6), from the continuity alone of the function \(\varphi(q)\) there already follows the existence of a function \(\lambda a^{(0)}-a^{(0)}\varphi(q)\) of least deviation for every continuous function \(f(q)\), and in order that for any continuous function \(f(q)\) there exist a unique function \(\lambda a^{(0)}-a^{(0)}\varphi(q)\) \((a^{(0)}\in H)\) of least deviation, it is necessary and sufficient that, for every \(a\ne\theta\) in \(H\), the function \(\lambda a-a\varphi(q)\) nowhere vanish on \(Q\), which is equivalent to the condition that for each \(\alpha\) at all points \(q\) of the compact set \(Q\) one have \(\varphi_\alpha(q)\ne\lambda\).
Thus, whatever the continuous function \(\varphi(q)\), choosing \(\lambda\) so that \(|\lambda|>\max_{q\in Q}\|\varphi(q)\|\) ensures uniqueness of the function of least deviation for any continuous function being approximated.
Lutsk State Pedagogical Institute
named after Lesya Ukrainka
Received
2 XII 1957
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