Abstract Generated abstract
This paper studies best Chebyshev approximation of continuous functions on a compact set with values in an infinite-dimensional commutative Hilbert ring by generalized polynomials with coefficients in that ring. Using the decomposition of ring elements into orthogonal idempotents and an operator-function formulation, it gives necessary and sufficient conditions for existence of a least-deviation polynomial for every continuous target function, showing that this requires only finitely many nonzero component coefficient functions. The paper also treats uniqueness, relating it to bounds on the number of zeros of nontrivial approximating polynomials and to Chebyshev systems in the scalar components, and discusses the case where least approximants exist only for a dense subclass. A final variant shows that approximants of the form involving a nonzero parameter can ensure existence, and under a simple spectral separation condition, uniqueness.
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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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