Relations between rational and polynomial approximations in Banach spaces
Notations. \((\mathbb{B},|\cdot |)\) is a Banach space over the field of real or complex numbers having the below described properties. There exists a linearly independent sequence \(\{g_n\}^\infty_{n=0}\) of elements which have a structure of a commutative algebra with unit element \(g_0\). Let \(\mathbb{L}\) be the linear envelope of \(\{g_n\}\) and let \(\mathbb{P}_n = L(g_0, \dots, g_n)\) be the linear envelope of the set of polynomials of degree not exceeding \(n\), whereby the degree \(\partial P\) of a polynomial \(P\) is the maximal index of vectors \(g_k\) with nonzero coefficients representing \(P\), \(\partial (\alpha g_0)=0\), \(\partial (PQ)= \partial P+ \partial Q\). It is possible to divide two polynomials \(H,P\in \mathbb{L}\) with a remainder as follows: There exists \(Q,S\in \mathbb{L}\) such that \(H=P Q+S\), \(\partial S\leq \partial P\), \(\partial P+ \partial Q= \partial H\). For every \(Q\in \mathbb{L}\) there exists \(T=T (Q)>0\) such that for every \(t\in (0,t)\), \((g_0- tQ)^{-1} = g_0+tQ+ \widehat o(t) \in \mathbb{B}\) where \(|\widehat o(t) |= o(t)\). Rational functions \(\mathbb{R}_{n,m}\) are defined by the product of a polynomial of degree not exceeding \(n\) by a reciprocal of a polynomial of degree not exceeding \(m\). The terms ``numerator'' and ``denominator'' used through out with double quotes in the paper, shall be used without quotes in this review. The least deviation of an element \(f\in \mathbb{B}\) from the subspaces \(\mathbb{P}_n\) and \(\mathbb{R}_{n,m}\) is defined as usual by \({\mathbf e}_n (f)= \inf_{P_n\in \mathbb{P}_n} |f_- P_n|\), \({\mathbf r}_{n,m} (f)= \inf_{R_{n,m} \in\mathbb{R}_{n,m}} |f-R_{n,m} |\). It is known that the best approximating element exists in any finite dimensional Banach space. \(\mathbb{Z}_+\) denotes the set of nonnegative integers. In the first section the author proves two theorems, generalizing some results of \textit{A. L. Levin} and \textit{V. M. Tikhomirov} [Dokl. Akad. Nauk SSSR 174, 279-282 (1967; Zbl 0184.29902)]. Theorem 1. Let \(f\in \mathbb{B},m,s \in\mathbb{Z}_+\) and a natural number \(k\geq s\) be fixed. Suppose that \({\mathbf e}_{-1} (f)=+ \infty\). If \({\mathbf e}_{k-s-1} (f)>{\mathbf e}_k (f)={\mathbf e}_{k+m} (f)>{\mathbf e}_{k+m+1} (f)\), then \({\mathbf e}_k (f) > {\mathbf r}_{k,m+s+1} (f)\). Theorem 2. Let \(f\in\mathbb{B}\), \(s\in\mathbb{Z}_+\) and natural numbers \(k \geq s\) be fixed. Suppose that \({\mathbf e}_{-1} (f)= +\infty\). If \({\mathbf e}_{k-s-1} (f)>{\mathbf e}_k(f) = {\mathbf r}_{k,m+s} (f)\), then \({\mathbf e}_k(f) = {\mathbf e}_{k+m} (f)\). The author discusses in some length the condition \({\mathbf e}_n(f) = {\mathbf r}_{n,n} (f)\), \(\forall n\in \mathbb{Z}_+\) and its implications on \(f\) in some concrete Banach spaces. Some interesting corollaries and remarks are also included in this section. The next section deals with the approximation of elements from \(\mathbb{B}\) by rational functions with fixed denominator \(Q_j\). The notation is \(\mathbb{R}^*_n(Q_j) = \{P_n Q_j^{-1} \mid P_n\in \mathbb{P}_n\}\) and \({\mathbf r}^*_n (f;Q_j)= \inf_{P_n\in \mathbb{P}_n} |f-P_n Q_j^{ -1} |\). Here the main result is Theorem 3. Let \(f\in \mathbb{B}\), \(Q_j\) a reversible polynomial of degree \(j\geq 1\), \(s\in \mathbb{Z}_+\), and let the natural numbers \(k\geq s\) and \(m\geq j\) be fixed. Assume \({\mathbf r}^*_{-1} (f;Q_j)= +\infty\). (i) if \({\mathbf r}^*_{k-s-1} (f;Q_j)> {\mathbf r}^*_k (f;Q_j) = {\mathbf r}^*_{r+m-j} (f;Q_j)> {\mathbf r}^*_{k+m-j+1} (f;Q_j)\) then \({\mathbf r}^*_k (f; Q_j) > {\mathbf r}_{k, m+s+1} (f)\), (ii) If \({\mathbf r}^*_{k-s-1} (f;Q_j) > {\mathbf r}^*_k (f;Q_j)= {\mathbf r}_{k,m+s} (f)\), then \({\mathbf r}^*_k (f;Q_j) = {\mathbf r}^*_{k+m-j} (f;Q_j)\). In the final section an additional property is assumed of the Banach space \(\mathbb{B}\) and is then denoted by \(\mathbb{B}^+\). For every \(Q\in\mathbb{L}\) such that \(Q^{-1} \in\mathbb{B}\) and for every \(U\in \mathbb{L}\) there exists a positive number \(T=T (Q, U)>0\) with the property that for every \(t\in(0,T)\) the relation \[ (Q-tU)^{-1} = Q^{-1}+ tUQ^{-2}+ \widehat o(t) \in\mathbb{B}, \] holds, where it is assumed that \(|\widehat o(t) |=o(t)\), for fixed \(Q\) and \(U\). In this section the author considers the best approximation \({\mathbf r}_{n,m} (f)\) as compared to the local best approximation with some fixed denominator. The dependence of the best approximation on the degree of the numerator is also considered. Theorem 4. Let \(f\in \mathbb{B}^+\), and let natural numbers \(n,m\) be fixed. Assume that the rational function \(P_kQ_s^{-1} \neq 0\) gives the best rational approximation of degree \((m,n)\), that is \({\mathbf r}_{n,m} (f)= |f- P_kQ_s^{-1} |\). Assume also \(\partial P_k=k \leq n\), \(\partial Q_s=s \leq m\), \(\lambda = \max \{n+s,m+k\}\). Finally the polynomials \(P_k\) and \(Q_s\) are assumed to be relatively prime. Then the following relation holds: \[ {\mathbf r}_{n,m} (f)= {\mathbf r}^*_\lambda (f;Q^2_s). \] Here two corollaries are worthwhile mentioning. Corollary 6. If in addition to the assumptions of Theorem 4, \(\partial P_k =n\) or \(\partial Q_s =m\). Then \({\mathbf r}_{n,m} (f)= {\mathbf r}^*_{n+m} (f;Q^2_s)\). Corollary 7. If in addition to the assumptions of Theorem 4, \(m+k > n+s\), then \({\mathbf r}^*_n (f;Q_s) = {\mathbf r}^*_{m+k-s} (f;Q_s)\). The author notes that it is not difficult to show that the degree of the best approximating function is achieved either in the numerator or in the denominator if \({\mathbf r}_{n-1,m-1} (f)>{\mathbf r}_{n,m} (f)\). This has been shown for particular Banach spaces by [\textit{E. W. Cheney} and \textit{A. A. Goldstein}, Math. Z. 95, 232-241 (1967; Zbl 0162.08501); \textit{A. L. Levin}, Mat. Zametki 9, 121-130 (1971; Zbl 0221.30037) and \textit{Kh. M. Makhmudov}, Mat. Zametki 45, No. 4, 89-94 (1989; Zbl 0684.30032)]. In the final section the author proves an approximation result in the Hardy space of analytic functions in the unit disk \({\mathcal D}\), that is in \(\mathbb{H}_2 ({\mathcal D})\) with the \(\mathbb{L}^2\) norm on the unit circle. Theorem 5. For any natural number \(n\) there exists a function \(f\in \mathbb{H}_2 ({\mathcal D}) \backslash \mathbb{P}_n\) for which \({\mathbf e}_n(f) = {\mathbf r}_{n,1} (f)\) hold. As a corollary one obtains that for any natural number \(n\) there exists a function \(f\in\mathbb{H}_2 ({\mathcal D})\), for which the relations \({\mathbf e}_{n-1} (f)> {\mathbf e}_n(f) = {\mathbf r}_{n,1} (f)={\mathbf e}_{n+1} (f)>{\mathbf e}_{n+2} (f)\). hold.
scientific article; zbMATH DE number 913006
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