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Differential properties of the operator of best approximation of continuous functions by rational fractions. (English. Russian original) Zbl 0655.41023
Math. Notes 41, 366-377 (1987); translation from Mat. Zametki 41, 654-672 (1987).
Let C[a,b] be the space of real continuous functions on the segment [a,b] with the uniform norm \(\| \cdot \|\), M be a Chebyshev subset of C[a,b], \(f\in C[a,b]\), \(E_ M(f)=\inf \{\| f-q\|:q\in M\}=\| f- \pi_ M(f)\|\) be the best approximation of the function f; \(\pi_ M:f\mapsto \pi_ M(f)\) be the operator of best approximation. For \(f\in C[a,b]\), \(\epsilon >0\) we consider \[ A_ 1(f,M,\epsilon)=\sup \{\| q-\pi_ M(f)\| +q\in M,\| q-f\| \leq E_ m(f)+\epsilon \}, \] \[ A_ 2(f,M,\epsilon)=\sup \{\| \pi_ M(g)-\pi_ M(f)\div g\in C[a,b],\| g-f\| \leq \epsilon \}, \] which are connected by the obvious relation: \(A_ 2(f,M,\epsilon /2)\leq A_ 1(f,M,\epsilon)\). Throughout what follows c(\(\cdot,...,\cdot)\) will denote positive constants, not necessarily equal, depending only on the parameters indicated; \(V_{\delta}(f)\) for \(\delta >0\), \(f\in C[a,b]\) is the ball map \(\{g\in C[a,b]:\| f-g\| \leq \delta \}\) in C[a,b] with center at the point f and radius \(\delta\). We shall say that the operator \(\pi_ M\) satisfies the strict uniqueness condition locally uniformly (is locally uniformly Lipschitz) at the point \(f\in C[a,b]\), if one can find a \(\delta =\delta (f,M)>0\) and a \(c=c(f,M)>0\) such that for all \(g\in V_{\delta}(f)\), \(\epsilon >0\), \(A_ 1(g,M,\epsilon)\leq c\epsilon\), \((A_ 2(g,M,\epsilon)\leq c\epsilon)\). As M we shall consider an n- dimensional Chebyshev subspace \(\psi_ n\) of the space C[a,b] (in this case we shall use the notation \(\pi_ n=\pi \beta_ n)\) and the set of classical rational fractions \(R_{n,m}\) with positive weight \(u\in C[a,b]\), where \(n,m\in Z_+\) (nonnegative integers). Let \(P_ n\) be the space of algebraic polynomials of degree no higher than n, \(P_{\mu}=\{0\}\) for \(\mu <0\), \[ R_{n,m}=\{up/q:p\in P_ n,q\in P_ m,q(x)>0\quad for\quad x\in [a,b]\}, \] \[ (u)=\pi_{n,m}=\pi_{R_{n,m}},\quad \rho_{n,m}(f)=E_{R_{n,m}}(f), \] \[ R^ 0_{n,m}=\{up/q\in R_{n,m}:p\quad and\quad q\quad relatively\quad prime\quad \| q\| =1\}. \] We study the locally uniform and differential properties of the operator \((u)\pi_{n,m}\). In particular, we characterize the set of points at which the operator \((u)\pi_{n,m}\) satisfies the strict uniqueness condition locally uniformly, the set of functions at which the operator \((u)\pi_{n,m}\) is unilaterally differentiable in any direction from C[a,b], and the set of functions at which the operator is Gateau differentiable.

MSC:
41A20 Approximation by rational functions
41A50 Best approximation, Chebyshev systems
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