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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
##### Keywords:
locally uniform strict uniqueness
Full Text:
##### References:
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