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Differential properties of the operator of best approximation by rational fractions. (English. Russian original) Zbl 0708.41016
Math. Notes 45, No. 1-2, 115-122 (1989); translation from Mat. Zametki 45, No. 2, 40-50 (1989).
S. N. Bernstein proved, under some conditions, the analytic dependence on t of the coefficients of the polynomial of best uniform approximation of the sum $$f+t\phi$$, $$0\leq t\leq 1$$, where f and $$\phi$$ are given analytic functions on a given interval [a,b]. The author analyzes the multiple differentiability in the direction of the operator $$\pi$$ of best uniform approximation by rational functions with a positive weight $$u\in C[a,b].$$
The operator $$\pi$$ is said to be k-times differentiable on an element $$f\in C[a,b]$$ in the direction $$\phi\in C[a,b]$$ if there are functions $$D^+_ 1,...,D^+_ k\in C[a,b]$$ such that $\pi (f+t\phi)=\pi (f)+\sum^{k}_{i=1}t^ iD^+_ i(f,\phi)+o(t^ k)\text{ for } t\to 0.$ Similarly, the k-times differentiability to the left and k-times differentiability is defined. Let $$B_{n,0}=C[a,b]$$ and $$B_{n,m}$$, $$m>0$$ $$n\geq 0$$, be the set of all functions continuous on [a,b] for which $$\pi (f)=u\cdot p/q$$, where p and q are mutually prime polynomials, $$p\neq 0$$, $$q>0$$, $$\| q\| =1$$ and deg p$$=n$$ or deg q$$=m$$. An example of the author’s results reads as follows: Let $$k\geq 2$$ and let f be any function having k continuous derivatives. Let $$f\in B_{m,n}$$ an let $$(f(x)-\pi (f,x))''\neq 0$$ in every point $$x\in [a,b]$$ for which the norm of f-$$\pi$$ (f) is attained. Then, the operator $$\pi$$ is k-times differentiable on the element f in any k-times continuously differentiable direction $$\phi$$ to the right.
Reviewer: J.Fuka
##### MSC:
 41A20 Approximation by rational functions
##### Keywords:
best uniform approximation; multiple differentiability
Full Text:
##### References:
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