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.

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 |

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\textit{A. V. Kolushov}, Math. Notes 45, No. 1--2, 115--122 (1989; Zbl 0708.41016); translation from Mat. Zametki 45, No. 2, 40--50 (1989)

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##### References:

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