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 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 2004, Volume 76, Issue 2, Pages 216–225 (Mi mz101)

Best Uniform Rational Approximations of Functions by Orthoprojections

A. A. Pekarskii

Belarusian State Technological University

Abstract: Let $C[-1,1]$ be the Banach space of continuous complex functions $f$ on the interval $[-1,1]$ equipped with the standard maximum norm $\|f\|$; let $\omega( \cdot )=\omega( \cdot ,f$ be the modulus of continuity of $f$; and let $R_n=R_n(f)$ be the best uniform approximation of $f$ by rational functions (r.f.) whose degrees do not exceed $n=1,2,\ldots$. The space $C[-1,1]$ is also regarded as a pre-Hilbert space with respect to the inner product given by $(f,g)=(1/\pi)\int_{-1}^1f(x)g(x)(1-x^2)^{-1/2} dx$. Let $z_n=ż_1,z_2,\ldots,z_n\}$ be a set of points located outside the interval $[-1,1]$. By $F( \cdot ,f,z_n)$ we denote an orthoprojection operator acting from the pre-Hilbert space $C[-1,1]$ onto its $(n+1)$-dimensional subspace consisting of rational functions whose poles (with multiplicity taken into account) can only be points of the set $z_n$. In this paper, we show that if $f$ is not a rational function of degree $\leqslant n$, then we can find a set of points $z_n=z_n(f)$ such that $\|f( \cdot )-F( \cdot ,f,z_n)\|\leqslant 12R_n\ln\frac3{\omega^{-1}(R_n/3)}$.

DOI: https://doi.org/10.4213/mzm101

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English version:
Mathematical Notes, 2004, 76:2, 200–208

Bibliographic databases:

UDC: 517.53

Citation: A. A. Pekarskii, “Best Uniform Rational Approximations of Functions by Orthoprojections”, Mat. Zametki, 76:2 (2004), 216–225; Math. Notes, 76:2 (2004), 200–208

Citation in format AMSBIB
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