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 Izv. RAN. Ser. Mat., 2001, Volume 65, Issue 5, Pages 73–90 (Mi izv357)

Best quadrature formulae on Hardy–Sobolev classes

K. Yu. Osipenko

Moscow State Aviation Technological University

Abstract: For functions in the Hardy–Sobolev class $H_\infty^r$, which is defined as the set of functions analytic in the unit disc and satisfying $f^{(r)}(z)|\leqslant 1$, we construct best quadrature formulae that use the values of the functions and their derivatives on a given system of points in the interval $(-1,1)$. For the periodic Hardy–Sobolev class $H_{\infty,\beta}^r$, which is defined as the set of $2\pi$-periodic functions analytic in the strip $|\operatorname{Im}z|<\beta$ and satisfying $|f^{(r)}(z)|\leqslant 1$, we prove that the rectangle rule is the best for an equidistant system of points, and we calculate the error in this formula. We construct best quadrature formulae on the class $H_{p,\beta}$, which is defined similarly to $H_{\infty,\beta}$, except that the boundary values of functions are taken in the $L_p$-norm. We also construct an optimal method for recovering functions in $H_p^r$ from the Taylor information $f(0),f'(0),…,f^{(n+r-1)}(0)$.

DOI: https://doi.org/10.4213/im357

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English version:
Izvestiya: Mathematics, 2001, 65:5, 923–939

Bibliographic databases:

MSC: 41A55

Citation: K. Yu. Osipenko, “Best quadrature formulae on Hardy–Sobolev classes”, Izv. RAN. Ser. Mat., 65:5 (2001), 73–90; Izv. Math., 65:5 (2001), 923–939

Citation in format AMSBIB
\Bibitem{Osi01} \by K.~Yu.~Osipenko \paper Best quadrature formulae on Hardy--Sobolev classes \jour Izv. RAN. Ser. Mat. \yr 2001 \vol 65 \issue 5 \pages 73--90 \mathnet{http://mi.mathnet.ru/izv357} \crossref{https://doi.org/10.4213/im357} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1874354} \zmath{https://zbmath.org/?q=an:1017.41020} \elib{http://elibrary.ru/item.asp?id=13361629} \transl \jour Izv. Math. \yr 2001 \vol 65 \issue 5 \pages 923--939 \crossref{https://doi.org/10.1070/IM2001v065n05ABEH000357} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-27144453793} 

• http://mi.mathnet.ru/eng/izv357
• https://doi.org/10.4213/im357
• http://mi.mathnet.ru/eng/izv/v65/i5/p73

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This publication is cited in the following articles:
1. Fang Gensun, Li Xuehua, “Optimal quadrature problem on Hardy–Sobolev classes”, J. Complexity, 21:5 (2005), 722–739
2. Fang Gensun, Li Xuehua, “Optimal quadrature problem on classes defined by kernels satisfying certain oscillation properties”, Numer. Math., 105:1 (2006), 133–158
3. Xue Hua Li, Gen Sun Fang, “Optimal quadrature problem on n-information for Hardy-Sobolev classes”, Acta. Math. Sin.-English Ser, 27:12 (2011), 2371
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