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On the best interval quadrature formulae for classes of differentiable periodic functions. (English) Zbl 1147.41008
The authors discuss the Kolmogorov problems on optimal quadrature formulae for class \(W^{r}F\) of differentiable periodic functions with rearrangement invariant set \(F\) of their derivative of order \(r\). They prove that for any fixed \(h \in [0,\frac{\pi}{n}]\) the interval quadrature formula having equidistant nodes and equal coefficients, \(b_{j}=\frac{2\pi}{n}\) is optimal for the class \(W^{r}F\). To this end a sharp inequality for antiderivatives of rearrangement of averaged monosplines is proved.

MSC:
41A55 Approximate quadratures
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[1] P.S. Aleksandrov, Combinatorial Topology, OGIZ, Moscow, 1947 (in Russian); P.S. Aleksandrov, Combinatorial Topology, vol. 1, Graylock Press, Albany, NY, 1956 (in English).
[2] Babenko, V.F., Nonsymmetric approximations in the spaces of summable functions, Ukrainian math. J., 34, 409-416, (1982), (in Russian)
[3] Babenko, V.F., Inequalities for rearrangements of differentiable periodic functions, Problems of approximation and integrating, dokl. USSR, 272, 1038-1041, (1983), (in Russian) · Zbl 0547.41027
[4] V.F. Babenko, On a certain problem of optimization of the approximate integration, Studies on Modern Problems of Summation and Approximation of Functions and their Applications, Dnepropetrovsk University, Dnepropetrovsk, 1984, pp. 3-13 (in Russian).
[5] Babenko, V.F., Widths and optimal quadrature formulae for classes of periodic functions with rearrangement invariant sets of derivatives, Anal. math., 13, 15-28, (1987) · Zbl 0652.41008
[6] Babenko, V.F., Widths and optimal quadrature formulae for convolution classes, Ukrainian math. J., 43, 1135-1148, (1991) · Zbl 0743.42001
[7] Borodachov, S.V., On optimization of interval quadrature formulae on some nonsymmetric classes of periodic functions, Bull. dnepropetrovsk univ. math., 4, 19-24, (1999), (in Russian)
[8] Borodachov, S.V., On optimization of interval quadrature formulae on some classes of absolutely continuous functions, Bull. dnepropetrovsk univ. math., 5, 28-34, (2000), (in Russian)
[9] N.P. Korneichuk, Extremal Problems of Approximation Theory, Nauka, Moscow, 1976, p. 320 (in Russian).
[10] N.P. Korneichuk, A.A. Ligun, V.G. Doronin, Approximation with Constraints, Naukova dumka, Kiev, 1982 (in Russian).
[11] Krasnosel’skii, M.A.; Rutickii, Ya.B., Convex functions and orlich spaces, (1958), Fizmatgiz Moscow, (in Russian)
[12] Krein, S.G.; Petunin, Yu.I.; Semenov, E.M., Interpolation of linear operators, (1978), Nauka Moscow, (in Russian) · Zbl 0499.46044
[13] Kuz’mina, A.L., Interval quadrature formulae with multiple node intervals, Izv. vuzov math., 7, 39-44, (1980), (in Russian) · Zbl 0464.41021
[14] Ligun, A.A., Exact inequalities for spline-functions and best quadrature formulae for some classes of functions, Math. zametki, 19, 913-926, (1976), (in Russian)
[15] Milovanovic, G.V.; Cvetkovic, A.S., Gauss – radau and gauss – lobatto interval quadrature rules for Jacobi weight function, Numer. math., 3, 102, 523-542, (2006) · Zbl 1114.65030
[16] Motornyi, V.P., On the best quadrature formula of the form \(\sum_{k = 1}^n p_k f(x_k)\) for certain classes of periodic differentiable functions, Izv. akad. nauk SSSR. ser. mat., 38, 583-614, (1974), (in Russian)
[17] Motornyi, V.P., On the best interval quadrature formula in the class of functions with bounded \(r\)th derivative, East J. approx., 4, 459-478, (1998)
[18] Omladich, M.; Pahor, S.; Suhadolc, S., On a new type of quadrature formulae, Numer. math., 25, 421-426, (1976) · Zbl 0314.65007
[19] Oskolkov, K.I., On optimality of quadrature formula with equidistant nodes on the classes of periodic functions, Dokl. akad nauk USSR, 249, 49-52, (1979), (in Russian) · Zbl 0441.41018
[20] Pittnauer, Fr.; Reimer, M., Interpolation mit intervallfunctionalen, Math. Z., 146, 7-15, (1976) · Zbl 0302.41002
[21] R.N. Sharipov, Best interval quadrature formulae for Lipschitz classes, Constructive Function Theory and Functional Analysis, vol. 4, Kazan University, Kazan, 1983, pp. 124-132 (in Russian). · Zbl 0566.41039
[22] Tribel, H., Theory of interpolation, function spaces, differential operators, (1980), Mir Moscow, (in Russian)
[23] Zhensykbaev, A.A., The best quadrature formula for some classes of periodic functions, Izv. akad nauk USSR, ser. math., 41, 1110-1124, (1977), (in Russian)
[24] Zhensykbaev, A.A., Monosplines of minimal norm and the best quadrature formulae, Uspehi math. nauk., 36, 107-159, (1981), (in Russian) · Zbl 0504.41024
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