# zbMATH — the first resource for mathematics

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:
 [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