Estimates of the error of interval quadrature formulas on some classes of differentiable functions

V.P. Motornyi (Oles Honchar Dnipro National University)
D.A. Ovsyannikov (Oles Honchar Dnipro National University)


The exact value of error of interval quadrature formulas
$$\int_0^{2\pi}f(t)dt -\frac{\pi}{nh}\sum_{k=0}^{n-1}\int_{-h}^hf(t+\frac {2k\pi}{n})dt = R_n(f;\vec{c_0};\vec{x_0};h)$$
obtained for the classes $$$W^rH^{\omega} (r=1,2,...)$$$ of differentiable periodic functions for which the modulus of continuity of the  $$$r -$$$th derivative is majorized by the given modulus of continuity $$$\omega(t)$$$. This interval quadrature formula coincides with the rectangles formula for the Steklov functions $$$f_h(t)$$$ and is optimal for some important classes of functions.


function; integral; interval quadrature formula; error; sum

Full Text:



Pittnauer Fr., Reimer M. "Interpolation mit Intervallfunktionalen", Math. Z., 1976; 146(1): pp. 7-15. (in German) doi:10.1007/BF01213713

Omladic M., Pahor S., Suhadolc A. "On a new type of quadrature formulas", Numer. Math., 1976; 25(4): pp. 421-426. doi:10.1007/BF01396338

Kuzmina A.L. "Interval quadrature formulas with multiple node intervals", Izv. vuzov, 1980; 7: pp. 39-44. (in Russian)

Sharipov R.N. "The best interval quadrature formulas for Lipschitz classes", Constr. Funct. Theory and Funct. Analysis. Izd-vo Kazansk. un-ta, 1983; 4: pp. 124-132. (in Russian)

Babenko V.F. "On one problem of optimal integration", Researches on modern problems of summation and approximation of functions, and their applications, Dnipropetrovsk, DGU, 1984; pp. 3-13. (in Russian)

Motornyi V.P. "On the best interval quadrature formula in the class of functions with bounded $$$r^{th}$$$ derivative", East J. Approx., 1998; 4(4): pp. 459-478.

Borodachev S.V. "Optimization of 'interval' quadrature formulas for $$$H^{\omega_{+},\omega_{-}}$$$ classes", Res. Math., 1998; 3: pp. 19-26. (in Russian)

Borodachev S.V. "On optimization of 'interval' quadrature formulas on some non-symmetric classes of periodic functions", Res. Math., 1999; 4: pp. 19-24. (in Russian)

Korniichuk M.P. "On extremum properties of periodic functions", Dokl. AN USSR, 1962; 8: pp. 993-998. (in Ukrainian)

Korneichuk N.P. Extremum problems in approximation theory, Moscow, 1976; 320 p. (in Russian)




  • There are currently no refbacks.

Copyright (c) 2020 V.P. Motornyi, D.A. Ovsyannikov

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Registered in

ISSN (Online): 2664-5009
ISSN (Print): 2664-4991