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

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