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The Corrected Two-Point Weighted Quadrature Formulae

The weighted version of the corrected two-point quadrature formula is derived. In the corrected two-point formula the integral is approximated both with the values of the integrand in nodes $x$ and $a+b-x$, and the values of its first derivative in the end-points of the interval $[a, b]$. Those formulae have a degree of exactness higher than the ordinary formulae. Estimates of error bounds under various regularity conditions for such formulae are established. As special cases, the corrected two-point formulae of Gauss type are obtained. Also, the corrected version of weighted trapezoid, weighted midpoint, weighted two-point Maclaurin and weighted two-point Newton-Cotes formulae are considered.

weighted function; corrected formula; two-point quadrature formula; w-harmonic sequences of functions; $L_p$ spaces; corrected Gauss formula; weighted corrected Maclaurin two-point formula; weighted corrected trapezoid formula; weighted corrected midpoint formula; weighted corrected Newton-Cotes two-point formula

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