engleski

General Three-point Quadrature Formulae

General three-point quadrature formulae of the type $$\int_a^b w(t)f(t){; ; d}; ; t=A(x) \left[f\left(x\right)+f(a+b-x)\right]+B(x) f\left(\frac{; ; a+b}; ; {; ; 2}; ; \right)+E(f, x, w), $$ are considered. The sequences of harmonic polynomials and $w-$harmonic functions are applied as a constructive tool for deriving such formulas. Some best possible and sharp error estimates are obtained for the functions whose higher order derivatives belong to $L_p[a, b]$ spaces. Considering the special case $w\equiv 1$, the generalizations of the well-known three-point formulae are obtained. The Simpson's formula ($x=a$), dual Simpson's formula ($x=\frac{; ; 3a+b}; ; {; ; 4}; ; $), dual Simpson's $3/8$ formula ($x=\frac{; ; 5a+b}; ; {; ; 6}; ; $) and Gauss-Legendre two-point ($B(x)=0$) formula ($x=\frac{; ; a+b}; ; {; ; 2}; ; -\frac{; ; b-a}; ; {; ; 2\sqrt{; ; 3}; ; }; ; $) are established as special cases.\\ For the general case of weighted function $w$, some new inequalities regarding quadrature formulae of Gaussian type are obtained.

harmonic polynomials; numerical integration; $w-$harmonic functions; $L_p$ spaces; inequalities; Gaussian quadrature; Simpson's rule; dual Simpson's rule; Maclaurin's rule

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