The Errors Analysis For Numerical Quadrature Formula On The Average Case Setting

In this paper, we obtain the average errors of the composite Simpson’s rule, the Gauss-Legendre quadrature formula and the numerical quadrature formula based on the extrema of the second kind Chebyshev polynomials in the r-fold integral Wiener space. We obtain that the saturation order is three for the composite Simpson’s rule and the Gauss-Legendre quadrature formula is a kind of universal operators which have a high accuracy for the different order smooth functions. At last, we show that the numerical quadrature formula based on extrema of the second kind Chebyshev polynomial and give the strong asymptotic orders of approximation errors when r=0,1,2.