Das asymptotische Verhalten der Peanokerne einiger interpolatorischer Quadraturverfahren. (Asymptotic behaviour of the Peanokernels of some interpolatory quadrature formulae) (Q1100371)

Let \(X=\{x_ 1,...,x_ n\}\subset I=[-1,1]\), \(\psi (x)=(x-x_ 1)...(x- x_ n)\), \(m=2[(n+1)/2]\), \(n\geq 2\), \(f\in C^{(m)}(I)\), and \(p_ f\) denotes the interpolating polynomial of f with respect to the knot set X. If we consider the interpolatory formula \(\int^{1}_{- 1}f(x)dx=\int^{1}_{-1}p_ f(x)dx+R_ X(f)\) where \(R_ X(f)\) is the remainder, then the author proves the following interesting representation of the remainder \[ R_ X(f)=\frac{1}{2^ n(m- 1)!}\int^{1}_{-1}(1-t^ 2)^{m-}\tilde P_ X(t)f^{(m)}(t)dt \] where \(\tilde P_ X(t)=\int^{1}_{-1}\frac{(1-u^ 2)^{m- 1}}{G(z)}H(z)du,\) \(z=z(t,u)=t+iu\sqrt{1-t^ 2}\), \(t\in (-1,1)\), \(G(z)=(2z)^ n\psi ((z+z^{-1})/2)\). For n even \[ H(z)=2^ n\int^{1}_{-1}\frac{(1-z^ 2)\psi (x)}{1-2xz+z^ 2}dx \] while for n odd \[ H(z)=2^ n\int^{1}_{-1}\frac{(x-z)\psi (x)}{1-2xz+z^ 2}dx. \] By means of the above result, the author determines the asymptotic behaviour of the Peano kernel for the quadrature formulas of Filippi, Polya and Clenshaw-Curtiss.