New minimality properties of Gaussian quadratures (Q762395)

This paper is concerned with dual quadrature. These occur if the Hermitian polynomial \((d/dx)H_ n(F,x)\) where \(F(x):=\int^{x}_{a}f(x)dx\) is integrated to give \(\bar Q_ nf=\sum^{n}_{i=1}[\underline A_ if(\underline x_ i)+\underline B_ iF(\underline x_ i)]\), and if required that \(B_ i=0\), \(i=1(1)n\), so that the familiar shape of a quadrature formula is obtained. \(\bar Q_ nf\) is not uniquely determined by \(B_ i=0\), \(i=1(1)n\) because its degree is 2n-2. Hence \(\bar Q_ nf\) can be considered to depend on e.g. \b{x}\({}_ 1\) as free parameter. If is now shown that \(\bar Q_ nf\) becomes identical with the well-known Gauss-quadrature formula \(Q_ nf\) if we impose one of the global conditions (for any symmetric weight function W(x)) (i) Determine \(\bar Q_ nf\) such that its variance with respect to the weights \b{A}\({}_ i\) is minimal. (ii) Determine \(\bar Q_ nf\) such that the diameter of the nodes \(d_ n:=\max_{i,k}| x_ i-x_ k|\) is minimal. Thus the Gaussian quadrature formulae prove to be the minimal formulae in the set of their dual quadrature formulae.