On Gauss-Kronrod quadrature formulae of Chebyshev type

Abstract:

We prove that there is no positive measure $d\sigma$ on (a, b) such that the corresponding Gauss-Kronrod quadrature formula is also a Chebyshev quadrature formula. The same is true if we consider measures of the form $d\sigma (t) = \omega (t)dt$, where $\omega (t)$ is even, on a symmetric interval $( - a,a)$, and the Gauss-Kronrod formula is required to have equal weights only for n even. We also show that the only positive and even measure $d\sigma (t) = d\sigma ( - t)$ on $( - 1,1)$ for which the Gauss-Kronrod formula has all weights equal if $n = 1$, or has the form $\smallint _{ - 1}^1f(t)d\sigma (t) = w\Sigma _{\nu = 1}^nf({\tau _\nu }) + {w_1}f(1) + w\Sigma _{\mu = 2}^nf(\tau _\mu ^\ast ) + {w_1}f( - 1) + R_n^K(f)$ for all $n \geq 2$, is the Chebyshev measure of the first kind $d{\sigma _C}(t) = {(1 - {t^2})^{ - 1/2}}dt$.