On Gauss-Kronrod Quadrature Formulae of Chebyshev Type

Sotirios E. Notaris
1992 Mathematics of Computation  
We prove that there is no positive measure da 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 do(t) = a(t)dt, where w(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 da(t) = da(-t) on (-1, 1) for which the Gauss-Kronrod formula has all weights equal if
more » ... ll weights equal if n -1, or has the form /I, f(t)da(t) = w J^., /(t") + to,/(l) + w ££=2 fil») + wi/(-l) + R%(f) for all n > 2, is the Chebyshev measure of the first kind doc(t) = (\ -t2)-ll2dt.
doi:10.2307/2153213 fatcat:fi22r7v7nnbclpd6cm7zsvac4m