On Gautschi's conjecture for generalized Gauss-Radau and Gauss-Lobatto formulae

TL;DR

This paper proves the positivity of weights in generalized Gauss-Radau and Gauss-Lobatto quadrature formulas, confirming Gautschi's conjecture and establishing convergence theorems for these Gaussian-type formulas.

Contribution

It confirms Gautschi's conjecture on weight positivity and derives convergence theorems for the generalized quadrature formulas involving derivatives.

Findings

Positivity of weights is established for the formulas.

Convergence theorems are proved for the quadrature formulas.

The results confirm the conjecture and extend understanding of these quadrature methods.

Abstract

Recently, Gautschi introduced so-called generalized Gauss-Radau and Gauss-Lobatto formulae which are quadrature formulae of Gaussian type involving not only the values but also the derivatives of the function at the endpoints. In the present note we show the positivity of the corresponding weights; this positivity has been conjectured already by Gautschi. As a consequence, we establish several convergence theorems for these quadrature formulae.