On multiple node Gaussian quadrature formulae

Abstract:

Let ${\mu _1}, \ldots ,{\mu _k}$ be odd positive integers and $n = \Sigma _{i = 1}^k({\mu _i} + 1)$. Let $\{ {\mu _i}\} _{i = 1}^n$ be an extended Tchebycheff system on $[a,b]$. Let L be a positive linear functional on $U \equiv {\operatorname {span}}(\{ {\mu _i}\} )$. We prove that L has a unique representation in the form \[ L(p) = \sum \limits _{i = 1}^k {\sum \limits _{j = 0}^{{\mu _i} - 1} {{a_{ij}}{p^{(j)}}({t_i}),\quad a < {t_1} < \cdots < {t_k} < b,} } \] for all $p \in U$. The proof uses the topological degree of a mapping $F:\overline D \subset {R^k} \to {R^k}$. The result is proved by showing that the equation $F(\underline {t}) = 0$ has a unique solution, which in turn is proved by showing that F has degree 1 and that for any solution $\underline {t}$ to the equation $F(\underline {t}) = 0$, $\det F\prime (\underline {t}) > 0$. We also give extensions to the cases when the $\{ {u_i}\}$ are a periodic extended Tchebycheff system and when L is a nonnegative linear functional.