On multiple node Gaussian quadrature formulae

Abstract

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