Abstract.In this paper the Christoffel numbers av n for ultraspherical weight functions wk , wx(x) = (\ -x ) ~ ' , are investigated. Using only elementary functions, we state new inequalities, monotonicity properties and asymptotic approximations, which improve several known results. In particular, denoting by dv " the trigonometric representation of the Gaussian nodes, we obtain for À e [0, 1] the inequalities and similar results for X <£ (0, 1). Furthermore, assuming that a\ ' remains in a fixed closed interval, lying in the interior of (0, n) as n -► oo , we show that, for every fixed A > -1/2 ,

No abstract

No abstract

Abstract.Recently Gautschi [5] has proved that the weight functions w e defined byw~(x) = '. 0 elsewhere , 0 < ~ < 1 are the only symmetric ones, apart from the Chebyshev weight function, for which there exists for every even n a Chebyshev rule in the strict sense, having n nodes and Gaussian degree 2n-1. In this note we show that for an odd number n of nodes the maximum degree of Chebyshev-type rules for we has a completely different behavior from that for even n.

A weighted quadrature formula is of Chebyshev type if it has equal coefficients and real (but not necessarily distinct) nodes. For a given weight function we study the set T(n,d) consisting of all Chebyshev-type formulas with n nodes and at least degree d. It is shown that in nonempty T(n,d) there exist two special formulas having "extremal" properties. This result is used to prove uniqueness and further results for £-optimal Chebyshev-type formulas. For the weight function w = 1, numerical investigations are carried out for n < 25.