On estimates for the weights in Gaussian quadrature in the ultraspherical case

In this paper the Christoffel numbers a(2)G for ultraspherical weight functions w,A, WA(X) = (1 x2 1/2 , are investigated. Using only elementary functions, we state new inequalities, monotonicity properties and asymptotic approximations, which improve several known results. In particular, denoting by 0(i) the trigonometric representation of the Gaussian nodes, we obtain for A E [0, 11 the inequalities n v,fn 2(n + A)2 sin2 (A) (AG 7r 2A1 3.) av,n 1/2, a(A)G 7r 2A (A) __________ ~~~~~ sin 0 ~ ~ in()' v,n n + A /f n 2(n + A)2sin 2 ) v,n A ( I A) [3(A + I )(A 2) + 4 sin 2 0(A)n (8(n?+ A)4 sin 46~ +O(n )

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