Some recent investigations (see e.g., Gerstner and Griebel [5], Novak and Ritter [9] and [10], Novak, Ritter and Steinbauer [11], Wasilkowski and Woźniakowski [18] or Petras [13]) show that the so-calledSmolyak algorithm applied to a cubature problem on the d-dimensional cube seems to be particularly useful for smooth integrands. The problem is still that the numbers of nodes grow (polynomially but) fast for increasing dimensions. We therefore investigate how to obtain Smolyak cubature formulae with a given degree of polynomial exactness and the asymptotically minimal number of nodes for increasing dimension d and obtain their characterization for a subset of Smolyak formulae. Error bounds and numerical examples show their good behaviour for smooth integrands. A modification can be applied successfully to problems of mathematical finance as indicated by a further numerical example.

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With the help of a new representation of the Stieltjes polynomial it is shown by using Bessel functions that the Stieltjes polynomial with respect to the ultraspherical weight function with parameter λ has only few real zeros for λ > 3 and sufficiently large n. Since the nodes of the GaussKronrod quadrature formulae subdivide into the zeros of the Stieltjes polynomial and the Gaussian nodes, it follows immediately that Gauss-Kronrod quadrature is not possible for λ > 3. On the other hand, for λ = 3 and sufficiently large n, even partially positive Gauss-Kronrod quadrature is possible.

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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 ,