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The classical approach to the theory of quadrature formulae is based on the concept of algebraic degree of precision (ADP). A quadrature formula ß, is considered to be "better" than Q2 if ADP(g, ) > ADP(ß2). However, there are many quadratures that use the same number of evaluations of the integrand and have the same ADP. Then, how should one compare such… (More)
Gauss-Lobatto quadrature formulae associated with symmetric weight functions are considered. The kernel of the remainder term for classes of analytic functions is investigated on elliptical contours. Sufficient conditions are found ensuring that the kernel attains its maximal absolute value at the intersection point of the contour with either the real or… (More)
Let pm(x) = P (λ) m (x)/P (λ) m (1) be the m-th ultraspherical polynomial normalized by pm(1) = 1. We prove the inequality |x|pn(x)−pn−1(x)pn+1(x) ≥ 0, x ∈ [−1, 1], for −1/2 < λ ≤ 1/2. Equality holds only for x = ±1 and, if n is even, for x = 0. Further partial results on an extension of this inequality to normalized Jacobi polynomials are given.