We consider a positive measure on [0,∞) and a sequence of nested spaces L0 ⊂ L1 ⊂ L2 · · · of rational functions with prescribed poles in [−∞, 0]. Let {φk}k=0, with φ0 ∈ L0 and φk ∈ Lk \ Lk−1, k = 1, 2, . . . be the associated sequence of orthogonal rational functions. The zeros of φn can be used as the nodes of a rational Gauss quadrature formula that is exact for all functions in Ln · Ln−1, a...

Let L be a quadratic imaginary field, inert at the rational prime p. Fix an integer n ≥ 3, and let M be the moduli space (in characteristic p) of principally polarized abelian varieties of dimension n equipped with an action by OL of signature (1, n−1). We show that each Newton stratum of M, other than the supersingular stratum, is irreducible.

Let {Q„)denote a sequenceof quadrature formulas, Q„(j) m Yfj-iW^fix^), such that ß„(/) -> P0 j(x) dx for all / G CTO, 1], Let 0 < e < \ and a sequence (aX_j.be given, where a, ä si ^ a, 5 • • • , and where a„ —> 0 as n —* c°. Then there exists a function / G CTO, l]and a sequence |nt-)"=i suchthat |/(x)| g 2(7,71(1 4e)|, and such that n,Kx)dx Q„k(1) = ak,k = 1,2, 3, ••• .

In this paper a new method for inverting the Laplace transform from the real axis is formulated. This method is based on a quadrature formula. We assume that the unknown function f(t) is continuous with (known) compact support. An adaptive iterative method and an adaptive stopping rule, which yield the convergence of the approximate solution to f(t), are proposed in this paper. MSC: 15A12; 47A5...

In this paper we establish Gauss-type quadrature formulas for weakly singular integrals. An application of the quadrature scheme is given to obtain numerical solutions of the weakly singular Fredholm integral equation of the second kind. We call this method a discrete product-integration method since the weights involved in the standard product-integration method are computed numerically.

We show, for each n > 1, that the (2ra + l)-point Kronrod extension of the n-point Gaussian quadrature formula for the measure do-^t) = (1 + 7)2(1 t2)^2dt/((l + -y)2 47t2), -K -y < 1, has the properties that its n + 1 Kronrod nodes interlace with the n Gauss nodes and all its 2ra + 1 weights are positive. We also produce explicit formulae for the weights.

We prove that to every rational function R(z) satisfying R(−z)R(z) = 1, there exists a symplectic Runge-Kutta method with R(z) as stability function. Moreover, we give a surprising relation between the poles of R(z) and the weights of the quadrature formula associated with a symplectic Runge-Kutta method.

The solution to the problem of factorization of the covariance function of a stationary, discrete-time process is obtained by using a Newton-Raphson procedure which converges quadratically in I1 provided the initial iterate is chosen suitably. The existence of a suitable initial iterate is guaranteed by an approximation result. An application to error localization in spectral factorization is s...

It is shown that the zeros of the Faber polynomials generated by a regular m-star are located on the m-star. This proves a recent conjecture of J. Bartolomeo and M. He. The proof uses the connection between zeros of Faber polynomials and Chebyshev quadrature formulas.

A positive quadrature formula with n nodes which is exact for polynomials of degree In — r — 1, 0 < r < « , is based on the zeros of certain quasi-orthogonal polynomials of degree n . We show that the quasi-orthogonal polynomials that lead to the positive quadrature formulae can all be expressed as characteristic polynomials of a symmetric tridiagonal matrix with positive subdiagonal entries. A...