Abs t r ac t -An account is given of the role played by moments and modified moments in the construction of quadrature rules, specifically weighted Newton-Cotes and Gaussian rules. Fast and slow Lagrange interpolation algorithms, combined with Gaussian quadrature, as well as linear algebra methods based on moment equations, axe described for generating Newton-Cotes formulae. The weaknesses and ...

We study asymptotic expansions, Filon-type methods and complex-valued Gaussian quadrature for highly oscillatory integrals with power-law and logarithmic singularities. We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand, the stationary points and the endpoints of the integral. A truncated asymptotic expansion...

In the companion paper (Bellet et al., 2021), a spherical harmonic subspace associated to Cubed Sphere has been introduced. This is further analyzed here. particular, it permits define new based quadrature. quadrature inherits rotational invariance properties of subspace. Contrary Gaussian quadrature, where set nodes and weights solution nonlinear system, only are unknown Despite this conceptua...

Abstract: For finite positive Borel measures supported on the real line we consider a new type of quadrature rule with maximal algebraic degree of exactness, which involves function derivatives. We prove the existence of such quadrature rules and describe their basic properties. Also, we give an application of these quadrature rules to the solution of a Cauchy problem without solving it directl...