Abstrakt
Gauss quadrature can be formulated as a method for approximating positive-definite linear functionals. Its mathematical context is extremely rich, with orthogonal polynomials, continued fractions and Padé approximation on one (functional analytic or approximation theory) side, and the method of moments,(real) Jacobi matrices, spectral decompositions and the Lanczos method on the other (algebraic) side.
The quadrature concept can therefore be developed in many different ways. After a brief review of the mathematical interconnections in the positive-definite case, this paper will investigate the question of a meaningful generalization of Gauss quadrature for approximation of linear functionals that are not positive definite.
For that purpose we use the algebraic approach, and, in order to build up the main ideas, recall the existing results presented in literature. Along the way we refer to the associated results expressed through the language of rational approximations.
As the main result, we present the form of generalized Gauss quadrature and prove that the quasi-definiteness of the underlying linear functional represents a necessary and sufficient condition for its existence.