Gauss quadrature for quasi-definite linear functionals
Gauss quadrature for quasi-definite linear functionals
复制标题
拟定线性泛函的高斯求积
DOI：
--
复制
发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Z. Strakoš
中科院分区：
文献类型：
--
作者：
S. Pozza;M. Pranic;Z. Strakoš
The 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 with 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 using 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 the Gauss quadrature for approximation of linear functionals which 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 the generalized Gauss quadrature and prove that the quasi-definiteness of the underlying linear functional represents the necessary and sufficient condition for its existence.
