Existence and uniqueness of Hermite-Birkhoff Gaussian quadrature formulas

CALCOLO ◽  
1989 ◽  
Vol 26 (1) ◽  
pp. 41-59 ◽  
Author(s):  
G. Nikolov
Keyword(s):  
Existence And Uniqueness ◽  
Gaussian Quadrature ◽  
Quadrature Formulas ◽  
Gaussian Quadrature Formulas

Kronrod Extensions of Gaussian Quadratures with Multiple Nodes

2006 ◽  
Vol 6 (3) ◽  
pp. 291-305 ◽  
Author(s):  
G.V. Milovanović ◽  
M.M. Spalević ◽  
L.J. Galjak
Keyword(s):  
Error Bounds ◽  
Existence And Uniqueness ◽  
Gaussian Quadrature ◽  
Weight Functions ◽  
Quadrature Formulas ◽  
Gaussian Quadrature Formulas ◽  
Gaussian Quadratures ◽  
Explicit Expressions

Abstract In this paper, general real Kronrod extensions of Gaussian quadrature formulas with multiple nodes are introduced. A proof of their existence and uniqueness is given. In some cases, the explicit expressions of polynomials, whose zeros are the nodes of the considered quadratures, are determined. Very effective error bounds of the Gauss — Turán — Kronrod quadrature formulas, with Gori — Micchelli weight functions, for functions analytic on confocal ellipses, are derived.

Padé approximation and orthogonal polynomials

1974 ◽  
Vol 10 (2) ◽  
pp. 263-270 ◽  
Author(s):  
G.D. Allen ◽  
C.K. Chui ◽  
W.R. Madych ◽  
F.J. Narcowich ◽  
P.W. Smith
Keyword(s):  
Power Series ◽  
Variational Method ◽  
Orthogonal Polynomials ◽  
Gaussian Quadrature ◽  
Quadrature Formulas ◽  
The Past ◽  
Gaussian Quadrature Formulas ◽  
Poles And Zeros ◽  
Formal Power ◽  
Determinant Theory

By using a variational method, we study the structure of the Padé table for a formal power series. For series of Stieltjes, this method is employed to study the relations of the Padé approximants with orthogonal polynomials and gaussian quadrature formulas. Hence, we can study convergence, precise locations of poles and zeros, monotonicity, and so on, of these approximants. Our methods have nothing to do with determinant theory and the theory of continued fractions which were used extensively in the past.

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