294 research outputs found
Discrete Fourier Analysis and Chebyshev Polynomials with Group
The discrete Fourier analysis on the
-- triangle is deduced from the
corresponding results on the regular hexagon by considering functions invariant
under the group , which leads to the definition of four families
generalized Chebyshev polynomials. The study of these polynomials leads to a
Sturm-Liouville eigenvalue problem that contains two parameters, whose
solutions are analogues of the Jacobi polynomials. Under a concept of
-degree and by introducing a new ordering among monomials, these polynomials
are shown to share properties of the ordinary orthogonal polynomials. In
particular, their common zeros generate cubature rules of Gauss type
Minimal Cubature rules and polynomial interpolation in two variables
Minimal cubature rules of degree for the weight functions
W_{\a,\b,\pm \frac12}(x,y) = |x+y|^{2\a+1} |x-y|^{2\b+1} ((1-x^2)(1-y^2))^{\pm
\frac12} on are constructed explicitly and are shown to be closed
related to the Gaussian cubature rules in a domain bounded by two lines and a
parabola. Lagrange interpolation polynomials on the nodes of these cubature
rules are constructed and their Lebesgue constants are determined.Comment: 23 page
Some Results on the Complexity of Numerical Integration
This is a survey (21 pages, 124 references) written for the MCQMC 2014
conference in Leuven, April 2014. We start with the seminal paper of Bakhvalov
(1959) and end with new results on the curse of dimension and on the complexity
of oscillatory integrals. Some small errors of earlier versions are corrected
- …