361 research outputs found
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
Bivariate Lagrange interpolation at the Padua points: the ideal theory approach
Padua points is a family of points on the square given by explicit
formulas that admits unique Lagrange interpolation by bivariate polynomials.
The interpolation polynomials and cubature formulas based on the Padua points
are studied from an ideal theoretic point of view, which leads to the discovery
of a compact formula for the interpolation polynomials. The convergence
of the interpolation polynomials is also studied.Comment: 11 page
Numerical hyperinterpolation over nonstandard planar regions
We discuss an algorithm (implemented in Matlab) that computes numerically total-degree bivariate orthogonal polynomials (OPs) given an algebraic cubature formula with positive weights, and constructs the orthogonal projection (hyperinterpolation) of a function sampled at the cubature nodes. The method is applicable to nonstandard regions where OPs are not known analytically, for example convex and concave polygons, or circular sections such as sectors, lenses and lunes
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
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