Positive spline operators and orthogonal splines

Orthogonal polynomials and diffusions operators

Generalizing the work of [5, 41], we give a general solution to the following problem: describe the triplets (Ω, g, μ) where g = (g(x)) is the (co)metric associated to the symmetric second order differential operator L(f) = 1 ρ ∑ ij ∂i(g ρ∂jf), defined on a domain Ω of R and such that L is expandable on a basis of orthogonal polynomials on L(μ), and dμ = ρ(x)dx is some admissible measure. Up to...

On Mixed Interpolating-Smoothing Splines and the ν-Spline

Algorithms on cone spline surfaces and spatial osculating arc splines

Developable surfaces are of considerable importance to many industry applications, e.g. sheet metal forming processes. The objective of this paper is to provide algorithms on the approximation of developable surfaces with pieces of right circular cones. Special emphasis is devoted to practical choices of free parameters and to error estimation. Furthermore, a new algorithm for the approximation...

Splines with non positive kernels

Non parametric regression methods can be presented in two main clusters. The one of smoothing splines methods requiring positive kernels and the other one known as Nonparametric Kernel Regression allowing the use of non positive kernels such as the Epanechnikov kernel. We propose a generalization of the smoothing spline method to include kernels which are still symmetric but not positive semi d...

On Positive and Copositive Polynomial and Spline

For a function f 2 L p ?1; 1], 0 < p < 1, with nitely many sign changes, we construct a sequence of polynomials P n 2 n which are copositive with f and such that kf ? P n k p C! ' (f; (n + 1) ?1) p , where ! ' (f; t) p denotes the Ditzian-Totik modulus of continuity in L p metric. It was shown by S. P. Zhou that this estimate is exact in the sense that if f has at least one sign change, then ! ...