A note on total degree polynomial optimization by Chebyshev grids

Variable degree polynomial splines are Chebyshev splines

Variable degree polynomial (VDP) splines have recently proved themselves as a valuable tool in obtaining shape preserving approximations. However, some usual properties which one would expect of a spline space in order to be useful in geometric modeling, do not follow easily from their definition. This includes total positivity (TP) and variation diminishing, but also constructive algorithms ba...

A Note on the Degree of Polynomial Approximation*

Let C be a rectifiable Jordan curve of the finite z plane. We shall say that a function f(z) belongs to the class Lip (C, j , a) if ƒ (z) is regular in the limited region bounded by C (which we shall call the interior of C), if f{z) is continuous in the corresponding closed region, and if the jth. derivative of f(z) is also continuous in this closed region and satisfies a Lipschitz condition wi...

Algebraic Degree of Polynomial Optimization

Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, we prove that the optimality conditions always hold on optimizers, and the coordinates of optimizers are algebraic functions of the coefficients of the input polynomials. We also give a general formula for the algebraic degree of the opti...

Note on a polynomial of

Chebyshev polynomial Kalman filter

A novel Gaussian state estimator named Chebyshev Polynomial Kalman Filter is proposed that exploits the exact and closed-form calculation of posterior moments for polynomial nonlinearities. An arbitrary nonlinear system is at first approximated via a Chebyshev polynomial series. By exploiting special properties of the Chebyshev polynomials, exact expressions for mean and variance are then provi...