Complex homogeneous splines on the torus

We construct functions M_α which are piecewise homogeneous polynomials on the (d+1)-dimensional torus U^d+1. These functions possess complete symmetry with respect to the independent variables. The symmetry and homogeneous relations for these functions are exploited to obtain a recurrence relation and explicit representations. Furthermore, we show that , where ω=e^12x/k, 0≤j_t≤k−1, are linearly independent. By restricting M_α to U^d, we obtain the complex analogue of polynomial box splines on a (d+1)-direction mesh on U^d, which is a multivariate analogue of B-splines on the circle studied by I.J. Schoenberg^[8].     

Hereditary optimal control problems: Numerical method based upon a padé approximation

In this paper, we consider a particular approximation scheme which can be used to solve hereditary optimal control problems. These problems are characterized by variables with a time-delayed argumentx(t – ). In our approximation scheme, we first replace the variable with an augmented statey(t) x(t - ). The two-sided Laplace transform ofy(t) is a product of the Laplace transform ofx(t) and an exponential factor. This factor is approximated by a first-order Padé approximation, and a differential relation fory(t) can be found. The transformed problem, without any time-delayed argument, can then be solved using a gradient algorithm in the usual way. Four problems are solved to illustrate the validity and usefulness of this technique.This research was supported in part by the National Aeronautics and Space Administration under NASA Grant NCC-2-106.