The Weierstrass Approximation Theorem on Linear Varieties: Poly- nomial Lyapunov Functionals for Delayed Systems

In 1885, Weierstrass first published a result [16] showing that real-valued polynomials can be used to approximate any continuous function on a compact interval to arbitrary accuracy with respect to the supremum norm. Various generalizations of the Weierstrass approximation theorem have focused on generalized mappings [15] and alternate topologies [4]. More recently, the Weierstrass theorem has found applications in numerical computation due to the ease with which polynomial functions are parameterized and evaluated. In this paper, we reexamine the Weierstrass theorem from the relatively new perspective of polynomial optimization. These problems consider optimization over C(X), the Banach space of continuous functions on X where X ⊂ R is compact. The structure of the problem is often a special case of