WEIERSTRASS'S THEOREM

This chapter provides an overview of Weierstrass's theorem. The basis of the theory of approximation of functions of a real variable is a theorem discovered by Weierstrass that is of great importance in the development of the whole of mathematical analysis. For continuous functions of a single real variable defined on the finite segment [ a, b ], it asserts that for any function f ( x ) continuous on [ a, b ], there exists a sequence of ordinary polynomials which converges uniformly to f ( x ) on [ a, b ]. This remarkable constructive and characteristic property of continuous functions applies also to all functions of many variables that are continuous in the closed bounded region G of a given multi-dimensional space. The chapter also explains mean approximation of integrable functions by polynomials.