Polynomial approximation of $C^M$ functions by means of boundary values and applications: A survey

We collect classical and more recent results on polynomial approximation of sufficiently regular real functions defined in bounded closed intervals by means of boundary values only. The problem is considered from the point of view of the existence of explicit formulas, interpolation to boundary data, bounds for the remainder and convergence of the polynomial series. Applications to some problems of numerical analysis are pointed out, such as nonlinear equations, numerical differentiation and integration formulas, special associated differential boundary value problems. Some polynomial expansions for smooth enough functions defined in rectangles or in triangles of ${\mathbb R}^2$ as well as in cuboids or in tetrahedrons in ${\mathbb R}^3$ and their applications are also discussed.