The theory of multi-dimensional polynomial approximation

We consider the problem of polynomial approximation to a real valued function f defined on a compact set X. An approximation theorem is proven in terms of the newly defined modulus of approximation. It is shown to imply a multidimensional Jackson type theorem which is stronger than previously known results even for the interval [-1, 1]. A strong multidimensional Bernstein type inverse theorem is also proven. We allow quite general approximation quasi-norms including L(q) for 0 < q less than or equal to infinity. We have found that the space of polynomials P on a compact set X induces a semimetric mu(P,X) which encapsulates the local structure of Ip. Any semimetric rho equivalent to mu(P,X) Suffices for the rough theory presented here. Many examples of sets X subset of R(N) and their metrics are presented.