The theory of multi-dimensional polynomial approximation

AbstractWe consider the problem of polynomial approximation to a real valued functionf defined on a compact set $$\mathbb{X}$$ . 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 $$\mathcal{L}^{q} $$ for 0<q≤∞.We have found that the space of polynomials ℙ on a compact setX induces a semimetric $$\mu _{\mathbb{P},\mathbb{X}} $$ which encapsulates the local structure of ℙ. Any semimetric ρ equivalent to $$\mu _{\mathbb{P},\mathbb{X}} $$ suffices for the rough theory presented here. Many examples of sets $$\mathbb{X} \subset \mathbb{R}^N $$ and their metrics are presented.