On the Nontriviality of Restricted Range Polynomial Approximation
Darell J. Johnson · SIAM Journal on Numerical Analysis · 1975
In this note, necessary and sufficient conditions (on the region within which one wishes to perform restricted range polynomial approximation) are given in order that the set of polynomials lying in the given specified region be nonempty and not a singleton set. The proof is constructive and allows one to construct a polynomial lying in a given region (if such is possible), and hence also gives one a degree of polynomials beyond which the restricted range polynomial approximation problem does not trivialize. The result is then applied to derive a Weierstrass theorem for the restricted range polynomial approximation problem.