The strong uniqueness constant in complex approximation

Abstract

Let f(x) belong to the set C(B) of continuous functions, possibly complex valued, on a compact subset, B, of a finite dimensional euclidean space. Let V be a finite dimensional subspace of C(B). v₀ ε V is a strongly unique best approximation to f if (1) ‖f − v‖ ≥ ‖f − v₀‖+γ ‖v − v₀∥ holds for some γ(f, B, V) > 0 and all v ε V. The largest γ (= γ*) such that (1) holds is the strong uniqueness constant for f. The strong uniqueness constant has previously been determined in essentially one case, namely, the approximation of a monomial by lower degree polynomials in the real case. When f(z)=zⁿ, B is the closed unit disc in the complex plane and V is the set of polynomials of degree at most n−1, γ*=1/n. We show that this fact is a trivial consequence of a simple result of Szász (1917). Some related results and problems are also discussed.