For a given function F(Q)
defined for Q ∈ S, the connection between these questions is investigated: (1) For
arbitrary 𝜖 > 0 (or possibly {𝜖i}, where 𝜖i corresponds to a compoment Si of S), does
there exist a function f of a specified class ℱ such that supQ𝜖s|F(Q) −f(Q)| < 𝜖 on
S (or 𝜖i on Si)?; (2) Given an admissible function 𝜖(Q), does there exist a function
f ∈ℱ such that |F(Q) −f(Q)|≦|𝜖(Q)| on S? A continuous function 𝜖(Q) defined on
S is admissible if for each zero Qβ there is a positive integer nβ such that
𝜖(Q)∕(Q − Qβ)nβ is bounded from zero in a deleted neighborhood of Qβ. A
typical result is: Corresponding to any F(z) analytic on a closed bounded
set S and to any admissible 𝜖(z), there exists a rational function r(z) with
its poles on a certain preassigned set such that |F(z) − r(z)|≦|𝜖(z)| on
S.
