For a given function F(Q)
defined for Q ∈ S, the connection between these questions is investigated: (1) For
arbitrary 𝜖 > 0 (or possibly {𝜖_{i}}, where 𝜖_{i} corresponds to a compoment S_{i} of S), does
there exist a function f of a specified class ℱ such that sup_{Q𝜖s}F(Q) −f(Q) < 𝜖 on
S (or 𝜖_{i} on S_{i})?; (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 Q_{β} there is a positive integer n_{β} such that
𝜖(Q)∕(Q − Q_{β})^{n}β is bounded from zero in a deleted neighborhood of Q_{β}. 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.
