Vol. 15, No. 4, 1965

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|𝜀(z)|-closeness of approximation

Annette Sinclair

Vol. 15 (1965), No. 4, 1405–1413

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 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.

Mathematical Subject Classification
Primary: 30.70
Received: 11 March 1964
Revised: 10 August 1964
Published: 1 December 1965
Annette Sinclair