Bernstein-type characterization of entire functions - Dovgoshey, O.A.; Prestin, J.; Shevchuk, I.O. (Reports of the National Academy of Sciences of Ukraine, 2023)
Let ε be the set of all entire functions on the complex plane C. Let us consider the class XE of all complex Banach spaces X such that X ⊇ ε . For (X, ⎥⎥ ⋅ ⎥⎥)∈XE and g ∈X we write En, X (g ) = inf {⎥⎥ g − p⎥⎥: p∈Πn }, where Πn is the set of all polynomials with degree at most n. We describe all X ∈XE for which the relation lim n→∞ (En, X( g ))1/n = 0 holds if and only if g ∈ ε.