Finite type functions as limits of exponential sums (Q1378979)

The author proves that for a function \(f\:\mathbb C\rightarrow\mathbb C\) the following conditions are equivalent: (i) \(f\) is an entire function \(f\) such that \(|f(x+iy)|\leq M\exp(|y|)\) for \(x+iy\in\mathbb C\); (ii) there exists a sequence \((f_{n})_{n=1}^\infty\) of finite exponential sums \(\sum_{\lambda\in[-1,1]} C_\lambda\exp(i\lambda x)\) such that \(f_{n}\longrightarrow f\) locally uniformly in \(\mathbb R\) and \(|f_{n}(x)|\leq M\) for \(x\in\mathbb R\), \(n=1,2,\dots\).