Entire [formula omitted]-functions of exponential type

Abstract

We characterize the entire functions of exponential type, whose restriction to the real line is in Lp, in different ways: by the usual classical Paley–Wiener growth estimates in the complex plane, by Bernstein inequalities using derivatives or differences, by Lp-growth properties of iterated derivatives or differences, and by support properties of the Fourier image. We also establish Paley–Wiener theorems for the Fourier series of a function on the circle.