On Oscillation of Functions with Bounded Spectral Band

In this paper, we consider extremal oscillatory properties of functions with bounded spectrum, i.e., with bounded support (in the sense of distributions) of the Fourier transform. For such functions f, we give criteria of extendability of }f} from the real axis to a function F on the complex plane with derivatives F ^(m) having no real zeros and without enlarging the width of spectrum. In particular, we give examples of functions $f$ from the real Paley–Wiener space such that every function f ^(m), m=0, 1,..., has a finite number of real zeros.