Non-oscillating Paley-Wiener functions

Abstract

Abstract. A non-oscillating Paley-Wiener function is a real entire function f of exponential type belonging to L2 (R) and such that each derivative f('~), n = 0, 1, 2,..., has only a finite number of real zeros. It is established that the class of such functions is non-empty and contains functions of arbitrarily fast decay on tt allowed by the convergence of the logarithmic integral. It is shown that the Fourier transform of a non-oscillating Paley-Wiener function must be infinitely differentiable outside the origin. We also give close to best possible asymptotic (as n ~ oo) estimates of the number of real zeros of the n-th derivative of a function f of the class and the size of the smallest interval containing these zeros.