Non-oscillating Paley-Wiener functions

A non-oscillating Paley-Wiener function is a real entire functionf of exponential type belonging toL^sub 2^(R) and such that each derivativef^sup (n)^,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 onR 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 (asn[arrow right]∞) estimates of the number of real zeros of then-th derivative of a functionf of the class and the size of the smallest interval containing these zeros.[PUBLICATION ABSTRACT]