On the oscillation of differential transforms. II. Characteristic series of boundary value problems

Abstract

1. Formulation of problem. G. Polya and N. Wiener [2](i) have recently made important contributions to the S. Bernstein problem concerning the relation between the frequency of oscillation of derivatives of high order and the analytic character of the function. Assumingf(x) of period 2w and denoting the number of sign changes of f(k)(x) in the period by Nk, they show that restrictions in the rate of growth of Nk when k-* Co, imply that the high frequency terms in the Fourier series of f(x) have small amplitudes. In particular, if Nk is bounded, Nk <-N for all k, then the high frequency terms are entirely missing and f(x) reduces to a trigonometric polynomial of degree at most N/2. Conversely, if f(x) is a trigonometric polynomial of degree K, then Nk = 2K for all large k. Their results are less precise when Nk is unbounded. While it is likely that Nk=O(k) is necessary and.sufficient for analyticity of f(x), this has not yet been proved, and the best they could do was to show that Nk=o(kl2) implies thatf(x) is an entire function. For these and similar questions G. Szego has devised a new method of attack, presented in the first paper of this.series [4]. This method showed itself capable of giving more precise information when Nk is unbounded. In particular, Szego could show that Nk<k(log k)-' implies thatf(x) is entire. The present paper is also closely related to the paper of Polya and Wiener, but proceeds in a different direction. We aim to preserve the essence of -the methods developed by these writers and to apply them to a wider range of problems. There are several features in the investigation of Polya and Wiener which suggest possible generalizations, in particular, the class of functions considered and the operations applied to them. Let T be an operator which takes functions f(x) of a certain class F into functions of the same class. Any function u(x) of F such that Tu(x) =Xu(x) will be called a characteristic function of T corresponding to the characteristic value X and any formal series Zflu,(x) will be called a characteristic series of T if its terms are characteristic functions. In this terminology we can describe the investigation of Polya and Wiener