The Levin-Golovin theorem for sobolev spaces

We obtain sufficient conditions for the basis property of the family of exponentials in the Sobolev spaceH ^s (0,a) in terms of the behavior of the generating function, which is an entire function of exponential type with zeros λ _n . This result is a generalization of the Levin-Golovin theorem on the basis property of the family of exponentials generated by a function of sine type inL ²(0,a). We apply the theorem obtained to the interpolation of entire functions of exponential type; this application is a generalization of the Kotel’nikov-Shannon theorem in signal theory. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 163–172, August, 2000.