Shannon Integrals and Applications to Reimann-Hilbert Problems

We say that a function f can be represented by a Shannon integral if it can be written as the integral of sin(xs)/(xs), with respect to some finite measure. We develop a theory of Shannon integral representations for positive definite functions and show that under mild conditions f can be written as sum of two positive definite functions that constitute solutions to an additive Reimann_Hilbert problem on real line.