Convolution, differential equations, and entire functions of exponential type

Abstract:

The Whittaker-Shannon interpolation formula, or “cardinal series", is a special case of the more general linear integro-differential equation with constant complex coefficients $\Sigma _{n = 0}^\infty {a_n}{f^{(n)}}(z) = \smallint f(z - t)d\mu (t)$ where the integral is taken over the whole real line with respect to the measure $\mu$. In this study, I show that many of these equations provide representations for particular classes of entire functions of exponential type. That is, every function in the class satisfies the equation and conversely every solution of the equation is a member of the class of functions. When the measure in the convolution integral above is chosen to be discrete, a particular form of the above type of equation is an equation of periodicity $f(z) = f(z + \tau )$. Following an extensive treatment of the general equation written above, the study concludes by offering a generalization in terms of these convolution equations of a classical theorem in complex analysis concerning periodic entire functions.