The discrete Paley-Wiener theorem

Abstract

then the function is called discrete entire (d.e.). If f is written in terms of its real and imaginary parts, Eq. (1) s h ows the initial analogy between regular analytic functions and d.e. functions, since it is equivalent to a pair of partial difference equations analogous to the Cauchy-Riemann equations. Discrete analytic functions were first developed by Ferrand [3] and then Duffin [2], and many similarities between d.e. functions and entire functions have been established recently by Zeilberger and others [5-71. In particular, in [Sj, Zeilberger established one form of a discrete Paley-Wiener theorem for d.e. functions of exponential growth, where a function is said to be of exponential growth (R, 5’) if there exists a constant C such that if(m, n)l < CRI”z’Slni for every pair of integers (m, n). In connecting d.e. functions of exponential growth with the Paley-Wiener theorem, two different approaches could be taken. The approach used by Zeilberger is to concentrate on the discrete nature of the function and make a connection with Fourier series, A different approach, which is explored in this paper, is to make a closer analogy with entire functions of exponential type themselves, and use a discrete analog of a Fourier transform along the real line. One distinct advantage of this approach is that the exponential growth of the function can be connected with the domain of integration in a manner that is strikingly analogous to the way that the exponential type of an entire function is connected with the domain of integration of its transform. In this paper, the initial emphasis will be on developing properties of the discrete exponential functions so as to approach the theory of transforms using the discrete exponentials. Then a very intriguing bilinear transformation will be used to connect this theory with Zeilbergcr’s result, achieving a direct analogy with the Paley-Wiener theorem. Finally, an application of the theorem 172