Poisson summation formula

Abstract

The Poisson summation (PS) formula describes the fundamental duality between periodization and decimation operators under the Fourier transform. In this chapter, the finite abelian group version of the PS formula is derived as a simple application of the character formulas of Chapter 3. The general case is equally simple to prove, but special care must be taken to provide conditions for convergence. The power of the PS formula is not so much with the formula itself, but rather that it highlights a construction that is basic to many derivations and algorithms in pure and applied mathematics and engineering. Common to these applications is the importance of periodization and of computing the Fourier transform of periodizations. In algebraic and analytic number theory this computation results in closed form expressions, the transformation equations for theta and zeta functions, and other important number theoretic special functions.