Extensions of the Poisson Summation Formula

The classical Poisson summation formula (1.1) and the corresponding distributional formula (1.2) have found extensive applications in various scientific fields. However, they are not universally valid. For instance, if φ( x ) is a smooth function, the left-hand side of (1.1) is generally divergent. Even when both sides of (1.1) converge absolutely, they may do so to different numbers. Indeed, in Example 3 we are faced with the embarrassing situation where the series on the left-hand side of (1.1) converges for Re s > 1 while that on the right-hand side converges only for Re s < 0. Our aim is to extend formulas (1.1) and (1.2) with the help of some new results in distributional theory. For instance, the evaluation of the distribution with zero mean as given by (3.1) at a test function φ( x ) yields the relation ∑ ∞ −∞ φ( k ) − ∫ ∞ −∞ φ( x ) dx. Both the series and the integral in this expression are generally divergent. The concept of the Cesàro limit is then used to obtain the finite difference of these two terms. Thereafter we extend the analysis to higher dimensions. Various innovative examples are presented to illustrate these concepts.