In number theory, the Poission summation formula is used in proving functional equations of theta functions (assuring that they are modular forms) and then via this proving functional equations for zeta- and L-functions. What is happening in both cases is that there is some fundamental duality (often self-duality) that is being expoited.

In coding theory there is also the MacWilliams identities which follow from a discrete version of Poisson summation. In some sense they are equivalent to certain restricted classes of functional equations for theta functions. Again duality and self-duality are behind these symmetries.

In number theory, the Poisson summation formula is used in proving functional equations of theta functions (assuring that they are modular forms) and then via this proving functional equations for zeta- and L-functions. What is happening in both cases is that there is some fundamental duality (often self-duality) that is being exploited.

In coding theory there is also the MacWilliams identities which follow from a discrete version of Poisson summation. In some sense they are equivalent to certain restricted classes of functional equations for theta functions. Again duality and self-duality are behind these symmetries.