What is poisson summation formula?

The Poisson summation formula is a mathematical technique that establishes the link between the discrete and the continuous Fourier transform. It states that the sum of a function defined on a lattice can be expressed in terms of the integral of its Fourier transform over the dual lattice.

In other words, if we have a function f(x) defined on a lattice Z, the Poisson summation formula relates the sum of this function over the lattice to the integral of its Fourier transform over the dual lattice Z*.

The Poisson summation formula has many applications in mathematics, physics, and engineering, such as signal processing, number theory, partial differential equations, and quantum mechanics. It is also used in the study of modular forms, which are a class of functions playing a fundamental role in many areas of mathematics and physics.

The Poisson summation formula has many different forms and generalizations, depending on the specific setting in which it is used. Some of the most well-known versions include the discrete Poisson summation formula, the continuous Poisson summation formula, the Poisson summation formula for general lattices, and the Poisson summation formula in higher dimensions.