<P> It has a variety of useful forms that are derived from the basic one by application of the Fourier transform's scaling and time - shifting properties . The formula has applications in engineering, physics, and number theory . The frequency - domain dual of the standard Poisson summation formula is also called the discrete - time Fourier transform . </P> <P> Poisson summation is generally associated with the physics of periodic media, such as heat conduction on a circle . The fundamental solution of the heat equation on a circle is called a theta function . It is used in number theory to prove the transformation properties of theta functions, which turn out to be a type of modular form, and it is connected more generally to the theory of automorphic forms where it appears on one side of the Selberg trace formula . </P> <P> Suppose f (x) is a differentiable function, and both f and its derivative f ′ are integrable . Then the Fourier transform of the derivative is given by </P> <Dl> <Dd> f ′ ^ (ξ) = 2 π i ξ f ^ (ξ). (\ displaystyle (\ widehat (f' \;)) (\ xi) = 2 \ pi i \ xi (\ hat (f)) (\ xi).) </Dd> </Dl>

Fourier transform of the square of a function