The Unitarity of the Fourier Transform by Ky Elasmi ’25 & Marcello Berger ’25

The Unitarity of the Fourier Transform: Generalizing Parseval’s Theorem through Functional Analysis by Ky Elasmi ’25 & Marcello Berger ’25, Monday September 23, 1:00 – 1:50pm, NSB 113, Wachenheim, Mathematics Colloquium

One formulation of Parseval’s theorem, a classic result in physics, states that the square of the average of a function over a period is equal to the sum of its Fourier coefficients squared. When generalized using the techniques of functional analysis, this result is known as Plancherel’s theorem, one formulation of which is that the Fourier transform of a function defines a unique unitary map with respect to the function space L^2(R). We will introduce some of the tools of functional analysis that are necessary to state the theorem and then give a short proof.