I stumbled upon the 2012 AMS release Complex Proofs of Real Theorems by Peter Lax and Lawrence Zalcman while browsing online a few days ago, and I finally got around to checking out the library’s copy today. It’s a short publication–part of the AMS University Lecture Series–of around 90 pages focused on (particularly elegant) applications of complex analysis to proving statements about functions of a real variable. I have only looked at a small fraction of the text, which is not saying much, but one result concerning the uniqueness of the Fourier transform on stood out for me.
The argument in the following proof of Fourier uniqueness was originally given D.J. Newman, Fourier uniqueness via complex variables, Amer. Math. Monthly, 81 (1974), 379-380. Proof:Suppose for all . Fix and define by
We now extend the domain of to the complex plane. For , define by
and if , define
It is clear that this defines a continuation of , also denoted by , which is bounded on . Moreover, is continuous on as a consequence of the Lebesgue dominated convergence theorem. I claim that is analytic in . By Morera’s theorem, it suffices to show that for any rectangle with positively oriented boundary. Without loss of generality, we can assume that (the argument for is completely analogous). Since is compact, , and therefore
we can apply Fubini’s theorem to obtain
, since is an analytic function of for fixed . Since is a bounded entire function, must be constant by Liouville’s theorem. I claim that . Indeed, by the Lebesgue dominated convergence theorem,
Since was arbtrary, we obtain
It follows from the Lebesgue differentiation theorem that a.e.
It is worth remarking that we invoked non-trivial results of the Lebesgue theory of integration through the proof in the forms of the dominated convergence theorem, Fubini’s theorem, and the Lebesgue differentiation theorem.