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.

If and

then .

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.

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