Went to an interesting talk today by Dr Pierre Portal about the interpretation of Fourier analysis within a wider mathematical analysis context. Although much of the discussion about Fourier type operator equations in spectral theory went over my head I did benefit from being reminded of two interesting examples of Fourier analysis in two quite different mathematical proofs: (1) the isoperimetric inequality (relating the length of a closed curve to the area it encloses*); and (2) Weyl’s equidistribution theorem (that the numbers 1q, 2q, 3q, … mod 1 are distributed uniformly on the circle for irrational q). The first proof uses standard results for the Fourier transform of distances in the L2 norm and the preservation of the dot product under Fourier transforms; while the second uses the approximation of a periodic indicator function by a continuous function. The latter in particular recalls proof strategies from probability theory in the space of bounded continuous functions, C[0,1], e.g. for the convergence of partial sums to Wiener measure.
Anyway, all good stuff and one day I hope to find the time to work through the above-mentioned proofs in detail!
*on seeing this inequality, which is strict for shapes other than circles, Emilie (who was sitting next to me) made the following concise observation: “no shit!” 🙂