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Let $\gamma$ be a smooth closed curve in the plane and let $(x(t), y(t))$ be a parametrization. The functions $x(t)$ and $y(t)$ are smooth and periodic, so each has a uniformly convergent Fourier series. I am interested in understanding what geometric properties of $\gamma$ can be conveniently expressed in terms of the Fourier coefficients. Can one say anything about the centroid? Curvature? What can one say about the curves obtained by taking partial sums of the Fourier series other than that they converge to $\gamma$?

The only result in this direction that I know about is the famous proof of the isoperimetric inequality using Fourier series. The isoperimetric inequality is expressed as an inequality between one variable integrals using Green's theorem, and the integral inequality is converted to a straightforward inequality between Fourier coefficients using Parseval's identity. I would be very interested if there are proofs of other theorems along these lines.

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see: Harmonic representation of closed curves (1992)

The centroid is given by the zeroth order Fourier coefficient, the two first harmonics give the orientation and scale of an elliptic approximation to the curve.

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There are two recent and very comprehensive monographs on this subject:

Groemer: Geometric applications of Fourier series


Koldobsky: Fourier analysis in convex geometry

which might interest you.

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