From a Random Polygon in $$\mathbb {R}^m$$ R m to an Ellipse: A Fourier Analysis of Iterated Circular Convolutions

Abstract

We consider random, skew, N-gons in $$\mathbb {R}^m$$ , having zero centroids. We investigate a normalized, midpoint averaging transformation, repeatedly applied to the original polygon, which we identify with a normalized circular convolution modulo N. The paper has two parts: the first part investigates the case of a repeated circular convolution applied to m-dimensional polygons without normalization, and the second part investigates the more difficult case with normalization. We prove that if the normalized convolution is repeated sufficiently many times, then the limiting shape is that of a planar affine-regular polygon, inscribed in an ellipse, embedded in a 2-dimensional subspace of $$\mathbb {R}^m$$ , spanned by a basis specified by the principal harmonic components of the discrete Fourier transforms of the initial vectors of vertex coordinates. We provide a compact analytical description of the limiting elliptical form, as well as the limiting plane, using the language of circular convolutions and the tools of discrete Fourier analysis. In this paper, we generalize to the higher-dimensional case, previous results obtained in the planar case.