## A graphical case study

At a given time t = a, the fundamental solution to the 2D wave equation is given by

R(x,y) = C (a^{2} – x^{2} – y^{2})^{-1/2}

for x^{2} + y^{2} less than a^{2}, and R(x,y) = 0 for x^{2} + y^{2} greater than a^{2}.

In the figures below we graph partial sums of the Fourier series of such a function, on the torus, of period 2 pi by 2 pi. We take a = 1.5. We use spherical summation of the Fourier series, over (j,k) with j^{2} + k^{2} less than or equal to N^{2}. In the first picture, N = 4 pi.

The spike in the center of the graph is the manifestation of the Pinsky phenomenon. This phenomenon was discovered by M. Pinsky for the spherical Fourier inversion on 3D Euclidean space, for a piecewise smooth function with jump across a sphere. The function R has just the right singularity for this phenomenon to arise in 2D.

The downward spike will retract and proceed to point upward as N is increased by (2/3) pi. Then the spike will turn back down. Its amplitude varies in this fashion, with period (4/3) pi, the spike becoming sharper and sharper as N increases.

In the next 5 pictures, we let N increase by even increments from 4 pi to (4+2/3) pi.

We note that the behavior of the approximation outside the support of R has become a bit choppier in the second picture than in the first. In the third and fourth pictures, it will be even choppier.

This choppiness does not arise for the corresponding spherical Fourier inversion on 2D Euclidean space. Its appearance reflects a difference between the behavior of Fourier inversion on the torus and on Euclidean space.

A partial explanation for this choppiness rests in the poor convergence one sees in an effort to apply the Poisson summation formula to analyze Fourier inversion on the torus in terms of Fourier inversion on Euclidean space.

However, for our function R, this choppiness “magically” clears up whenever N is an integral multiple of (2/3) pi. I call this

### Serendipitous Fourier Inversion.

A full analysis of this serendipity involves somewhat delicate estimates. This is presented in a recent paper, with title as given above.