Double Fourier series of functions with simple singularities

A graphical case study

This paper investigates the spherical summation of the Fourier series of two functions on the 2-dimensional torus. The first is f, the characteristic function of a disk, and the second is R, the value at a specified positive time of the fundamental solution to the wave equation. The first function is bounded, with a jump discontinuity across a circle, while the second is supported in a disk but blows up at the boundary like the -1/2 power of the distance from the boundary.

On this web page we present some of the graphs of these partial sums of Fourier series. More graphs are given in the paper itself.

Our first three pictures are of partial sums of the double Fourier series of the function f. We sum over lattice points (j,k) for which j2 + k2 is less than or equal to N2. In the first graph we take N = 10.

In the second graph we take N = 15. The convergence pointwise to f of its Fourier series was proven by Brandolini and Colzani [1]. In Taylor's paper [4] there appear estimates that guarantee that the convergence is slower at the center than elsewhere (except at the singular set of f).

In the third graph we take N = 25. The relatively slow convergence at the center is not so clearly visible here, but the Gibbs phenomenon is becoming apparent. The behavior of the approximation outside the disk is somewhat choppier than it would be for Fourier inversion on 2-dimensional Euclidean space, in place of the torus. More on that on the next page.

The relatively slow convergence of the Fourier series of f at the center is a pale shade of the Pinsky phenomenon, which becomes marked for Fourier inversion for the characteristic function of a ball in 3D. This phenomenon was analyzed for Fourier inversion in 3D Euclidean space in [2]. In [3] there was an alternative analysis in terms of focusing of waves. The pointwise behavior of the Fourier series of such a function on the 3-dimensional torus was obtained in [4]. This analysis was extended in a number of ways in [5] and [6].

The appropriate analogue of the Pinsky phenomenon for Fourier inversion in 2 dimensions is provided by R, which is singular on the boundary of a disk with a blow up like the -1/2 power of the distance to the boundary. Click here to see pictures of partial sums of the Fourier series of R, and further discussion.

References

  1. L. Brandolini and L. Colzani, Localization and convergence of eigenfunction expansions, J. Fourier Anal. 5 (1999), 431-447.
  2. M. Pinsky, Pointwise Fourier inversion and related eigenfunction expansions, Comm. Pure Appl. Math. 47 (1994), 653-681.
  3. M. Pinsky and M. Taylor, Pointwise Fourier inversion: a wave equation approach, J. Fourier Anal. 3 (1997), 647-703.
  4. M. Taylor, Pointwise Fourier inversion on tori and other compact manifolds, J. Fourier Anal. 5 (1999), 449-463.
  5. M. Taylor, Eigenfunction expansions and the Pinsky phenomenon on compact manifolds, J. Fourier Anal. 7 (2001), 507-522.
  6. M. Taylor, The Gibbs phenomenon, the Pinsky phenomenon, and variants for eigenfunction expansions, Comm. PDE 27 (2002), 179-196.