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Completeness of the Eigenfunctions of the Laplacian

We have referred often to the completeness of the eigenfunctions of the Laplacian in a lattice cell. This means that (in an appropriate sense) you can build any function periodic with respect to a lattice out of lattice waves.

As a small verification of this, we took the complex-valued identity function f(z) = z, and computed its Fourier coefficients with respect to a square lattice. The sum of thirty terms in the series produces the coloring the complex plane shown below.

[the identity function]

If the approximation were exact, we would expect to see perfect copies of the color wheel on a completely black background, like a quilt sewn together out of swatches just like the original figure. This figure is imperfect in two ways: there are subtle wiggles, as if the original wheel had acquired waves; that imperfection comes from the fact that we included only 30 terms. The second problem is more fundamental.

The periodic extension of the identity function is discontinuous. For example, the value of the extension at the point p should be both p and -p (the latter to match what happens in the interval from p to 3p, which is supposed to be a repeat of the piece from -p to p). Since our approximation is a continuous function, no matter how many terms we include, it cannot exactly match a discontinuous one. But from the picture, you can see that it's trying very hard to match both. The position corresponding to the number p is colored white, representing 0, halfway between p and -p. This is similar to a well-known property of Fourier series on an interval: at a jump discontinuity they converge to the average of the two one-sided limits.

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Communications in Visual Mathematics, vol 1, no 1, August 1998.
Copyright © 1998, The Mathematical Association of America. All rights reserved.
Created: 08 Jul 1998 --- Last modified: Sep 30, 2003 10:00:52 AM
Comments to: CVM@maa.org