ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 42, 41 Helmberg, Gilbert, Universität Innsbruck, Austria A corner-point Gibbs phenomenon for Fourier series in two dimensions. Let f be the function periodic with period 2π in x and y which extends the indicator function of the parallelogram

A = {(x, y) : 0 ≤ π, y/c ≤ x ≤ y/c + π} (0 ≠ c ∈ ℝ).

The partial sums S_{2M+1,2M=1}[f] of the Fourier series of f, evaluated at (\frac{ππ}{2M+1}, \frac{πy}{2M+1}), converge for M → ∞ to a sum of integrals of the functions

\frac{sin t}{t}, \frac{sin s}{s} \frac{sin t}{t}, \frac{cos s}{s} \frac{cos t}{t} over domains depending on x, y and c. This causes the partial sums to display near the point (0, 0) an overshoot (depending on c) of up to 37,4 % of half the jump size.