## Proceedings of the Centre for Mathematics and its Applications

### A counterexample to localization for multiple fourier series

#### Abstract

An explicit description is given of a real-valued function $f$ on 2 $[- \pi, \pi]$ which is zero in a neighbourhood of 0 but for which the square partial Fourier sums $S_n f$ satisfy $lim$ $sup_n S_n f(0,0) = \infty$. Furthermore, the function is infinitely differentiable everywhere except along the y-axis where it is continuous. Also its support is contained in a square at dis·tance $\pi/2$ from 0 and the square may be chosen to have arbitrarily small sides. Finally, neither of the axes intersect the interior of the support of $f$.

#### Article information

Dates
First available in Project Euclid: 18 November 2014