## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- Miniconference on Harmonic Analysis. M Cowling, C Meaney, and W Moran, eds. Proceedings of the Centre for Mathematical Analysis, v. 15. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1987), 201 - 207

### 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

**Permanent link to this document**

https://projecteuclid.org/
euclid.pcma/1416336468

**Mathematical Reviews number (MathSciNet)**

MR935603

**Zentralblatt MATH identifier**

0633.42006

#### Citation

Price, John; Shepp, Larry. A counterexample to localization for multiple fourier series. Miniconference on Harmonic Analysis, 201--207, Centre for Mathematical Analysis, The Australian National University, Canberra AUS, 1987. https://projecteuclid.org/euclid.pcma/1416336468