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VOL. 15 | 1987 A counterexample to localization for multiple fourier series
John Price, Larry Shepp

Editor(s) Michael Cowling, Christopher Meaney, William Moran


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 $f$.


Published: 1 January 1987
First available in Project Euclid: 18 November 2014

zbMATH: 0633.42006
MathSciNet: MR935603

Rights: Copyright © 1987, Centre for Mathematical Analysis, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.


Vol. 15 • 1 January 1987
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