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CONVERGENCE ALMOST EVERYWHERE OF CERTAIN PARTIAL SUMS OF FOURIER INTEGRALS

Published online by Cambridge University Press:  20 March 2003

ANTHONY CARBERY ,
DIRK GORGES ,
GIANFRANCO MARLETTA  and
CHRISTOPH THIELE
ANTHONY CARBERY
Affiliation:
The University of Edinburgh, James Clerk Maxwell Building, King's Buildings, Mayfield Road, Edinburgh EH9 3JZ [email protected]
DIRK GORGES
Affiliation:
c'o Alwine Gorges, Am Kreuzchen 20, 54292 Trier, [email protected]
GIANFRANCO MARLETTA
Affiliation:
34 Springfield Road, Brighton BN1 6DA [email protected]
CHRISTOPH THIELE
Affiliation:
Department of Mathematics, University of California, Los Angeles, CA 90055-1555, [email protected]
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Abstract

Suppose that $R$ goes to infinity through a second-order lacunary set. Let $S_R$ denote the $R$th spherical partial inverse Fourier integral on ${\rm I\!R}^d$. Then $S_R f$ converges almost everywhere to $f$, provided that $f$ satisfies \[ \int \widehat{f}(\xi)\log\log(8+|\xi|)^2\,d\xi < \infty. \]

Keywords

42B1542B2542B08
Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 35 , Issue 2 , March 2003 , pp. 225 - 228
Copyright
© The London Mathematical Society 2003

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Footnotes

The first and second authors were supported by the European Commission TMR Network ‘Harmonic Analysis’. The third author was supported by EPSRC grant GR/J65594.