International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 63, Pages 3423-3434
On the asymptotic behavior of the second moment of the Fourier
transform of a random measure
1Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa (FCT/UNL), Quinta da Torre, Caparica 2829-516, Portugal
2Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa (CMAF/UL), Lisboa 1649-003, Portugal
Received 21 October 2002
Copyright © 2004 Manuel L. Esquível. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The behavior at infinity of the Fourier transform of the random
measures that appear in the theory of multiplicative chaos of
Mandelbrot, Peyrière, and Kahane is an area quite unexplored.
For context and further reference, we first present an overview
of this theory and then the result, which is the main objective of
this work, generalizing a result previously announced by Kahane. We establish an estimate for the asymptotic behavior of
the second moment of the Fourier transform of the limit random
measure in the theory of multiplicative chaos. After looking at the behavior at infinity of the Fourier
transform of some remarkable functions and measures, we prove a
formula essentially due to Frostman, involving the Riesz kernels.