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# Group Actions and Singular Martingales II, The Recognition Problem

Published:2004-04-01
Printed: Apr 2004
• Joseph Rosenblatt
• Michael Taylor
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## Abstract

We continue our investigation in [RST] of a martingale formed by picking a measurable set $A$ in a compact group $G$, taking random rotates of $A$, and considering measures of the resulting intersections, suitably normalized. Here we concentrate on the inverse problem of recognizing $A$ from a small amount of data from this martingale. This leads to problems in harmonic analysis on $G$, including an analysis of integrals of products of Gegenbauer polynomials.
 MSC Classifications: 43A77 - Analysis on general compact groups 60B15 - Probability measures on groups or semigroups, Fourier transforms, factorization 60G42 - Martingales with discrete parameter 42C10 - Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

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