On a property of entire characteristic functions of finite order with real zeros (Q1594391)

The authors continue their investigations of phase retrieval on l.c.a. groups started in [J. Theor. Probab. 8, No. 3, 717-725 (1995; Zbl 0837.60007)]. The problem is whether a given probability measure \(\mu\) on a l.c.a. group is determined uniquely (up to shifts and central symmetry) by \(|\widehat\mu|\). In the present paper the authors state without proof that for a measure \(\mu\in M^1(\mathbb{R})\) with \(\widehat\mu\) an entire function of finite order with real zeros, the measure \(\mu\times\nu\in M^1(\mathbb{R}\times G)\) is determined uniquely for any \(\nu\in M^1(G)\) for any l.c.a. group \(G\).