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Reconstruction of functions from their triple correlations. (English) Zbl 1040.42009

In the first part of this paper, the authors consider functions \(f\in L^1(\mathbb{R})\). The \(k\)-deck of \(f\) with \(k= 2,3,\dots\) is a higher-order autocorrelation function defined by \[ N_f(x_1,\dots, x_{k-1})= \int f(t)\,f(t+ x_1)\cdots f(t+ x_{k-1})\,dt. \] Using Fourier technique, it is shown that in many cases a function \(f\geq 0\) can be reconstructed up to translations if the \(3\)-deck \(N_f\) is given. Negative results of this reconstruction problem are presented, too. Note that this problem is a generalization of the phase retrieval problem where one has to reconstruct \(f\) knowing only the modulus of its Fourier transform.
In the second part, the authors consider the corresponding discrete reconstruction problem for \(n\)-periodic sequences. Here discrete Fourier transforms are applied.

MSC:

42A99 Harmonic analysis in one variable
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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