## 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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