On approximating two distributions from a single complex-valued function

We consider the problem of approximating two, possibly unrelated probability distributions from a single complex-valued function $ψ$ and its Fourier transform. We show that this problem always has a solution within a specified degree of accuracy, provided the distributions satisfy the necessary regularity conditions. We describe the algorithm and construction of $ψ$ and provide examples of approximating several pairs of distributions using the algorithm.

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