On approximating two distributions from a single complex-valued function

TL;DR

This paper presents a method to approximate two different probability distributions simultaneously using a single complex-valued function and its Fourier transform, with proven accuracy under certain conditions.

Contribution

It introduces a novel algorithm for constructing a complex function that approximates two distributions simultaneously, expanding the tools for distribution approximation.

Findings

The algorithm can approximate various pairs of distributions with specified accuracy.

The method is applicable under certain regularity conditions of the distributions.

Examples demonstrate the effectiveness of the approximation technique.

Abstract

We consider the problem of approximating two, possibly unrelated probability distributions from a single complex-valued function $\psi$ 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 $\psi$ and provide examples of approximating several pairs of distributions using the algorithm.