Conditioning moments of singular measures for entropy optimization. I

TL;DR

This paper introduces a nonlinear exponential transform using the Fantappiè transform to enable entropy optimization for reconstructing measures, including singular ones, from moments in higher dimensions.

Contribution

It extends the classical Cauchy transform approach to higher dimensions with the Fantappiè transform, allowing entropy-based methods to handle singular measures.

Findings

Transform enables entropy optimization for singular measures.

Provides algorithms for moment-based reconstruction.

Lays groundwork for numerical experiments in future work.

Abstract

In order to process a potential moment sequence by the entropy optimization method one has to be assured that the original measure is absolutely continuous with respect to Lebesgue measure. We propose a non-linear exponential transform of the moment sequence of any measure, including singular ones, so that the entropy optimization method can still be used in the reconstruction or approximation of the original. The Cauchy transform in one variable, used for this very purpose in a classical context by A.\ A.\ Markov and followers, is replaced in higher dimensions by the Fantappi\`{e} transform. Several algorithms for reconstruction from moments are sketched, while we intend to provide the numerical experiments and computational aspects in a subsequent article. The essentials of complex analysis, harmonic analysis, and entropy optimization are recalled in some detail, with the goal of making the main results more accessible to non-expert readers. Keywords: Fantappi\`e transform; entropy optimization; moment problem; tube domain; exponential transform