Mathematics – Functional Analysis
Scientific paper
2012-03-14
Mathematics
Functional Analysis
Submitted to Indagnationes Mathematicae, I. Gohberg Memorial issue
Scientific paper
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. Cauchy's 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 reconstruction from moments algorithms are sketched, remaining for a subsequent article to provide the numerical experiments and computational aspects. The essentials of entropy optimization, complex and harmonic analysis are recalled in some detail with the aim at facilitating the non-expert reader the access to the main results. Keywords: Fantappi\`e transform; entropy optimization; moment problem; tube domain; exponential transform
Budišić Marko
Putinar Mihai
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