From the introduction: The moments of a measure \(\mu\) supported by the interval \([0,1]\) correspond, after an exponential change of variables, to the values of the Fourier-Laplace transform \(\widehat \mu\) at all nonnegative integers. This very simple idea provides a natural inversion formula for the Laplace transformation which exploits only simple facts about the classical Hausdorff moment problem. For instance, it is well known how to read properties of the measure \(\mu\) (such as the positivity, absolute continuity with particular weight functions with respect to the Lebesgue measure, etc.) in terms of positivity conditions involving only the values of \(\widehat\mu\) at integral points. It was S. Bergman and W. T. Martin [Duke Math. J. 6, 389-407 (1940; Zbl 0024.02703)] who worked out the details of a similar transformation in two real variables, in the case of spaces of square summable functions. The Hilbert space methods which are inherent to the Bergman spaces studied by these two authors constitute the simplest and best understood framework of relating moment problems to interpolation problems via a Fourier transformation.
The aim of the present note is to extend the Bergman and Martin correspondence to two more restrictive classes of measures and to analytic functions defined in a quadrant or, respectively, in a tube domain over a quadrant. Meanwhile a great amount of work has been done in the harmonic analysis in tube domains. This considerably simplifies parts of our paper. A second technical source for this note comes from an inverse problem for the principal function of a hyponormal operator with rank-one self-commutator. This inverse problem can be interpreted as a moment problem in two real variables [see M. Putinar, J. Funct. Anal. 94, No. 2, 288-307 (1990; Zbl 0735.47008); J. Funct. Anal. 136, No. 2, 331-364 (1996; Zbl 0917.47014)]. Besides its obvious operator-theoretic nature, this moment problem has some interesting analytic counterparts. For instance, the extremal solutions of the corresponding truncated moment question are in bijective correspondence with all semialgebraic sets of the unit square, given by a single polynomial inequality. The finite determination of certain functions carried by these semialgebraic sets constitutes the object of the final part of the note. When completing the cycle of correspondences proposed in the sequel, we end up with a natural class of analytic functions, a completely solvable interpolation problem for this class (in contrast to the similar, much harder and still mysterious, Nevanlinna-Pick problem in two variables) and some canonical relations between hyponormal operators, semialgebraic sets and the interpolation problem. Throughout our investigation the classical results of M. G. Krein and his school were a solid comparison basis. However, most of the two-variable analogous aspects in this comparison remain still unknown.
The note is divided into three parts. The first part recalls and expands from earlier works of ours [loc.cit.] some formulae which solve the \(L\)-problem of moments in two variables with the help of hyponormal operators. The second part contains the necessary preparation in Fourier analysis and its relevance for our interpolation problem. In the last part of the paper we recall some classical dual convex optimization results which give the characterization of extremal solutions of the truncated interpolation problem. Some resulting open problems are briefly discussed at the end of the paper.