Finite term relations for the exponential orthogonal polynomials

The exponential orthogonal polynomials encode via the theory of hyponormal operators a shade function $g$ supported by a bounded planar shape. We prove under natural regularity assumptions that these complex polynomials satisfy a three term relation if and only if the underlying shape is an ellipse carrying uniform black on white. More generally, we show that a finite term relation among these orthogonal polynomials holds if and only if the first row in the associated Hessenberg matrix has finite support. This rigidity phenomenon is in sharp contrast with the theory of classical complex orthogonal polynomials. On function theory side, we offer an effective way based on the Cauchy transforms of $g, \overline{z}g, \ldots, \overline{z}^dg$, to decide whether a $(d+2)$-term relation among the exponential orthogonal polynomials exists; in that case we indicate how the shade function $g$ can be reconstructed from a resulting polynomial of degree $d$ and the Cauchy transform of $g$. A discussion of the relevance of the main concepts in Hele-Shaw dynamics completes the article.Comment: 33 page

Reconstruction of algebraic-exponential data from moments

Let $G$ be a bounded open subset of Euclidean space with real algebraic boundary $\Gamma$. Under the assumption that the degree $d$ of $\Gamma$ is given, and the power moments of the Lebesgue measure on $G$ are known up to order $3d$, we describe an algorithmic procedure for obtaining a polynomial vanishing on $\Gamma$. The particular case of semi-algebraic sets defined by a single polynomial inequality raises an intriguing question related to the finite determinateness of the full moment sequence. The more general case of a measure with density equal to the exponential of a polynomial is treated in parallel. Our approach relies on Stokes theorem and simple Hankel-type matrix identities