Math. Model. Nat. Phenom.
Volume 15, 2020
|Number of page(s)||25|
|Published online||14 February 2020|
Finite term relations for the exponential orthogonal polynomials*
Department of Mathematics, KTH,
2 Department of Mathematics, University of California, Santa Barbara, CA 93106-3080, USA.
3 School of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK.
** Corresponding author: firstname.lastname@example.org
Accepted: 14 May 2019
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,z̅g,…,z̅dg, 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.
Mathematics Subject Classification: 33C47 / 32A26 / 47B20 / 47B35
Key words: Complex orthogonal polynomials / exponential transform / finite term relation / hyponormal operator / quadrature domain
© EDP Sciences, 2020
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