Bergman polynomials on an archipelago: Estimates, zeros and shape reconstruction
2009 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 222, no 4, 1405-1460 p.Article in journal (Refereed) Published
Growth estimates of complex orthogonal polynomials with respect to the area measure supported by a disjoint union of planar Jordan domains ( called, in short, an archipelago) are obtained by a combination of methods of potential theory and rational approximation theory. The study of the asymptotic behavior of the roots of these polynomials reveals a surprisingly rich geometry, which reflects three characteristics: the relative position of an island in the archipelago, the analytic continuation picture of the Schwarz function of every individual boundary and the singular points of the exterior Green function. By way of explicit example, fine asymptotics are obtained for the lemniscate archipelago vertical bar z(m)-1 vertical bar < r(m), 0 < r < 1, which consists of m islands. The asymptotic analysis of the Christoffel functions associated to the same orthogonal polynomials leads to a very accurate reconstruction algorithm of the shape of the archipelago, knowing only finitely many of its power moments. This work naturally complements a 1969 study by H. Widom of Szego orthogonal polynomials on an archipelago and the more recent asymptotic analysis of Bergman orthogonal polynomials unveiled by the last two authors and their collaborators.
Place, publisher, year, edition, pages
2009. Vol. 222, no 4, 1405-1460 p.
Bergman orthogonal polynomials, Disjoint Jordan domains, Zeros of, polynomials, Shape reconstruction, Equilibrium measure, Green function, Strong asymptotics, Geometric tomography, orthogonal polynomials, planar domains, complex plane, approximation, moments, curves
IdentifiersURN: urn:nbn:se:kth:diva-19073DOI: 10.1016/j.aim.2009.06.010ISI: 000273016100008ScopusID: 2-s2.0-68349144684OAI: oai:DiVA.org:kth-19073DiVA: diva2:337120
QC 201005252010-08-052010-08-052011-02-07Bibliographically approved