Bergman orthogonal polynomials and the Grunsky matrix

Bernhard Beckermann (1) , Nikos Stylianopoulos (2)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 Department of Mathematics and Statistics, University of Cyprus

By exploiting a link between Bergman orthogonal polynomials and the Grunsky matrix, probably first observed by Kühnau in 1985, we improve some recent results on strong asymptotics of Bergman polynomials outside the domain G of orthogonality, and on entries of the Bergman shift operator. In our proofs we suggest a new matrix approach involving the Grunsky matrix, and use well-established results in the literature relating properties of the Grunsky matrix to the regularity of the boundary of G, and the associated conformal maps. For quasiconformal boundaries, this approach allows for new insights for Bergman polynomials.

Mots clés: Bergman orthogonal polynomials; Faber polynomials; Conformal mapping; Grunsky matrix; Bergman shift; Quasiconformal mapping

Domaines: Analyse classique [math.CA]; Variables complexes [math.CV]; Analyse fonctionnelle [math.FA]

Bernhard Beckermann, Nikos Stylianopoulos. Bergman orthogonal polynomials and the Grunsky matrix. Constructive Approximation, 2018, 47 (2), pp.211-235. ⟨10.1007/s00365-017-9381-7⟩. ⟨hal-01325026v2⟩