Power matrices for Faber polynomials and conformal welding

Abstract

The power matrix is the matrix of the coefficients of the power series at 0 of powers of an analytic function. Composition corresponds to matrix multiplication. We generalize the power matrix by replacing power series with Faber polynomial expansions. We show that composition corresponds to multiplication of the generalized power matrices, for both simply and doubly connected domains. We apply this to give some matrix product formulas for the coefficients of conformal welding maps of an analytic homeomorphism of an analytic curve. In particular, in some sense one can solve for the coefficients of the conformal welding maps in terms of the generalized power matrix of the analytic homeomorphism.

Keywords:

• Faber polynomials
• power matrix
• conformal welding
• univalent functions

AMS Subject Classifications::

• 30C55
• 30C35
• 30B99