Effectiveness of equivalent sets of polynomials in Faber regions
Suppose that the transformation \[ z= \phi(t)= t+ \sum^\infty_{n= 0}\alpha_n/t^n= t+ M(1/t), \] is conformal for \(T_0<|t|< \infty\). For fixed \(\gamma> T_0\), let \(C\) be a simple regular closed curve which is the image of the circe \(|t|= \gamma\). The curve \(C\) is called the Faber curve \(C_\gamma\), and the interor of \(C\) are denoted by \(D(C)\) and is called a Faber region. The Faber polynomials \(\{P_n(z)\}\) associated with the transformation \(\phi(t)\) are defined as follows \[ [t- M'(1/t)/t]/[t+ M(1/t)- z]= \sum^\infty_{n= 0}P_n(z)/t^n. \] The associated basic set \(\{p_n(z)\}\) is said to be effective in a Faber set \(D(C)\) if every function \(f\) analytic in \(D(C)\) may be represented in a form \[ f(z)\sim \sum^\infty_{n= 0}A_n p_n(z). \] In this paper the effectiveness properties, in the Faber regions, of some basic sets of equivalent polynomials are investigated.
scientific article; zbMATH DE number 1322878