# Double bases from generalized Faber polynomials with complex-valued coefficients in weighted Lebesgue spaces

## TL;DR

This paper investigates the basis properties of generalized Faber polynomials with complex coefficients in weighted Lebesgue spaces, establishing conditions under which they form a basis in these function spaces.

## Contribution

It extends the theory of Faber polynomials by analyzing their basis properties with complex coefficients in weighted Lebesgue spaces on regular curves.

## Key findings

- Generalized Faber polynomials form a basis in weighted Smirnov spaces under Muckenhoupt condition.
- Double systems of these polynomials are studied for basis properties in weighted Lebesgue spaces.
- Conditions for basis formation are established for complex-valued coefficient systems.

## Abstract

In the paper it is considered the generalized Faber polynomials defined inside and outside a regular curve on the complex plane. The weighted Smirnov spaces corresponding to bounded and unbounded regions are defined. It is proved that the generalized Faber polynomials forms a basis in weighted Smirnov spaces, if the weight function satisfies the Muckenhoupt condition on the regular curve. The double system of generalized Faber polynomials with complex-valued coefficients is also considered and the basis properties of such a system in weighted Lebesgue spaces over regular curves are studied.