Quadrature Domains and the Faber Transform

We present a framework for reconstructing any simply connected, bounded or unbounded, quadrature domain $Ω$ from its quadrature function $h$. Using the Faber transform, we derive formulae directly relating $h$ to the Riemann map for $Ω$. Through this approach, we obtain a complete classification of one point quadrature domains with complex charge. We proceed to develop a theory of weighted quadrature domains with respect to weights of the form $ρ_a(w)=|w|^{2(a-1)}$ when $a > 0$ ("power-weighted" quadrature domains) and the limiting case of when $a=0$ ("log-weighted" quadrature domains). Furthermore, we obtain Faber transform formulae for reconstructing weighted quadrature domains from their respective quadrature functions. Several examples are presented throughout to illustrate this approach both in the simply connected setting and in the presence of rotational symmetry.