Weighted Quadrature Domains and the Faber Transform

This thesis develops the theory of weighted quadrature domains in parallel with the Faber transform as a tool for their analysis and explicit construction. Under this framework,e obtain a number of existence, uniqueness, and classification results for classical and weighted quadrature domains. In the classical setting, we derive explicit formulae relating the Riemann map of a simply connected quadrature domain to its quadrature function via the Faber transform. This reduces the inverse and direct problems to a problem of solving finitely many algebraic equations. Applying these results, along with tools from logarithmic potential theory, we obtain a complete classification of one-point quadrature domains with complex charge. We then introduce power-weighted quadrature domains (PQDs) - domains admitting a quadrature identity with respect to the weight ρa(w) = ∣w∣2(a-1) for some a>0. A central structural result is that a simply connected domain is a PQD if and only if the ath power of the outer factor of its Riemann map extends to a rational function. This characterization yields Faber transform formulae analogous to the classical case, which we apply to partially classify one-point and monomial PQD families. Novel boundary phenomena - including the formation of boundary "corners" at the origin with angles that are integer multiples of π/a - are exhibited. Next, we develop the theory of log-weighted quadrature domains (LQDs) - domains admitting a quadrature identity with respect to the weight ρ0 = ∣w∣-2, the limiting case a→0⁺ of ρa |w|2(a-1). The non-integrable singularity at the origin introduces new phenomena: when the domain contains the origin, the quadrature function is no longer uniquely determined, but only up to the addition of a point charge q/w. Despite this loss of uniqueness, we show that a simply connected domain is an LQD if and only if the outer factor of its Riemann map extends to the exponential of a rational function. Classification results for null, monomial, and one-point LQD families are obtained, and the connection to classical quadrature domains via the exponential map is established. Finally, we introduce algebraic quadrature domains (AQDs), defined with respect to weights of the form ρR = |R'|², where R is a non-constant rational function. This class subsumes both classical quadrature domains (R(w)=w) and integer power-weighted quadrature domains (R(w)=wn/n). We derive representation formulae in terms of the Faber transform and present several examples.