A quadrature domain for arc-length (or just quadrature domain) is a planar Jordan region Ω with rectifiable boundary Γ and a positive Borel measure μ supported on a compact set κ ⊂ Ω , such that

holds for all functions f analytic on Ω and continuous on Γ. An identity such as 0.1 is called a quadrature identity.

We first pose an extremal problem regarding harmonic measure, and show that the extremal domain is a quadrature domain. Next, we look at the process for determining the Riemann mapping function of a quadrature domain that satisfies 0.1 for a given measure μ. This mapping function is the solution for a minimal-perimeter extremal problem, and is related to the eigenvector associated with the largest eigenvalue of the Hankel matrix of the moments of the measure. A numerical example is presented, to give some idea of the accuracy of numerical solutions to this problem.

We also look at the problem of determining the positive measure μ for a given domain. In the case that a quadrature identity can be found, we show how applying that identity to certain functions can result in some new proofs of old inequalities, such as Hilbert's inequality on quadratic forms.