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We define Szego coordinates on a finitely connected smoothly bounded planardomain which effect a holomorphic change of coordinates on the domain that canbe as close to the identity as desired and which convert the domain to aquadrature domain with respect to boundary arc length. When these Szegocoordinates coincide with Bergman coordinates, the result is a doublequadrature domain with respect to both area and arc length. We enumerate a hostof interesting and useful properties that such double quadrature domainspossess, and we show that such domains are in fact dense in the realm ofbounded finitely connected domains with smooth boundaries.

Szegö Coordinates, Quadrature Domains, and Double Quadrature Domains