ABSTRACT
Not only is the unit disc the most famous of area quadrature domains, it is also the most famous of boundary arc length quadrature domains. When a holomorphic function that extends continuously to the boundary is averaged over the unit circle with respect to arc length measure, the value of the function at the origin is obtained. More generally, a finitely connected domain in the plane bounded by n nonintersecting C1
smooth curves is called an arc length quadrature domain if the average of a holomorphic function that extends continuously to the closure of the domain over the boundary with respect to arc length measure is a finite linear combination of values of the function and its derivatives at finitely many points in the domain. The points and the coefficients are fixed in this quadrature identity even though the holomorphic function is allowed to vary. In this chapter, we will restrict our attention to arc length quadrature domains with C∞ smooth boundaries, and we will call such domains smooth arc length quadrature domains. In this context, a bounded domain Ω with C∞ smooth boundary is called a smooth arc length quadrature domain if there exist points {zj}Nj=1 in Ω, complex constants cjk, and nonnegative integers mj such that∫
h ds =
cjkh (k)(zj) (23.1)
for all h in the Hardy spaceH2(bΩ). In this chapter, we will show that arc length quadrature domains are to the Szego˝ kernel as area quadrature domains are to the Bergman kernel. We will also consider domains that are like the unit disc in that they are both area and arc length quadrature domains. Once again, these results can be viewed as improvements upon the Riemann mapping theorem.
