Domains Allowing Exact Quadrature Identities for Harmonic Functions — An Approach Based on P.D.E.

Abstract

The integral of a harmonic function u over a ball in Rd centered at xo equals the volume of the ball times u(xo); this is the simplest “quadrature identity” of the type here under discussion. For certain domains other than balls analogous quadrature identities exist whereby the integral is exactly expressible as a finite linear combination of point evaluations. Such domains can be studied by converting the “quadrature” property into an equivalent boundary value problem. A new and convenient method for effecting this conversion is presented. It is based on techniques borrowed from the theory of partial differential equations and Sobolev spaces, which others have already successfully used in such areas as complex analysis and potential theory, polynomial and rational approximation, etc. As applied to quadrature identities, the strength of this method is its flexibility: it is adaptable to multidimensional problems, unbounded domains, quadrature identities of more general types, etc.