Integration of polyharmonic functions

Abstract:

The results in this paper are motivated by two analogies. First, $m$-harmonic functions in $\mathbf {R}^{n}$ are extensions of the univariate algebraic polynomials of odd degree $2m-1$. Second, Gauss’ and Pizzetti’s mean value formulae are natural multivariate analogues of the rectangular and Taylor’s quadrature formulae, respectively. This point of view suggests that some theorems concerning quadrature rules could be generalized to results about integration of polyharmonic functions. This is done for the Tchakaloff-Obrechkoff quadrature formula and for the Gaussian quadrature with two nodes.