On Pizzetti’s formula

Abstract

We give an extended Pizzetti asymptotic formula, namely, we show that if ϕ is smooth in an open set that contains the origin in ${\mathbb{R}}^{\mathit{n}}$ and if p is a harmonic homogeneous polynomial of degree k, then $$\begin{array}{}\text{(0.1)}& {{\displaystyle \int }}_{\mathbb{S}}\mathit{\varphi }\mathrm{(}\mathit{\epsilon }\mathit{\omega }\mathrm{)}\mathit{p}\mathrm{(}\mathit{\omega }\mathrm{)}\mathrm{d}\mathit{\sigma }\mathrm{(}\mathit{\omega }\mathrm{)}\sim \mathit{C}{\displaystyle \sum }_{\mathit{m}=0}^{\infty }{\displaystyle \frac{{\mathrm{\Delta }}^{\mathit{m}}\mathit{p}\mathrm{(}\mathbf{\nabla }\mathrm{)}\mathit{\varphi }\mathrm{(}\mathbf{0}\mathrm{)}}{{\mathit{W}}_{\mathit{n}\mathrm{,}\mathit{k}\mathrm{,}\mathit{m}}}}{\mathit{\epsilon }}^{\mathit{k}+2\mathit{m}}\mathrm{,}\end{array}$$ as $\mathit{\epsilon }\to 0^{+}$ for some constants C and ${\mathit{W}}_{\mathit{n}\mathrm{,}\mathit{k}\mathrm{,}\mathit{m}}$ that are given in the text. We also show that the formula never holds, for all ϕ, if p is not harmonic.

We consider two procedures, one algebraic and the other analytic, to find the part of an object – a formal power series or a smooth function – that is a radial multiple of a given polynomial and show that the two constructions yield the same results for harmonic polynomials but not otherwise.

We also consider mean value type results for solutions of partial differential equations, including a version of Morera’s theorem that applies to locally integrable functions.