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An extension of the Pizzetti formula for polyharmonic functions. (English) Zbl 0980.31004
Summary: We represent the integral over the unit ball $$B$$ in $$R^n$$ of any polyharmonic function $$u(x)$$ of degree $$m$$ as a linear combination with constant coefficients of the integrals of its Laplacians $$\Delta^ju$$ $$(j=0,\dots,m-1)$$ over any fixed $$(n-1)$$-dimensional hypersphere $$S(p)$$ of radius $$\rho$$ $$(0\leq\rho\leq 1)$$. In case $$\rho=0$$ the formula reduces to the classical Pizetti formula. In particular, the cubature formula derived here integrates exactly all algebraic polynomials of degree $$2m-1$$.

MSC:
 31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
Keywords:
polyharmonic functions
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