Abstract
We represent the integral over the unit ball B in R n of any poly-harmonic function u(x) of degree m as a linear combination with constant coefficients of the integrals of its Laplacians Δj u (j = 0,...,m - 1) over any fixed(n - 1)-dimensional hypersphere S(ρ) of radius ρ (0 ≤ ρ ≤ 1). In case ρ = 0 theformula reduces to the classical Pizzetti formula. In particular, the cubature formula derived here integrates exactly all algebraic polynomials of degree 2m - 1.
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