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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
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