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Cubature formulae for polyharmonic functions. (English) Zbl 0988.41017
Haussmann, Werner (ed.) et al., Recent progress in multivariate approximation. Proceedings of the 4th international conference, Witten-Bommerholz, Germany, September 24-29, 2000. Basel: BirkhĂ¤user. ISNM, Int. Ser. Numer. Math. 137, 49-74 (2001).
Let $$B(r)$$ be the closed ball in $$\mathbb{R}^d$$ of radius $$r$$ and center at zero. A function $$u$$ is said polyharmonic of degree $$m$$ in a given domain $$D\subset \mathbb{R}^d$$ if $$\Delta ^mu=0$$ on $$D$$. The author constructs cubature formulae based on integrals of the function $$v(x)$$ and its normal derivatives $$\delta$$/$$\delta \nu$$ over $$n$$ distinct $$(d-1)$$-dimensional hyperspheres $$S_i$$, $$i=1,\dots ,n$$ of radius $$\rho_i$$, that is of the form $\int_ {B(r)}v(x) dx \approx \sum_ {i=1}^n \sum _ {k=0}^ {m_i-1}A_ {ik} \int _ {S_i}\frac {\delta ^k} {\delta \nu ^k}v(x) d\sigma (x).$ Another formula that, instead of the normal derivatives, involves the iterated Laplacians $$\Delta ^j v$$, $$j=1\dots m_i -1$$ is constructed.The above cubature formulae integrate exactly wide classes of functions defined as the harmonic span of given radial functions, and including the class of polyharmonic functions of fixed order. Cubature formulae of mixed type and Gaussian type, that is attaining maximal polyharmonic degree of precision, are also considered.
For the entire collection see [Zbl 0972.00049].