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Uniqueness of the Gaussian quadrature for a ball. (English) Zbl 0979.41020
In a previous paper the authors [Numer. Math. 80, No. 1, 39-59 (1998; Zbl 0911.65015)] investigated quadrature formulae for the unit disk $$D=\{(x,y); x^2+y^2\leq 1\}$$ which are based on integrals over $$n$$ chords. Among all variety of quadrature formulae that use $$n$$ chords, the following result was established: “The Gaussian quadrature formula $\iint_D f(x,y)dx dy \approx \sum^n_{k=1} A_k\int^{+\sqrt {1-\eta_k^2}}_{-\sqrt{1- \eta_k^2}}f (\eta_k,y)dy \tag{*}$ where $$A_k= {\pi\over n+1}\sin {k\pi\over n+1}$$ $$(k=1,\dots,n)$$ has a highest degree of precision with respect to the set of algebraic polynomials of two variables of total degree $$2n-1$$. Here $$\eta_k$$, $$k=1,\dots,n$$, stand for the well-known zeros of the Chebychev polynomial of second kind of degree $$n$$”. In the present paper they prove the uniqueness (up to rotation) of the quadrature formula (*) with this extremal property. The authors give also a general multivariate analogue of (*) for numerical weighted integration of functions over the unit ball $$B^d$$ in $$\mathbb{R}^d$$ using integrals over the intersection of $$B^d$$ with $$n$$ hyperplanes $$(n$$ Radon projections of these functions likely as used in computer tomography). The uniqueness of this formula is proved when taking the weight equal to 1.