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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.

41A55 Approximate quadratures
65D32 Numerical quadrature and cubature formulas
Full Text: DOI
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