Birkhoff quadrature formulae based on the zeros of Jacobi polynomials. (English) Zbl 1068.41050
Author’s abstract: ”The main object of this paper is to construct a Birkhoff quadrature formula of the form
\[
\int _{-1}^1 f(x)dx \approx \sum _{i=1}^nA_if^\prime (x_i)+\sum _{i=1}^{n-1}B_if (x_i^*)+\sum _{j=0}^kC_j\left (f^{(j)} (1)+(-1)^jf^{(j)}(-1) \right ),
\]
which is exact for the polynomials of degree \(\leq 2n+2k+1\). We construct the formula when the nodes \(\{ x_i \} _1^n \) and \(\{ x_i ^*\} _1^{n-1}\) are the zeros of the ultraspherical polynomials \(P_n^{(k)}(x)\) and \(P_n^{(k)^\prime }(x)\), respectively.”
Reviewer: Luigi Gatteschi (Torino)
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