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Numerical integration as a finite matrix approximation to multiplication operator. (English) Zbl 1503.65050

Gaussian quadrature for the integral \(\int_\Omega f(x)w(x)dx\) is captured by the Golub-Meurant formula as the top left element in the matrix \(f(J_n)\) where \(J_n\) is the Jacobi matrix for the underlying orthonormal polynomial sequence. This matrix is the truncated form of the matrix representation of the multiplication operator \(M(x):f(x)\mapsto xf(x)\), with respect to the orthogonal polynomial basis. In this paper the idea is extended considerably in three essential ways. (1) One may replace \(M(x)\) by \(M(g(x)): f(x)\mapsto g(x)f(x)\) for a bounded \(g\) in a separable Hilbert space. (2) The polynomial basis can be replaced by any, not necessary orthogonal, basis. (3) \(\Omega\) can be a subset of \(\mathbb{R}^d\). The result will always be a scalar integration problem for \(f(x)\). The interpretation that can be given to the entries of the matrix \(f(M(g))\) is explored. Convergence and other properties are discussed. The paper is very readable with one-liner proofs referring to known literature. The possibilities are illustrated with some numerical examples that need symbolic computation when the condition is bad.

MSC:

65D30 Numerical integration
65D32 Numerical quadrature and cubature formulas
65F60 Numerical computation of matrix exponential and similar matrix functions
47N40 Applications of operator theory in numerical analysis
47A58 Linear operator approximation theory

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References:

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