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- W2078873941 abstract "This paper is concerned with the approximate evaluation of $int_{ - 1}^1 {k(x)f(x)d_x } $, where k is Lebesgue integrable and f is at least Riemann integrable, and preferably smooth. The integral is approximated by a rule of the form $sumnolimits_{i = 1}^n {w_{ni} (k)f(x_{ni} )} $, where the points $x_{ni} $ are chosen in some prescribed way, and the weights $w_{ni} (k)$ are such that the rule is exact if f is any polynomial of degree $ < n$. For suitable choices of the points and suitable functions k, the rule is shown to converge to the exact integral, and the companion rule $sum {|w_{ni} (k)|f(x_{ni} )} $ to$int {|k(x)|f(x)dx} $, for all Riemann integrable functions f. (The companion property ensures, for example, that the weights have the asymptotic positivity property if k is positive.) Error bounds are given that guarantee rapid convergence of the rule if f is smooth. Finally, by establishing a connection with mean convergence of Lagrangian interpolation some negative theorems are proved. In particular, it is shown that for every choice of the points ${ x_{ni} } $ there exist $k in L_1 $ and $f in C$ such that the quadrature rule diverges." @default.
- W2078873941 created "2016-06-24" @default.
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- W2078873941 date "1982-04-01" @default.
- W2078873941 modified "2023-09-27" @default.
- W2078873941 title "Properties of Interpolatory Product Integration Rules" @default.
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- W2078873941 doi "https://doi.org/10.1137/0719027" @default.
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