The problem of constructing unsaturated quadrature formulae on an interval

V. N. Belykh

Sbornik. Mathematics, Tome 210 (2019) no. 1, pp. 24-58

Résumé

Unsaturated quadrature formulae are constructed which are well conditioned on the finite interval $I=[-1,1]$ with $L_p[I]$-weight function, $1$. A specific feature of such formulae is the absence of the principal error term, which ensures that they can be automatically readjusted (with an increased number of nodes) to any excessive (extraordinary) amount of smoothness of the integrands. All the key parameters of quadratures (the nodes, the coefficients and the condition number) are evaluated within a single general approach based on the solution of a number of special boundary-value problems in the theory of meromorphic functions in the unit disc. For particular weight functions, which have important applications, algorithms for evaluating all the parameters of the quadratures efficiently are put forward. For $C^\infty$-smooth integrands, an answer is given with an absolutely sharp exponential error estimate. The sharpness of the estimate is secured by the asymptotic behaviour of the Alexandrov $n$-width of a compact set of $C^\infty$-smooth functions, which goes to zero exponentially (as the number of nodes goes off to infinity). Bibliography: 32 titles.

Détail

Keywords: unsaturation, roundoff error, well conditioning, exponential convergence.
Mots-clés : quadrature formula