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Vol 210, No 1 (2019)
Controllability and second-order necessary conditions for optimality
Abstract
3-26
The problem of constructing unsaturated quadrature formulae on an interval
Abstract
Unsaturated quadrature formulae are constructed which are well conditioned on the finite interval $I=[-1,1]$ with $L_p[I]$-weight function, $1< p< \infty$. 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.
27-62
Admissible changes of variables for Sobolev functions on (sub-)Riemannian manifolds
Abstract
63-112
Interpolation and absolutely convergent series in Frechet spaces
Abstract
113-154
Galerkin approximations for the Dirichlet problem with the $p(x)$-Laplacian
Abstract
155-174