Best approximation of analytic functions from information about their values at a finite number of points

On the numerical recovery of a holomorphic mapping from a finite set of approximate values

“…The optimal algorithm for p = +∞ (see [20]) behaves in a similar way with respect to perturbations δw. Consequently, though very satisfactory from the theoretical point of view, such optimal algorithms are badly conditioned and are not suited for practical use when perturbations are present.…”

“…By using (24) in (23) we get (20). Finally, in the view of (25), (21) holds ifλ 0 = 0, so that we assume that 0 < λ 0 ≤λ 0 .…”

“…[13,18,20]), that explicitly provide an interpolant Rf for f . For instance, the optimal algorithm of [13] for p = 2 is given by…”

Optimal quadrature algorithms inH p spaces

“…By the Smolyak's Theorem (see [5] and [12]) it follows that for every N, eJff there exists a linear optimal error algorithm qS~ which is of the form In this paper we shall show that the estimate (1.15) holds for every D. We shall then study a particular class of information operators N* which are almost optimal. We also exhibit the linear algorithm 05* using N* such that e(4,*) is no greater than the right hand side of (t.16).…”

Optimal interpolation of convergent algebraic series

“…Note that class W studied in this paper consists of analytical functions with bounded derivatives of all orders at point x = 0, while paper [4] studies the class of analytical and bounded functions in the unit circle.…”