Sample numbers and optimal Lagrange interpolation in Sobolev spaces
This paper studies approximations in weighted Sobolev spaces with respect to the Chebyshev norm. For this purpose we may choose a space embedded in the continuous functions over a compact domain which is a Banach space, and we wish to approximate functions given only at a finite number of nodes. The sampling numbers which are also computed in this article (this is the main point of the work in the paper) are the smallest errors for algorithms based on this number of knots for the worst errors with respect to the choice of approximants in the unit ball of the mentioned spaces. The sampling numbers are closely related to the well-known theory of the so-called \(n\)-widths. The authors are interested in optimal recovery from such spaces when the weight functions are integrable on \((-1,1)\). They then choose Chebyshev points of the first kind as interpolation nodes and show that those are optimal if Lagrange interpolation is used. A variation of this theme is the search for the optimal interpolation algorithm if the Chebyshev points are used \textbf{and} the end-points of the unit interval, and this is also investigated by the authors.
optimal Lagrange interpolation algorithm
scientific article; zbMATH DE number 7393769