On Interpolation and Integration in Finite-Dimensional Spaces of Bounded Functions
Per‐Gunnar Martinsson, Vladimir Abramovich Rokhlin, Mark Tygert · 2005
We observe that, under very mild conditions, an n-dimensional space of functions (with a finite n) admits numerically stable n-point interpolation and integration formulae. The proof relies entirely on linear algebra, and is virtually independent of the domain and of the functions to be interpolated. Approximation of functions and construction of quadrature formulae constitute an extremely well-developed area of numerical analysis; in most situations one is likely to encounter in practice, existing tools are satisfactory. Much of the research concentrates on obtaining powerful results under strong assumptions — designing interpolation and quadrature formulae for smooth functions on subspaces of R n, manifolds, etc. In this note, we make a very general observation that, given a finite set of bounded functions f1, f2,..., fn−1, fn (either real- or complex-valued) defined on a set S, there exists an interpolation formula that is exact on all linear combinations of f1, f2,..., fn−1, fn, is numerically stable, and is based on n nodes in S (to be denoted x1, x2,..., xn−1, xn). If, in addition, S is a measure space, and the functions f1, f2,..., fn−1, fn are integrable, then there exists a quadrature