Abstract (EN)

Let F be a compact set of a Banach space X. This paper analyses the ``Generalized Empirical Interpolation Method'' (GEIM) which, given a function f∈F, builds an interpolant Jn[f] in an n-dimensional subspace Xn⊂X with the knowledge of n outputs (σi(f))ni=1, where σi∈X′ and X′ is the dual space of X. The space Xn is built with a greedy algorithm that is \textit{adapted} to F in the sense that it is generated by elements of F itself. The algorithm also selects the linear functionals (σi)ni=1 from a dictionary Σ⊂X′. In this paper, we study the interpolation error maxf∈F∥f−Jn[f]∥X by comparing it with the best possible performance on an n-dimensional space, i.e., the Kolmogorov n-width of F in X, dn(F,X). For polynomial or exponential decay rates of dn(F,X), we prove that the interpolation error has the same behavior modulo the norm of the interpolation operator. Sharper results are obtained in the case where X is a Hilbert space.