Multivariate approximation by translates of the Korobov function on Smolyak grids - Dinh Dũng, Charles A. Micchelli (Journal of Complexity, 2016)
For a set W ⊂ Lp(Td), 1 < p < ∞, of multivariate periodic functions on the torus Td and a given function ϕ ∈ Lp(Td), we study the approximation in the Lp(Td)-norm of functions f ∈ W by arbitrary linear combinations of n translates of ϕ. For W = Ur p (Td) and ϕ = κr,d, we prove upper bounds of the worst case error of this approximation where Ur p (Td) is the unit ball in the Korobov space Kr p(Td) and κr,d is the associated Korobov function. To obtain the upper bounds, we construct approximation methods based on sparse Smolyak grids. The case p = 2, r > 1/2, is especially important since Kr 2 (Td) is a reproducing kernel Hilbert space, whose reproducing kernel is a translation kernel determined by κr,d. We also provide lower bounds of the optimal approximation on the best choice of ϕ.