Multivariate approximation by translates of the Korobov function on Smolyak grids

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For a set W ⊂ Lₚ(ᵈ), 1 < p < ∞, of multivariate periodic functions on the torus ᵈ and a given function φ∈ Lₚ(ᵈ), we study the approximation in the Lₚ(ᵈ)-norm of functions f ∈ W by arbitrary linear combinations of n translates of φ. For W = Uʳₚ(ᵈ) and φ= κᵣ,d, we prove upper bounds of the worst case error of this approximation where Uʳₚ(ᵈ) is the unit ball in the Korobov space Kʳₚ(ᵈ) 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 Kʳ₂(ᵈ) 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.