Multivariate approximation by translates of the Korobov function on Smolyak grids

2012/12/26 by Dung, Dinh, Micchelli, Charles
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1212.6160

Abstract

For a set \mathbbW ⊂ L_p(\bT^d), 1 < p < ∞, of multivariate periodic functions on the torus \bT^d and a given function φ∈ L_p(\bT^d), we study the approximation in the L_p(\bT^d)-norm of functions f ∈ \mathbbW by arbitrary linear combinations of n translates of φ. For \mathbbW = U^r_p(\bT^d) and φ= κ_r,d, we prove upper bounds of the worst case error of this approximation where U^r_p(\bT^d) is the unit ball in the Korobov space K^r_p(\bT^d) and κ_r,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 K^r₂(\bT^d) is a reproducing kernel Hilbert space, whose reproducing kernel is a translation kernel determined by κ_r,d. We also provide lower bounds of the optimal approximation on the best choice of φ.

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