Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Multivariate approximation by translates of the Korobov function on
  Smolyak grids   For a set $\mathbb{W} \subset L_p(\bT^d)$, $1 < p < \infty$, of multivariate
periodic functions on the torus $\bT^d$ and a given function $\varphi \in
L_p(\bT^d)$, we study the approximation in the $L_p(\bT^d)$-norm of functions
$f \in \mathbb{W}$ by arbitrary linear combinations of $n$ translates of
$\varphi$. For $\mathbb{W} = U^r_p(\bT^d)$ and $\varphi = \kappa_{r,d}$, 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
$\kappa_{r,d}$ 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 $K^r_2(\bT^d)$ is a reproducing
kernel Hilbert space, whose reproducing kernel is a translation kernel
determined by $\kappa_{r,d}$. We also provide lower bounds of the optimal
approximation on the best choice of $\varphi$.

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The categories of this paper are: quant-ph
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The categories of this paper are: solv-int
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The categories of this paper are: econ
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The categories of this paper are: nucl-th
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The categories of this paper are: funct-an
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The categories of this paper are: hep-lat
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The categories of this paper are: nlin
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The categories of this paper are: adap-org
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: math-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: eess
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The categories of this paper are: q-alg
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The categories of this paper are: stat