Multivariate approximation for analytic functions with Gaussian kernels

Abstract We approximate d -variate analytic functions defined on R d which belong to a tensor product reproducing kernel Hilbert space. The kernel of this space is Gaussian with non-increasing positive shape parameters γ j 2 for j = 1 , 2 , … , d . The error of approximation is defined in the L 2 sense with the standard Gaussian weight. We study the worst case error of algorithms that use at most n arbitrary linear functionals on d -variate functions. We prove that for arbitrary shape parameters there are algorithms enjoying exponential convergence, but the exponent of exponential convergence depends on d and goes to zero as d approaches infinity. We study the absolute and normalized error criteria, in which the information complexity n ( e , d ) is defined as the minimal number of linear functionals which are needed to find an algorithm whose worst case error is at most e or at most e times the norm of the approximation operator. We study different notions of tractability which describe how the information complexity behaves as a function of d and ln e − 1 . We find necessary and sufficient conditions on various notions of tractability in terms of shape parameters  γ j 2 . Surprisingly enough, these conditions are the same for the absolute and normalized error criteria although the norm of the approximation operator may be exponentially small in d making the normalized error much harder than the absolute one. In particular, for any positive t and κ we find conditions on γ j 2 for which lim d + e − 1 → ∞ ln n ( e , d ) d t + [ ln e − 1 ] κ = 0 . This holds • for all  positive γ j 2    if max ( t , κ ) > 1 , • for t = 1 and κ = 1  iff lim j γ j 2 = 0 , • for t 1 and κ = 1  iff lim j ln ( j ) ∕ ln ( 1 ∕ γ j 2 ) = 0 , • for t ≤ 1 and κ 1  iff lim j → ∞ j ( 1 − κ ) ∕ κ ∕ ln ( 1 ∕ γ j 2 ) = 0 .