3 research outputs found
Approximation in Hermite spaces of smooth functions
We consider -approximation of elements of a Hermite space of
analytic functions over . The Hermite space is a weighted
reproducing kernel Hilbert space of real valued functions for which the Hermite
coefficients decay exponentially fast. The weights are defined in terms of two
sequences and of positive
real numbers. We study the th minimal worst-case error of all algorithms that use information
evaluations from the class which only allows function
evaluations to be used.
We study (uniform) exponential convergence of the th minimal worst-case
error, which means that converges to
zero exponentially fast with increasing . Furthermore, we consider how the
error depends on the dimension . To this end, we study the minimal number of
information evaluations needed to compute an -approximation by
considering several notions of tractability which are defined with respect to
and . We derive necessary and sufficient conditions
on the sequences and for obtaining
exponential error convergence, and also for obtaining the various notions of
tractability. It turns out that the conditions on the weight sequences are
almost the same as for the information class which uses
all linear functionals. The results are also constructive as the considered
algorithms are based on tensor products of Gauss-Hermite rules for multivariate
integration. The obtained results are compared with the analogous results for
integration in the same Hermite space. This allows us to give a new sufficient
condition for EC-weak tractability for integration
Integration and approximation in cosine spaces of smooth functions
We study multivariate integration and approximation for functions belonging
to a weighted reproducing kernel Hilbert space based on half-period cosine
functions in the worst-case setting. The weights in the norm of the function
space depend on two sequences of real numbers and decay exponentially. As a
consequence the functions are infinitely often differentiable, and therefore it
is natural to expect exponential convergence of the worst-case error. We give
conditions on the weight sequences under which we have exponential convergence
for the integration as well as the approximation problem. Furthermore, we
investigate the dependence of the errors on the dimension by considering
various notions of tractability. We prove sufficient and necessary conditions
to achieve these tractability notions.Comment: arXiv admin note: text overlap with arXiv:1506.0860
A note on Korobov lattice rules for integration of analytic functions
We study numerical integration for a weighted Korobov space of analytic
periodic functions for which the Fourier coefficients decay exponentially fast.
In particular, we are interested in how the error depends on the dimension .
Many recent papers deal with this problem or similar problems and provide
matching necessary and sufficient conditions for various notions of
tractability. In most cases even simple algorithms are known which allow to
achieve these notions of tractability. However, there is a gap in the
literature: while for the notion of exponential-weak tractability one knows
matching necessary and sufficient conditions, so far no explicit algorithm has
been known which yields the desired result.
In this paper we close this gap and prove that Korobov lattice rules are
suitable algorithms in order to achieve exponential-weak tractability for
integration in weighted Korobov spaces of analytic periodic functions