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MiS Preprint
20/2007
Generalized Tractability for Linear Functionals
Michael Gnewuch and Henryk Woźniakowski
Abstract
We study approximation of continuous linear functionals $I_d$ defined over reproducing kernel weighted Hilbert spaces of $d$-variate functions. Let $n(\varepsilon,I_d)$ denote the minimal number of function values needed to solve the problem to within $\varepsilon$. There are many papers studying polynomial tractability for which $n(\varepsilon,I_d)$ is to be bounded by a polynomial in $\varepsilon^{-1}$ and $d$. We study generalized tractability for which we want to guarantee that either $n(\varepsilon,I_d)$ is not exponentially dependent on $\varepsilon^{-1}$ and $d$, which is called weak tractability, or is bounded by a power of $T(\varepsilon^{-1},d)$ for $(\varepsilon^{-1},d)\in \Omega\subseteq [1,\infty)\times\mathbb{N}$, which is called $(T,\Omega)$-tractability. Here, the tractability function $T$ is non-increasing in both arguments and does not depend exponentially on $\varepsilon^{-1}$ and $d$.
We present necessary conditions on generalized tractability for arbitrary continuous linear functionals $I_d$ defined on weighted Hilbert spaces whose kernel has a decomposable component, and sufficient conditions on generalized tractability for multivariate integration for general reproducing kernel Hilbert spaces. For some weighted Sobolev spaces these necessary and sufficient conditions coincide. They are expressed in terms of necessary and sufficient conditions on the weights of the underlying spaces.
