Tractability of Multivariate Approximation Defined over Hilbert Spaces with Exponential Weights

Abstract

We study multivariate approximation defined over tensor product Hilbert spaces. The domain space is a weighted tensor product Hilbert space with exponential weights which depend on two sequences $\boldsymbol{a}=\{a_j\}_{j\in\mathbb{N}}$ and $\boldsymbol{b}=\{b_j\}_{j\in\mathbb{N}}$ of positive numbers, and on a bounded sequence of positive integers $\boldsymbol{m}=\{m_j\}_{j\in\mathbb{N}}$. The sequence $\boldsymbol{a}$ is non-decreasing and the sequence $\boldsymbol{b}$ is bounded from below by a positive number. We find necessary and sufficient conditions on $\boldsymbol{a},\boldsymbol{b}$ and $\boldsymbol{m}$ to achieve the standard and new notions of tractability in the worst case setting.