Tractability of Multivariate Approximation Defined over Hilbert Spaces with Exponential Weights

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 $\mathit{a}=\{a_j{\}}_{j\in \mathbb{N}}$ and $\mathit{b}=\{b_j{\}}_{j\in \mathbb{N}}$ of positive numbers, and on a bounded sequence of positive integers $\mathit{m}=\{m_j{\}}_{j\in \mathbb{N}}$. The sequence $\mathit{a}$ is non-decreasing and the sequence $\mathit{b}$ is bounded from below by a positive number. We find necessary and sufficient conditions on $\mathit{a},\mathit{b}$ and $\mathit{m}$ to achieve the standard and new notions of tractability in the worst case setting.