Approximation numbers of Sobolev and Gevrey type embeddings on the sphere and on the ball -- Preasymptotics, asymptotics, and tractability

TL;DR

This paper analyzes the approximation numbers of Sobolev and Gevrey embeddings on spheres and balls, providing asymptotic behaviors, independence of dimension constants, and tractability results for high-dimensional approximation problems.

Contribution

It offers new asymptotic and preasymptotic estimates for approximation numbers, establishing tractability conditions independent of dimension for Sobolev and Gevrey spaces.

Findings

Approximation problems are weakly tractable if r > 1.

Gevrey space embeddings are uniformly weakly tractable for all positive parameters.

No curse of dimensionality occurs for Sobolev embeddings in the studied setting.

Abstract

In this paper, we investigate optimal linear approximations ($n$-approximation numbers ) of the embeddings from the Sobolev spaces $H^r\ (r>0)$ for various equivalent norms and the Gevrey type spaces $G^{\alpha,\beta}\ (\alpha,\beta>0)$ on the sphere $\Bbb S^d$ and on the ball $\Bbb B^d$, where the approximation error is measured in the $L_2$-norm. We obtain preasymptotics, asymptotics, and strong equivalences of the above approximation numbers as a function in $n$ and the dimension $d$. We emphasis that all equivalence constants in the above preasymptotics and asymptotics are independent of the dimension $d$ and $n$. As a consequence we obtain that for the absolute error criterion the approximation problems $I_d: H^{r}\to L_2$ are weakly tractable if and only if $r>1$, not uniformly weakly tractable, and do not suffer from the curse of dimensionality. We also prove that for any $\alpha,\beta>0$, the approximation problems $I_d: G^{\alpha,\beta}\to L_2$ are uniformly weakly tractable, not polynomially tractable, and quasi-polynomially tractable if and only if $\alpha\ge 1$.