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

作者：Chen, Jia; Wang, Heping^*

来源：Journal of Complexity, 2019, 50: 1-24.

DOI：10.1016/j.jco.2018.08.002

• Approximation numbers
• Sobolev spaces
• Gevrey type spaces
• Preasymptotics and asymptotics
• Tractability

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 S-d and on the ball 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 and the dimension d. We emphasize 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 : Hr -> 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) -> L-2 are uniformly weakly tractable, not polynomially tractable, and quasi-polynomially tractable if and only if alpha >= 1.

• 出版日期2019-2
• 单位首都师范大学