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

摘要

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. (C) 2018 Elsevier Inc. All rights reserved.