2017/01/13 by Jia Chen, Heping Wang, Chen, Jia +1
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.1701.03545
openalex publication_date 2017/01/13 · openalex created_date 2017/01/26 · openalex updated_date 2026/07/28
In this paper, we investigate optimal linear approximations (n-approximation numbers ) of the embeddings from the Sobolev spaces Hr (r>0) for various equivalent norms and the Gevrey type spaces Gα,β (α,β>0) on the sphere \Bbb Sd and on the ball \Bbb Bd, where the approximation error is measured in the L2-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 Id: Hr→ L2 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 α,β>0, the approximation problems Id: Gα,β→ L2 are uniformly weakly tractable, not polynomially tractable, and quasi-polynomially tractable if and only if α≥ 1.