Preasymptotics and asymptotics of approximation numbers of anisotropic Sobolev embeddings

2016/07/07 by Jia Chen, Heping Wang, Chen, JIa +1
Computer Science · Engineering · Mathematics · #46E35 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Mathematical Approximation and Integration #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1607.01865

openalex publication_date 2016/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

In this paper, we obtain the preasymptotic and asymptotic behavior and strong equivalences of the approximation numbers of the embeddings from the anisotropic Sobolev spaces W₂^{\bf R}(\Bbb T^d) to L₂(\Bbb T^d). We also get the preasymptotic behavior of the approximation numbers of the embeddings from the limit spaces W₂^∞(\Bbb T^d) of the anisotropic Sobolev spaces W₂^{\bf R}(\Bbb T^d) to L₂(\Bbb T^d). We show that both the above embedding problems are intractable and do not suffer from the curse of dimensionality.

Related

• Strong equivalences of approximation numbers and tractability of weighted anisotropic Sobolev embeddings
• Approximation numbers of Sobolev and Gevrey type embeddings on the sphere and on the ball -- Preasymptotics, asymptotics, and tractability
• Counting via entropy: new preasymptotics for the approximation numbers of Sobolev embeddings
• How anisotropic mixed smoothness affects the decay of singular numbers of Sobolev embeddings
• Sharp estimates for approximation numbers of non-periodic Sobolev\n embeddings