Approximation in periodic Gevrey spaces

Abstract

The well-known Gevrey classes consist of C∞-functions on Rd whose derivatives satisfy certain growth conditions. For periodic functions these conditions can be expressed in terms of Fourier coefficients. Motivated by this observation, we introduce Gevrey spaces on the d-dimensional torus and study approximation numbers of their embeddings into L2. Our special emphasis is on the dependence on the dimension of the underlying domain, which is an important aspect in the numerical treatment of high-dimensional problems. In particular, we determine the exact rate of decay of the approximation numbers, together with optimal asymptotic constants, and establish preasymptotic estimates. Finally we translate our findings into the language of tractability notions from information-based complexity.