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

Abstract

In this paper, we investigate optimal linear approximations (napproximation numbers ) of the embeddings from the Sobolev spaces H r (r > 0) for various equivalent norms and the Gevrey type spaces G α,β (α, β > 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 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 I d : H r → 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 α, β > 0, the approximation problems I d : G α,β → L 2 are uniformly weakly tractable, not polynomially tractable, and quasi-polynomially tractable if and only if α ≥ 1. r/d exist, having the same value for various norms. We also prove that for 0 < α < 1,, lim n→∞ e βγn α/d a n (I d : G α,β (S d ) → L 2 (S d )) = 1.

Tensor power sequences and the approximation of tensor product operators

“…Comparable estimates for the case of τ being the sequence of approximation numbers of the embedding H s mix T d ֒→ L 2 T d are established in Theorem 4.9, 4.10, 4.17 and 4.20 of [KSU15]. See [CW16], [KMU16] or [CW17] for other examples. An interesting consequence of these preasymptotic estimates is the following tractability result.…”

How anisotropic mixed smoothness affects the decay of singular numbers for Sobolev embeddings

“…However, only [40] is really close to our setting. Closer to us are the papers by Cobos, Kühn, Sickel [7] (L ∞ approximation), by Krieg [26] (dominating mixed smoothness, periodic and nonperiodic), by Wang et al [3,4,22] (anisotropic Sobolev spaces, Sobolev spaces on the sphere) as well as the papers by Mieth [34] and Novak [37] (approximation on general domains).…”

Strong equivalences of approximation numbers and tractability of weighted anisotropic Sobolev embeddings

“…Recently, Kühn and many other authors investigated and obtained strong equivalences, preasymptotics, asymptotics of the approximation numbers and tractability of the classical isotropic Sobolev embeddings, Sobolev embeddings of dominating mixed smoothness, Gevrey space embeddings, anisotropic Sobolev embeddings on the torus T d = [0, 2π] d (see [11,12,10,1]), and Sobolev embeddings and Gevrey type embeddings on the sphere S d and on the ball B d (see [2]). In [24] Werschulz and Woźniakowski investigated tractability of weighted isotropic Sobolev embeddings.…”