Hilbert numbers of Sobolev spaces with mixed smoothness in the sup-norm

Abstract

In this paper, we study Hilbert numbers of embedding of Sobolev space of mixed smoothness \(H^{s,r}_{\textrm{mix}}({{\mathbb {T}}}^d)\) on the torus \({{\mathbb {T}}}^d\) into \(L_\infty ({{\mathbb {T}}}^d)\) and the Wiener class \(\mathcal {A}({{\mathbb {T}}}^d)\). We obtain the exact asymptotic order of Hilbert numbers of these embeddings and the asymptotic constant for the embedding into \(\mathcal {A}({{\mathbb {T}}}^d)\). We also obtain the asymptotic constant of Hilbert numbers of embedding of Gaussian weighted Sobobev space \(H^s({{\mathbb {R}}}^d,\gamma )\) into \(\mathcal {A}({{\mathbb {R}}}^d,\gamma )\) which is a counterpart of the Wiener class \(\mathcal {A}({{\mathbb {T}}}^d)\).