Entropy Numbers of Finite Dimensional Mixed-Norm Balls and Function Space Embeddings with Small Mixed Smoothness

Abstract

AbstractWe study the embedding $$\mathrm {id}: \ell _p^b(\ell _q^d) \rightarrow \ell _r^b(\ell _u^d)$$ id : ℓ p b ( ℓ q d ) → ℓ r b ( ℓ u d ) and prove matching bounds for the entropy numbers $$e_k(\mathrm {id})$$ e k ( id ) provided that $$0<p<r\le \infty $$ 0 < p < r ≤ ∞ and $$0<q\le u\le \infty $$ 0 < q ≤ u ≤ ∞ . Based on this finding, we establish optimal dimension-free asymptotic rates for the entropy numbers of embeddings of Besov and Triebel–Lizorkin spaces of small dominating mixed smoothness, which gives a complete answer to an open problem mentioned in the recent monograph by Dũng, Temlyakov, and Ullrich. Both results rely on a novel covering construction recently found by Edmunds and Netrusov.