Approximation and entropy numbers in Besov spaces of generalized smoothness

Abstract

We determine the exact asymptotic behaviour of entropy and approximation numbers of the limiting restriction operator J : B p , q 1 s , ψ ( R d ) → B p , q 2 s ( Ω ) , defined by J ( f ) = f | Ω . Here Ω is a non-empty bounded domain in R d , ψ is an increasing slowly varying function, 0 < p < ∞ , 0 < q 1 , q 2 ⩽ ∞ , s ∈ R , and B p , q 1 s , ψ ( R d ) is the Besov space of generalized smoothness given by the function t s ψ ( t ) . Our results improve and extend those established by Leopold [Embeddings and entropy numbers in Besov spaces of generalized smoothness, in: Function Spaces, Lecture Notes in Pure and Applied Mathematics, vol. 213, Marcel Dekker, New York, 2000, pp. 323–336].