Sobolev Spaces and their Relatives: Local Polynomial Approximation Approach

Entropy numbers in function spaces with mixed integrability

“…Nearest to us is the new section [4, §30] in the second edition of [3] extending [2] to spaces with mixed integrability. It might well be possible that a combination of piecewise polynomial approximations originating from [5,6,8] with these integral representations paves the way to deal directly with entropy numbers and widths, for example approximation numbers, in some spaces with mixed integrability.…”

“…They devised the method of piecewisepolynomial approximation which proved to be useful and which has been often used in modified and refined versions. We refer in particular to the anisotropic generalization in [7] and the recent report [8] where one finds also many references. If is a bounded domain in R n and s 1 ∈ R, s 2 ∈ R and p 1 , p 2 , q 1 , q 2 ∈ (0, ∞] then id: B s 1 p 1 q 1 ( ) → B s 2 p 2 q 2 ( ) is compact if, and only if,…”

Smoothness spaces of higher order on lower dimensional subsets of the Euclidean space

“…In the setting of the Euclidean space, is the main object of the theory of local polynomial approximation and, in particular, it gives a unified framework for the description of various spaces of smooth functions, see for example the survey .…”

“…Remark Such characterization of Besov spaces is fairly standard, see for example , for the case when and , for the case of n ‐sets.…”

“…There are very few approaches to spaces of higher order smoothness even on such kind of sets, in spite of the fact that the first order smoothness spaces have been extensively studied in different situations. One of the goals of the paper is to show the advantage of Calderón's approach or, more precisely, its local polynomial approximation interpretation in , and , in defining Sobolev type spaces in more general setting; see the related discussion in Section .…”