Approximation Order of Kolmogorov, Gelfand, and Linear Widths for Sobolev Embeddings in Euclidean Measure Spaces

In this paper we completely solve the problem of finding the upper approximation order with respect to the Kolmogorov, Gelfand, and linear widths for the embedding of the Sobolev spaces Wα ,p W α , p and Wα ,p₀ W 0 α , p into the Lebesgue space Lqν L ν q . Here, ν ν is a Borel probability measure with support contained in the open unit cube of the m -dimensional Euclidean space, and we cover the entire range of parameters 1≤ p,q≤ ∞ 1 ≤ p , q ≤ ∞ . We determine the exact values of the approximation orders solely in terms of the Lq L q -spectrum of ν ν . For the lower approximation order, we generally obtain only bounds; however, in the case q=∞ q = ∞ we identify the lower order exactly in terms of the lower Minkowski dimension. We also provide sufficient regularity conditions on the Lq L q -spectrum that ensure the upper and lower approximation orders coincide. Finally, we clarify intrinsic links between approximation orders and the fractal-geometric notions of the upper and lower Minkowski dimensions of the support of ν ν .