Exact asymptotic order for generalised adaptive approximations

Abstract

In this note, we present an abstract approach to study asymptotic orders for adaptive approximations with respect to a monotone set function $J$ defined on dyadic cubes. We determine the exact upper order in terms of the critical value of the corresponding $J$-partition function, and we are able to provide upper and lower bounds in terms of fractal-geometric quantities. With properly chosen $J$, our new approach has applications in many different areas of mathematics, including the spectral theory of Kreĭn–Feller operators, quantisation dimensions of compactly supported probability measures, and the exact asymptotic order for Kolmogorov, Gel'fand and linear widths for Sobolev embeddings into the Lebesgue space $L_{\nu }^p$.