Exact asymptotic order for generalised adaptive approximations

Marc Kesseböhmer; Aljoscha Niemann

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arXiv · 2312.16644

Exact asymptotic order for generalised adaptive approximations

Marc Kesseböhmer·Aljoscha Niemann

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Abstract

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

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Marc Kesseböhmer, Aljoscha Niemann. 2023-12-27. Exact asymptotic order for generalised adaptive approximations. https://arxiv.org/abs/2312.16644

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