Kolmogorov widths of an intersection of anisotropic finite-dimensional balls: case $2\le q_j<\infty$
A. A. Vasil'eva
TL;DR
This work establishes sharp order estimates for the Kolmogorov $n$-widths of the intersection $M=\bigcap_{\alpha\in A} \nu_{\alpha} B^{\bar k}_{\bar p_\alpha}$ embedded in the anisotropic space $l^{\bar k}_{\bar q}$ with $2\le q_j<\infty$. The authors derive a unifying upper and lower bound framework expressed through a combinatorial optimization over affine-hull configurations ${\cal N}_m, {\cal Z}_m$ (and their relaxed versions) and a pivotal function $\Phi$ evaluated at a derived parameter vector $\bar\theta(\cdot)$; the width is asymptotically equivalent to a minimax combination of radii $\nu_\alpha$ and $\Phi$. The analysis handles both finite and general index sets $A$, leveraging a careful analysis of generalized positions, complementarity of affine spaces, and a Dubovitskii–Milyutin-type variational argument to identify minimizing configurations. The results extend prior results on low-dimensional anisotropic balls to general dimension and provide a tool for understanding widths of intersections of anisotropic function classes, with potential applications to multi-parameter approximation problems in finite-dimensional anisotropic norms.
Abstract
Order estimates for the Kolmogorov $n$-widths of $\cap _{α\in A}ν_αB^{\overline{k}}_{\overline{p}}$ in $l^{\overline{k}} _{\overline{q}}$ are obtained; here $\overline{q}=(q_1, \, \dots, \, q_d)$, $2\le q_j<\infty$, $j=1, \, \dots, \, d$.