Kolmogorov widths of the class $W_1^1$
Yuri Malykhin
TL;DR
The paper resolves the longstanding open problem of determining the exact decay rate of Kolmogorov widths for the univariate Sobolev class $W_1^1[0,1]$ in $L_q[0,1]$ for $2<q<\infty$, proving $d_n(W_1^1[0,1],L_q[0,1]) \asymp c(q) n^{-1/2}\log n$. The authors extend the dyadic Haar decomposition by a simultaneous control of all levels via a tailored duality-averaging framework, inspired by Gluskin and related work, to obtain matching upper and lower bounds. A central technical contribution is a lemma developed around step functions $\chi_t$ and an isotropic construction of Haar coefficients, which delivers the necessary lower bound by bounding duality quantities $I_1,I_2,I_3$ through Marcinkiewicz–Paley type inequalities. Collectively, the result closes the logarithmic gap in the sharp order of decay for this classical case, providing precise width estimates and informing optimal subspace approximations in $L_q$ spaces for $2<q<\infty$.
Abstract
We prove that $d_n(W^1_1,L_q)\asymp n^{-1/2}\log n$, $2<q<\infty$. This completes the study of orders of decay of Kolmogorov widths for the classical case of the univariate Sobolev classes of integer smoothness.