Kolmogorov width of classes of smooth functions on the sphere \(\mathbb S^{d-1}\)
The main result of the paper is an extension of the well-known theorem of Kashin on the widths of the Sobolev classes of periodic functions to the case of functions defined on the unit sphere \(S^{d-1} =\{x \in {\mathbb R}^d: \| x\| _2=1\}\). For \(r>0\), let \(B_p^r\) be the class of functions \(f:S^{d-1} \to {\mathbb R}\) representable in the form \[ f(x)=\int_{S^{d-1}}\phi(y)F_r(xy)d\sigma(y), \qquad \| \phi\| _p \leq 1, \] where \(d\sigma(y)\) is the normalized Lebesgue measure on \(S^{d-1}\), \[ F_r(xy)=\sum_{k-1}^\infty (k(k+2\lambda))^{-{r \over 2}}{\Gamma(\lambda)(k+\lambda) \over 2\pi^{\lambda+1}} P_k^\lambda(xy), \] \(\lambda={d-2 \over 2}\), and \(P_k^\lambda(t)\) is the ultraspherical polynomial. It is proved that for \(r\geq(d-1)^2\), \[ d_N(B_p^r, L^q) \asymp \begin{cases} N^{-{r \over d-1}+{1 \over p}-{1\over 2}} & \text{if } 1\leq p\leq 2 <q\leq \infty, \cr N^{-{r \over d-1}}& \text{if } 2\leq p <q \leq \infty \end{cases} \] Kashin's result corresponds to \(d=2\). Some similar results were claimed earlier by \textit{A. Kushpel} [J. Complexity 16, 424--458 (2000; Zbl 0960.41018)], but the authors show that his proof contains an essential mistake.
scientific article; zbMATH DE number 1910743
scientific article