Shape-preserving widths of weighted Sobolev-type classes of positive, monotone, and convex functions on a finite interval
Let \(I\) be a finite interval in \({\mathbb R}\), and let \(\rho (t)= \text{dist}\{t, \partial I\}\), \(t\in I\). For \(1\leq p\leq\infty\) and \(0\leq\alpha<\infty\), denote by \(W^r_{p,\alpha}(I)\), \(r\in{\mathbb N}\), the Sobolev-type class of all real-valued functions \(f\) on \(I\) such that \(f^{(r-1)}\in AC_{\text{loc}}(I)\) and \(\|f^{(r)}\rho^{\alpha}\|_{L_p(I)}\leq 1\). Let \(\Delta_{\tau }^{s}f(t)=\sum_{k=0}^{s}(-1)^{s-k}\left( ^{s}_{k}\right) f(t+k\tau)\), \(\{t, t+s\tau\}\subset I\), \(s=0, 1,\dots\), be the \(s\)th difference of the function \(f\), with step \(\tau>0\), and denote by \(\Delta^{s}_{+}W^{r}_{p,\alpha}(I)\), \(s=0, 1,\dots\), the subclasses of functions \(f\in W^{r}_{p,\alpha}\) for which \(\Delta^{s}_{\tau}f(t)\geq 0\), for all \(\tau\geq 0\) such that \([t, t+s\tau]\subseteq I\). Denote by \(\Delta^s_{+}L_q\) the subset of all functions \(y\in L_q(I)\) such that \(\Delta^s_{+} y(t)\geq 0\), \(\tau\geq 0\). In a previous paper the authors [J. Approximation Theory 113, 266-297 (2001; Zbl 0992.41013)] have investigated the behavior of the Kolmogorov widths \(d_n(\Delta^s_{+}W^{r}_{p,\alpha})_{L_q}\) and of the linear widths \(d_n(\Delta^s_{+}W^{r}_{p,\alpha})_{L_q}^{\text{lin}}\). In this paper they obtain for \(s=0,1,2\), two-sided estimates of the shape-preserving widths \[ d_n(\Delta^{s}_{+}W^{r}_{p,\alpha}, \Delta^s_{+}L_q)_{L_q}:= \inf_{M^n\in{\mathcal{M}}^n}\sup_{x\in \Delta^s_{+}W^r_{p,\alpha}}\inf_{y\in M^n\cap \Delta^s_{+}L_q} \|x-y\|_{L_q}, \] where \({\mathcal{M}}^n\) is the set of all linear manifolds \(M^n\) in \(L_q\) such that \(\dim M^n\leq n\), and satisfying \(M^n\cap \Delta^s_{+}L_q\neq\emptyset\).
scientific article; zbMATH DE number 1883398
scientific article