Infinite-dimensional widths in the spaces of function. I
For a positive integer \(r\) and \(p\in[1,+\infty]\) consider the following Sobolev function classes over \(\mathbb{R}\): \[ W^ r_ p(\mathbb{R}):=\{f\in L^ p(\mathbb{R}):f^{(r-1)}\text{ l.a.c. on }\mathbb{R},\quad f^{(r-1)}\in L^ p(\mathbb{R})\} \] \[ B^ r_ p(\mathbb{R}):=\{f\in W^ r_ p(\mathbb{R}):\| f^{(r)}\|_ p\leq 1\} \] where l.a.c.:=locally absolutely continuous and \(\|\centerdot\|_ p\) is the usual \(L^ p(\mathbb{R})\)-norm. Let \(S_{r-1}\) denotes the space of cardinal polynomial splines of degre \(r- 1\) with simple integer knots, i.e. \[ S_{r-1}:=\{s\in C^{r- 2}(\mathbb{R}):s\mid_{(j,j+1)}\in{\mathcal P}_{r-1},\;j\in\mathbb{Z}\} \] where \({\mathcal P}_{r-1}\) is the polynomial function class of degree \(r-1\). Also, for \(f\in W^ r_ 2(\mathbb{R})\), let \(s_{r-1}(f;x)\) be the function which interpolates \(f(x)\) at points \(\{j+a_ r\}\), \(j\in\mathbb{Z}\), \(a_ r={1\over 4}(1+(-1)^{r-1})\) from \(S^{r-1}\), i.e. \(s_{r-1}(f;j+a_ r)=f(j+a_ r)\), \(j\in\mathbb{Z}\). In a previous paper by the author and Sun Yongsheng, it was shown that \[ \sup_{f\in B^ r_ 2(\mathbb{R})}\| f- s_{r-1}(f)\|_ 2=\sup_{f\in B^ r_ 2(\mathbb{R})}\| f-s_{2r- 1}(f)\|_ 2=\pi^{-r}. \] In the present paper the author considers the problem of computing and comparing the quantities \(E(B^ r_ p(\mathbb{R});S_{r-1})_ p\) and \(E(B^ r_ p(\mathbb{R});S_{2r-1})_ p\) where \[ E(B^ r_ p(\mathbb{R});A)_ p:=\sup_{f\in B^ r_ p(\mathbb{R})}\inf_{g\in A}\| f-g\|_ p \] for any set \(A\) of functions on \(\mathbb{R}\).
scientific article; zbMATH DE number 21552