Relative average widths of Sobolev spaces in \(L_2({\mathbb R}^d)\)
Let \(X(\mathbb{R}^{d})\) be a normed space of real valued functions on \( \mathbb{R}^{d}(d\in \mathbb{Z}_{+}),\) having the norm \(\left\| \cdot \right\| _{X}\) and let \(L\) be a subspace of \(X(\mathbb{R}^{d}).\) For two centrally symmetric subsets \(W\) and \(V\) of \(X(\mathbb{R}^{d})\) the quantity \(\overline{K}_{\sigma }(W,V,X(\mathbb{R}^{d}))=\inf\limits_{\overline{\dim }L\leq \sigma }\sup\limits_{f\in W}\inf\limits_{g\in V\cap L}\left\| f-g\right\| _{X}\) is called the average \(\sigma \)-width of \(W\) relative to \(V\) in the sense of Kolmogorov (shortly, relative average \(\sigma \)-\(K\) width). Here \(\overline{\dim }L\) denote the average dimension of \(L\) and \(\sigma>0.\) When \(V=X(\mathbb{R}^{d})\) one obtains the \(\sigma \)-\(K\) width of \(W\) which is denote by \(\overline{d}_{\sigma }(W,X(\mathbb{R}^{d})).\) The authors give the smallest number \(M>0\) for which the following equality \[ \overline{K}_{\sigma }(W,V,L_{2}(\mathbb{R}^{d}))=\overline{d}_{\sigma }(W,L_{2}(\mathbb{R}^{d})), \] holds in the case \(W=U(\mathcal{R}_{2}^{a})\) where \(U(\mathcal{R}_{2}^{a})\) is the closed unit ball in the space \(\mathcal{R}_{2}^{a}\) of Riesz potentials and \(V=M(\mathcal{R}_{2}^{a})\) is the closed ball of radius \(M\) in this space. In this case one obtains \(M=1\) (Th.1.1). The problem is also solved for the case \(W=U(\mathcal{K}_{2}^{a}),\) \(V=M(\mathcal{K}_{2}^{a}),\) where \(\mathcal{K}_{2}^{a}\) is the space of Bessel potentials, and one obtains \(M=1-(1+4\pi (\Gamma \left( \frac{d}{2}+1\right) \sigma )^{\frac{2}{d}})^{-\frac{\alpha }{2}}\) (Th.1.2). Finally, for \(0< \beta < \alpha \) one obtains: \[ \overline{K}_{\sigma }(U(W_{2}^{a})\cap B(L_{2}(\mathbb{R}^{d})),U(W_{2}^{\beta })\cap B(L_{2}(\mathbb{R}^{d})),L_{2}(\mathbb{R}^{d})) =\overline{d}_{\sigma }(U(W_{2}^{a}),L_{2}(\mathbb{R}^{d}), \] where \(B(L_{2}(\mathbb{R}^{d}))\) is the clossed unit ball in \(L_{2}(\mathbb{R }^{d})\) and \(W_{2}^{a}\) may be \(\mathcal{R}_{2}^{a}\) or \(\mathcal{K}_{2}^{a}.\)
scientific article; zbMATH DE number 5355168