The Gel'fand \(n\)-widths of multivariate periodic Besov classes
Let \(L_{p}(T^{d})\), \(1\leq p\leq\infty\), be the usual periodic \(L_{p}\)-norm space on \(T^{d}\) and let \(\Omega_{k}(f,t)_{p}\) be the \(k\)-order modulus of smoothness of \(f\in L_{p}(T^{d})\). If \(k>\alpha>0\) and \(1\leq p,\theta \leq\infty\) then the periodic Besov spaces \(B_{p\theta}^{\alpha}=B_{p\theta }^{\alpha}(T^{d})\) consist of all functions \(f\in L_{p}(T^{d})\) such that the norm \(\left\| f\right\| _{B_{p\theta}^{\alpha}}=\left\| f\right\| _{p}+| f| _{B_{p\theta}^{\alpha}}\) is finite. Here \[ | f| _{B_{p\theta}^{\alpha}}=\left( \int_{0}^{\infty}\left( \frac{\Omega _{k}(f,t)_{p}}{t^{\alpha}}\right) ^{\theta}\frac{dt}{t}\right) ^{1/\theta },\quad \text{if }1\leq\theta<\infty, \] and \[ | f| _{B_{p\theta}^{\alpha}}=\sup_{t>0}\frac{\Omega_{k}(f,t)_{p}}{t^{\alpha} },\quad \text{if }\theta=\infty. \] If \(S_{p\theta}^{\alpha}\) is the unit ball in \(B_{p\theta}^{\alpha},\) the authors obtain the following evaluations for Gel'fand \(n\)-widths of \(S_{p\theta}^{\alpha}\) in \(L_{p}(T^{d})\): if \(1\leq\theta,q,p\leq\infty\) and \(\alpha>2d,\) then \(d^{n}(S_{p\theta}^{\alpha},L_{p}(T^{d}))\asymp n^{-\frac{\alpha }{d}}\) if \(1\leq q\leq p\leq\infty;\) \(\asymp n^{-\frac{\alpha}{d}}\) if \(1\leq p\leq q\leq2;\) \(\asymp n^{-\frac{\alpha}{d}+\frac{1}{2}-\frac{1}{q}}\) if \(1\leq p\leq2\leq q\leq\infty,\) \(\asymp n^{-\frac{\alpha}{d}+\frac{1}{p}-\frac{1}{q}}\) if \(2\leq p\leq q\leq\infty\), where \(A\asymp B\) means that there exists \(C\) independent of \(f\) and \(n\) such that \(A/C\leq B\leq CA.\)
scientific article; zbMATH DE number 1997660