Linear average and stochastic \(n\)-widths of Besov embeddings on Lipschitz domains
For a given bounded linear operator \(T:X\to Y\), denote by \(\Delta_{\mu}(T_n)=\| T(\cdot)-T_n(\cdot)\|_{L^2(\mu,Y)}\) the average error of \(T\) approximated by \(T_n:X\to Y\) with \(\dim T_n(X)\leq n\), with respect to a probability Borel measure \(\mu\) on the unit ball of \(X\), and \(a_n^{\text{avg}}(T)=\inf_{T_n,\mu} \Delta_{\mu}(T_n)\) the linear \(n\)-width average of \(T\). Similarly, the linear stochastic \(n\)-width is defined as \(a_n^{\text{ran}}(T)=\inf_{T^\omega_n,\mu} \Delta_{\max}(T^\omega_n)\) with \(\Delta_{\max}(T^\omega_n)=\sup_{\| f\|_X\leq 1}(E(\| T(f)-T_n^\omega(f)\|_Y^2))^{1/2}\) if \(\dim T^\omega_n(X)\leq n\), and \(T^\omega_n\) is randomly chosen. The authors consider the compact embeddings \(I:B_{q_0}^{s+t}(L^{p_0}(\Omega))\to B_{q_1}^{s}(L^{p_1}(\Omega))\) of Besov spaces into a bounded Lipschitz domain \(\Omega\subset{\mathbb R}^d\), with \(t>d(1/p_0-1/p_1)_+\). They show that \(a_n^{\text{avg}}(I)\) and \(a_n^{\text{ran}}(I)\) behave like powers \(n^{-\alpha(t)}\), where \(\alpha(t)\) is a linear function of \(t\). For instance, if \(1\leq p_0\leq p_1\leq 2\), then \(\alpha(t)=t/d-(1/p_0-1/p_1)\). The proof is based on a characterization of Besov spaces by means of wavelet coefficients.
scientific article; zbMATH DE number 5639029
scientific article