The \(N\)-widths of spaces of holomorphic functions on bounded symmetric domains of tube type
Let \(X\) be a normed linear space and \(W\) a subset of \(X\). The Kolmogorov \(N\)-width \(d_N(W,X)\) of \(W\) in \(X\) is defined by: \(d_N(W,X)= \inf [\sup\{\inf (\|v-w\|\); \(w\in X_N)\); \(v\in W\}\); \(X_N\subset X\) an \(N\)-dimensional subspace of \(X]\). The linear (resp. Bernstein, Gelfand) \(N\)-width of \(W\) in \(X\) is also defined as a means of measuring the degree to which \(W\) can be approximated by \(N\)-dimensional subspaces of \(X\). The authors consider the Hardy (resp. Bergman) spaces \(H_2(D)\) (resp. \(A_2(D))\) of holomorphic functions on a bounded symmetric domain \(D\) of tube type with the Shilov boundary \(\Sigma\), denoting by \(B(H_2(D))\) the closed unit ball in \(H_2(D)\). For an integer \(\ell\geq 0\) the authors define the \(\ell\)-th radial derivative \(R^\ell f\) of a holomorphic function \(f\) on \(D\). For \(0<\rho<\ell\), denote by \(W\) the set of holomorphic functions \(f\) on \(D\) such that \(R^\ell f\in B(H_2 (D))\) and set \(X=C(\rho\Sigma)\) the space of continuous functions on \(\rho \Sigma\). The authors show that the linear and Gelfand \(N\)-width of \(W\) in \(X\) coincide and then compute the exact value by using the expression \(D=\{w\in V^\mathbb{C}; |w|<1\}\), where \(V\) is a simple Jordan algebra and \(V^\mathbb{C}\) is the complexification of \(V\). The authors consider the similar situations for Hardy space \(H_p(D)\) for \(1\leq p\leq\infty\) and show that Kolmogorov, linear, Gelfand and Bernstein \(N\)-widths all coincide for the Hardy-Sobolev space \(W= H_{p, \ell}(D)\) in \(X=L^p(\Sigma_\rho)\) and then compute the exact value. Similar results are obtained for Bergman-Sobolev space \(A_{p,\ell}(D)\).
scientific article; zbMATH DE number 1455761