Best approximation in Hardy spaces and by polynomials, with norm constraints
Let \(G\) be either the disk or the annulus. Let \(I\) be a subset of the boundary of \(G\) so that both \(I\) and its complement \(J\) in the boundary of \(G\) have positive Lebesgue measure. Let \(f\) be a Lebesgue integrable function on \(I\) and let \(c\) and \(M > 0\) be given numbers. The problem of this study is to find a solution \(g\) so that the \(L_p\)-norm over \(I\) of \(f-g_c\) is equal \[ \min_g\{\| f-g_c\|_{L_p(I)} , g \in H^p(G) ; \|g -c\|_{L_p(J )}\leq M \}; \] where \( H^p(G) \) represents the Hardy space associated with \(p\) and \(G\). Truncated versions of the problem are analyzed with solutions chosen from finite dimensional spaces of polynomials or rational functions. The solutions are presented in terms of truncated Toeplitz operators. The results are illustrated with numerical examples. The results are related to inverse problems for PDEs.
subsets of boundary of the disk or annulus
scientific article; zbMATH DE number 6160907
scientific article