Asymptotic behavior of the singular numbers of the imbedding operator of the Hardy class \(H^ 2\) into the weight space \(L_ 2(D_ r,\mu)\)
Let \(H^ 2\) be a Hardy class in the unit circle D. We denote \(D_ r=\{z:\) \(| z| <r\}\) and \(T_ r\) is the boundary of \(D_ r\). Let \(\mu\) be an arbitrary measure on \(\bar D_ r\) and \(\mu_ 0\) its restriction on \(T_ r\). In this note it is proved that the singular numbers \(\{s_ n\}\) of the imbedding operator I of \(H^ 2\) into \(L_ 2(\bar D_ r,\mu)\) satisfies the relation \(\lim_{n\to \infty} s_ n(I)r^{1/2-n}=\sqrt{g(\mu_ 0)},\) where \(g(\mu_ 0)=\exp [(2\pi r)^{-1}\int_{T_ r}n((d\mu_ 0/| dz|)| dz|].\)
scientific article; zbMATH DE number 3923362