An elementary description of ideals localization methods

N. K. Nikolskii

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 17, Tome 170 (1989), pp. 207-232

Résumé

The paper is a short survey of a part of the theory of divisorial ideals for algebras (and spaces) of holomorphlc functions $X$ determined by growth conditions near the boundary: $X=X(\{\lambda_n\})\stackrel{def}{=}\{\,f\in\mathrm{Hol}\,(\Omega): |f(z)|\leqslant c\lambda_n(z), z\in\Omega; c=c_f, n=n_f\,\}$ where $\Omega\subset\mathbb{C}$, $\lambda_n$ are positive in $\Omega$. All methods used to prove divisoriality are classified into three groups: direct canonical products method by Weieratrass and Hadamard; approximate identity method by L. Schwartz and A. Beurling; spectral (resolvent, with estimations) method by L. Waelbroeck, L. Hörmander et al. Some observations and propositions seem to be new. Formally speaking, the paper can be considered as part II of survey [1].