By A.M. Jerbashian

This can be a precise publication regarding the speculation of services of a-bounded variety within the half-plane of the advanced airplane, that is built by means of software of the Liouville integro-differential operator.

In addition, the booklet includes advancements of a number of effects equivalent to the Phragmen-Lindelof precept and Nevanlinna Factorization within the Half-Plane, and provides a brand new, identical definition of the classical Hardy areas within the half-plane.

The final bankruptcy of the ebook provides an software of the built concept in addition to M.M.Djrbashian?s conception of Nevanlinna style sessions within the disc within the spectral thought of linear operators. it is a resolution of an issue many times acknowledged through M.G.Krein and being of particular curiosity for a very long time.

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**Additional info for Functions of a-Bounded Type in the Half-Plane**

**Example text**

1 W^-"log6,(^,C) = -W^-"Oa(^,C) r»+oo p\rj\ (")yo J-Mli{w-()-t + t,]'+' for w e C\{C -{- ih; 0 < h < +00}. It is obvious that the right-hand side integral in this equality is absolutely convergent. 2) for w G C\{C + ih : 0

2) for any u, v G (—oo,+oo). Hence, changing the order of integration we get + 00 W-''\og\ba{u / + iv,0\du -OO r\ri\ — ^ / {\rj\-\t\rsign {t + v)dt^U\rjlv). For calculation of the last integral, observe that sign {t+v) — sign v for \v\ > \r]\ and —\rj\ __ 0, and by S~ the remaining part of the same hne, where W^-"log|6a(^,C)l < 0 . __

1) k with zeros {wk} in the lower half-plane G~. 2) k the zeros Zk — W^^ (/c = 1, 2 , . . ) of which lie in the upper half-plane G+. l. For p < 0 we set G~ — {w \ \v^w < p) and G^ — {w \ Im tt; < p}. 1. 1°. 3) k=l for a given a G (—1, +oo). Then for any p < 0 the infinite product oo • oo N k=l ^ k=l J 25 absolutely and uniformly convergent in the closed half-plane G'^, and the function Ba{w, {wk}i^) is holomorphic in G~, where {wk}^ are its zeros. 2°. 3). Poof. 1°. Let p < 0 be a fixed number.