Продолжение Целых Функций Вполне Регулярного Роста И Правый Обратный Для Оператора Представления Аналитических Функций Рядами Квазиполиномов

О Разложении В Ряды Экспонент Функций, Аналитических На Выпуклых Локально Замкнутых Множествах

“…In [4,5,11] was investigated a problem of the determination of the coefficients of the expansions of all f ∈ A(Q), where Q is a convex bounded domain in C, in following setting. Let K ⊂ C be a convex set and suppose that L is an entire function on C with zero set (λ j ) j∈N and with the indicator H Q + H K , where H Q and H K is the support function of Q resp.…”

“…of K. By Λ 1 (Q) we denote a Fréchet space of all number sequence (c j ) j∈N such that the series j∈N c j exp(λ j •) converges absolutely in A(Q). In [4,5,11] were established the necessary and sufficient conditions under which a sequence of the coefficients (c j ) j∈N ∈ Λ 1 (Q) in a representation f = j∈N c j exp(λ j •) can be selected in such way that they depend continuously and linearly on f ∈ A(Q). In other words, in [4,5,11] was solved the problem of the existence of continuous linear right inverse for the representation operator R : Λ 1 (Q) → A(Q), c → j∈N c j exp(λ j •).…”

“…In [4,5,11] were established the necessary and sufficient conditions under which a sequence of the coefficients (c j ) j∈N ∈ Λ 1 (Q) in a representation f = j∈N c j exp(λ j •) can be selected in such way that they depend continuously and linearly on f ∈ A(Q). In other words, in [4,5,11] was solved the problem of the existence of continuous linear right inverse for the representation operator R : Λ 1 (Q) → A(Q), c → j∈N c j exp(λ j •). Note that in [4,5] a formula for continuous linear right inverse for R (if it exists) was not obtained.…”

“…In contrast to [4,5] here we do not use the structure theory of locally convex spaces. As in [11], we reduce the problem of existence a continuous linear right inverse for the representation operator to one of an extension of input function L to an entire functionL on C 2N satisfying some upper bounds. With the help ofL we construct a continuous linear left inverse for the transposed map to R. Using∂-technique, we obtain that the existence of such extensionL is equivalent to two conditions, namely, to the existence of two families of plurisubharmonic functions, first of which is associated only with Q and second is associated with K and Q.…”