On a convolution in the spaces of functions which are analytic in
disk domains (in Ukrainian)

T.I. Zvozdetskyi; S.S. Linchuk

Let ${\cal A}_R$ ($0 < R \leq \infty$) be the space of all analytic in the disk $z \in \Bbb C:|z| < R$ functions with topology of compact convergence, $L$ a linear continuous functional on ${\cal A}_R$ and~${\cal I}_{\alpha}$ the~operator of generalized integration which is determined by a sequence $\{\alpha_n:n \geq 0 \}$ of a non-zero complex numbers. The necessary and sufficient conditions on the sequence $\{\alpha_n:n \geq 0\}$ for existing of non-trivial continuous convolution $*$ for operator ${\cal I}_{\alpha}+L$ with $({\cal I}_{\alpha}+L)f=1*f$ in the space ${\cal A}_R$ are found.

1. Dimovski I.H. Convolutional calculus. – Sofia: Publ. House Bulg. Acad. Sci., 1982. – 198p.

2. Berg L. Generalized convolutions Math. Nachr. – 1976. – V.72. – P.239–245.

3. Dimovski I.H. Convolutions for the right inverse linear operators of the general linear differential operator of the first order Sofia: Bulg. Math. Publ. – 1976. – V.2, №1. – P.82–86.

4. Dimovski I.H., Mineff D.M. Convolution, multipliers and commutants for the backward shift operator Плиска Бълг. мат. студ. – 1981. – №4. – C.128–136.

5. Dimovski I.H. Representation formulas for the commutants of integer powers of Gelfond-Leontiev integration operators Math. and education in Math., 1982. – Sofia, 1982. – P.166–172.

6. Dimovski I.H., Kiryakova V.S. Convolution and commutant of Gelfond-Leontiev operator of integration Конструктивная теория функций: Тр. междунар. конф. (Варна, 1–5 июня 1981г.). – София, 1983. – С.288–294.

7. Линчук Н.Е. Представление коммутантов оператора обобщенного интегрирования Гельфонда-Леонтьева Изв. вузов. Математика. – 1985. – №5. – C.72–74.

8. Лiнчук С.С. Про побудову згорток у просторах аналiтичних функцiй Крайовi задачi з рiзними виродженнями i особливостями: Зб. наук. пр. – Чернiвцi, 1990. – C.138–142.

9. Коробейник Ю.Ф. Операторы сдвига на числовых семействах. – Ростов-на-Дону: Изд-во Рост. ун-та, 1983. – 160c.

10. Шефер Х. Топологические векторные пространства. – М.: Мир, 1971. – 359с.

	On a convolution in the spaces of functions which are analytic in disk domains (in Ukrainian)
Author	T.I. Zvozdetskyi, S.S. Linchuk Department of Mathematical Analysis, Yuriy Fedkovych Chernivtsi National University, 2 Kotsiubynskoho St., Chernivtsi 274012, Ukraine, Tel.: +380 372 59 84 88
Abstract	Let ${\cal A}_R$ ($0 < R \leq \infty$) be the space of all analytic in the disk $z \in \Bbb C:\|z\| < R$ functions with topology of compact convergence, $L$ a linear continuous functional on ${\cal A}_R$ and~${\cal I}_{\alpha}$ the~operator of generalized integration which is determined by a sequence $\{\alpha_n:n \geq 0 \}$ of a non-zero complex numbers. The necessary and sufficient conditions on the sequence $\{\alpha_n:n \geq 0\}$ for existing of non-trivial continuous convolution $$ for operator ${\cal I}_{\alpha}+L$ with $({\cal I}_{\alpha}+L)f=1f$ in the space ${\cal A}_R$ are found.
Keywords	analytic functions in disk, compact convergence topology, linear continuous functional, complex numbers
DOI	doi:10.30970/ms.9.1.78-89
Reference	1. Dimovski I.H. Convolutional calculus. – Sofia: Publ. House Bulg. Acad. Sci., 1982. – 198p. 2. Berg L. Generalized convolutions Math. Nachr. – 1976. – V.72. – P.239–245. 3. Dimovski I.H. Convolutions for the right inverse linear operators of the general linear differential operator of the first order Sofia: Bulg. Math. Publ. – 1976. – V.2, №1. – P.82–86. 4. Dimovski I.H., Mineff D.M. Convolution, multipliers and commutants for the backward shift operator Плиска Бълг. мат. студ. – 1981. – №4. – C.128–136. 5. Dimovski I.H. Representation formulas for the commutants of integer powers of Gelfond-Leontiev integration operators Math. and education in Math., 1982. – Sofia, 1982. – P.166–172. 6. Dimovski I.H., Kiryakova V.S. Convolution and commutant of Gelfond-Leontiev operator of integration Конструктивная теория функций: Тр. междунар. конф. (Варна, 1–5 июня 1981г.). – София, 1983. – С.288–294. 7. Линчук Н.Е. Представление коммутантов оператора обобщенного интегрирования Гельфонда-Леонтьева Изв. вузов. Математика. – 1985. – №5. – C.72–74. 8. Лiнчук С.С. Про побудову згорток у просторах аналiтичних функцiй Крайовi задачi з рiзними виродженнями i особливостями: Зб. наук. пр. – Чернiвцi, 1990. – C.138–142. 9. Коробейник Ю.Ф. Операторы сдвига на числовых семействах. – Ростов-на-Дону: Изд-во Рост. ун-та, 1983. – 160c. 10. Шефер Х. Топологические векторные пространства. – М.: Мир, 1971. – 359с.
Pages	78-89
Volume	9
Issue	1
Year	1998
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On a convolution in the spaces of functions which are analytic in disk domains (in Ukrainian)