Let a function $f$ with real poles that form a monotone bounded sequence be expanded in a Wolff-Denjoy series with positive coefficients. The main result of the paper states that, if we subtract the "linear part" from the function $1/f$, then the remaining "fractional part" is also expanded in a Wolff-Denjoy series (its poles are also real and the coefficients of the series are negative). An application of the result to operator theory is given.
Keywords: Wolff-Denjoy series, closed operator, left inverse operator, functional calculus
Received September 12, 2019
Revised November 13, 2019
Accepted November 18, 2019
Adolf Ruvimovich Mirotin, Dr. Phys.-Math. Sci., Prof., Francisk Skorina Gomel State University, g. Gomel, 246699 Belarus,
e-mail: amirotin@yandex.ru
Aleksandr Alekseevich Atvinovskii, Cand. Sci. (Phys.-Math.), Francisk Skorina Gomel State University, g. Gomel, 246699 Belarus,
e-mail: aatvinovskiy@gmail.com
REFERENCES
1. Sibilev R.V. A uniqueness theorem for Wolff–Denjoy series. St. Petersburg Math. J., 1996, vol. 7, no. 1, pp. 145–168.
2. Leont’eva T.A. The representation of analytic functions by series of rational functions. Math. Notes, 1967, vol. 2, no. 4, pp. 695–702. doi: 10.1007/BF01093644
3. Leont’eva T.A. Representation of functions analytic in a closed domain by series of rational functions. Math. Notes, 1968, vol. 4, no. 2, pp. 606–611. doi: 10.1007/BF01094960
4. Sherstyukov V.B. Expanding the reciprocal of an entire function with zeros in a strip in a Krein series. Sb. Math., 2011, vol. 202, no. 12, pp. 1853–1871. doi: 10.1070/SM2011v202n12ABEH004210
5. Akhiezer N. On some inversion formulae for singular integrals. Izv. Akad. Nauk SSSR Ser. Mat., 1945, vol. 9, no. 4, pp. 275–290 (in Russian).
6. Nevanlinna R. Asymptotische Entwicklungen Beschr$\ddot{\mathrm{a}}$nkter Funktionen und das Stieltjessche Momentenproblem. Ann. Acad. Sci. Fenn., Ser. A, 1922, vol. 18, no. 5, pp. 1–53.
7. Donoghue W.F. Distributions and Fourier transforms, N Y; London: Acad. Press, 1969, Ser. Pure Appl. Math. (Book 32), 315 p. ISBN: 0122206509 .
8. Mirotin A.R., Atvinovskii A.A. Analog for the Wiener lemma for Wolff-Denjoy series, 8 p., Available at: ArXiv: arXiv:1908.08029v1 [math.FA] 21 Aug 2019.
9. Atvinovskii A.A. On integral representation of a class of analytical functions. Izv. Gomel. Gos. Univ. im. F. Skoriny, 2011, vol. 67, no. 4, pp. 3–7 (in Russian).
10. Krein M.G., Nudelman A.A. The Markov Moment Problem and Extremal Problems. Leipzig: Max Planck Institute, 1977, 417 p. doi: 10.1090/mmono/050 . Original Russian text published in Krein M.G., Nudel’man A.A. Problema momentov Markova i ekstremal’nye zadachi. Moscow: Nauka Publ., 1973. 551 p.
11. Dunford N., Schwartz J.T. Linear operators Part 1: General Theory. N Y: Interscience, 1988, 872 p. ISBN: 978-0-471-60848-6 . Translated to Russian under the title Lineinye operatory, T. 1: Obshchaya teoriya. Moscow: Inostr. Lit. Publ., 1962.
12. Mirotin A.R., Atvinovskii A.A. Inversion of a linear combination of values of the resolvent of a closed operator. Problemy Fiziki, Matematiki i Tekhniki, 2014, no. 3 (20), pp. 77–79 (in Russian).
13. Atvinovskii A.A., Mirotin A.R. On some functional calculus of closed operators in a Banach space. Russian Math. (Iz. VUZ), 2013, vol. 57, no. 10, pp. 1–12. doi: 10.3103/S1066369X13100010
14. Atvinovskii A.A., Mirotin A.R. On some functional calculus of closed operators in a Banach space. II. Russian Math. (Iz. VUZ), 2015, vol. 59, no. 5, pp. 1–12. doi: 10.3103/S1066369X15050011
15. Mirotin A.R. The inverse of operator monotonic functions of negative operators on Banach spaces. Trudy Instituta Matematiki (Minsk), 2004, vol. 12, no. 1, pp. 104–108. (in Russian)
16. Ivanov V.K., Vasin V.V., Tanana V.P. Theory of linear ill-posed problems and its applications. Inverse and Ill-Posed Problems Series, Utrecht: VSP, 2002, 281 p. ISBN: 90-6764-367-X/hbk . Original Russian text published in Ivanov V.K., Vasin V.V., Tanana V.P. Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya. Moscow: Nauka Publ., 1978, 208 p.
Cite this article as: A.R.Mirotin, A.A.Atvinovskii. On multiplicative inversion for Wolff-Denjoy series, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 4, pp. 147–154.
