Abstract
The theory of the one-sided invertibility of discrete composite convolution operators in the space of sequences summed with exponential weight is constructed. In this part we consider those situations when Φ-theory of the operators is the same as the theory of their one-sided invertibility. The kernels and co-kernels of the operators are described. Constructions of the one-sided inverse operators are given.
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Gohberg I., Krein M.G.: On a dual integral equation and its traspose. I (Russian). Teoret. Prikl. Mat. Vyp. 1, 58–81 (1958)
Feldman I.A.: On the asymptotic behavior of solutions of certain systems of integral equations (Russian). Dokl. Akad. Nauk SSSR 154, 57–60 (1964)
Feldman I.A.: The asymptotics of solutions of systems of integral equations of Wiener-Hopf type. Sib. Math. J. 4, 596–615 (1965)
Feldman, I.A.: Asymptotic behavior of solutions of a system of Wiener-Hopf equations (Russian). Studies in Algebra and Math. Anal., pp. 147–152, Izdat “Karta Moldovenjaske”, Kishinev (1965)
Gohberg, I., Feldman, I.A.: Convolution Equations and Projection Methods for their Solution. In series: Translations of Mathematical Monographs, vol. 41. Amer. Math. Soc., Providence, RI (1974)
Krein M.G.: Integral equations on the half-line with a kernel depending on the difference of the arguments. Am. Math. Soc. Transl. 22, 163–288 (1962)
Gelfand, I.M., Raikov, D.A., Shilov, G.E.: Commutative Normed Rings. Chelsea Publishing Company, Bronx, New York (translation from Russian of the book published in 1960 by Fizmatgiz, Moscow) (1960)
Prössdorf S.: Some Classes of Singular Equations. In series: Mathematical Studies vol. 17. North-Holland Publishing Company, Amsterdam (1978)
Dybin, V.B.: Well-posed Problems for Singular Integral Equations (Russian). Rostov. Gos. Univ., Rostov-on-Don, 160 pp. ISBN: 5-7507-0030-5 (1988)
Dybin, V.B.: One-dimensional singular integral equations with coefficients that vanish on countable sets (Russian). Izv. Akad. Nauk SSSR Ser. Math. 51 (1987), 936–961; translation in Math. USSR-Izv. 31 (1988), 245–271
Dybin, V.B.: The Wiener-Hopf equation and the Blaschke product (Russian), Mat. Sb. 181 (1990), 779–803; translation in Math. USSR-Sb. 70 (1991), 205–230
Pasenchuk A.E.: Abstract Singular Operators. Novocherkassk, Izd. NPU (1993)
Dybin, V.B., Grudsky, S.M.: Introduction to the Theory of Toeplitz Operators with Infinite Index. In series: Operator Theory: Advances and Applications, vol. 137, Birkhäuser, Basel–Boston–Berlin (2002)
Dybin, V.B.: Equation of convolution on a material straight line in space of the functions summarized with exponential by scales. Parts 1–2. Vestnik-RUDN, Series: Mathematics, Computer science, Physics. 2011: No. 2, pp. 16–27, No. 3. pp. 39–48. (2011)
Dybin, V.B., Dzhirgalova, S.B.: The operator of discrete convolution in space {α, β} p , 1 ≤ p ≤ ∞. News of high schools Northern Caucasus region. Nat. Sciences. The appendix. 2003. No. 9. pp. 3–16 (2003)
Dybin, V.B., Dzhirgalova, S.B.: Compound discrete convolutions in space {α, β} p , 1 ≤ p ≤ ∞. Part I. RSU, Dep. VINITI 14.01.03. No. 90 (2003)
Dybin, V.B., Dzhirgalova, S.B.: Compound discrete convolutions in space {α, β} p , 1 ≤ p ≤ ∞. Part II. RSU, Dep. VINITI 12.11.03. No. 1946 (2003)
Gahov F.D., Chersky Y.I.: Equation of Convolution Type. Nauka, Moscow (1978)
Dybin, V.B., Dzhirgalova, S.B.: Scalar compound discrete convolutions in space {α, β} p , 1 ≤ p ≤ ∞. One-sided convertibility. Izv. Severo-Kavkaz. Nauchn. Centra Vyssh. Shkoly Ser. Estestv. Nauk. Special issue. Pseudo-differential equations and some problems of mathematical physics. pp. 56–63 (2005)
Dybin, V.B., Pasenchuk, A.E.: Discrete convolutions in the quarter plane with a vanishing symbol (Russian). Part I. Izv. Severo-Kavkaz. Nauchn. Tsentra Vyssh. Shkoly Ser. Estestv. Nauk. 1977/3, 7–10 (1977); Part II: ibidem 1979/4, 11–14 (1979)
Markushevich A.I.: Theory of the Analytical Functions, vol. I. Nauka, Moscow (1967)
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V. B. Dybin: This work was completed with the support of Southern Federal University, internal grant 05/6-89.
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Dybin, V.B., Dzhirgalova, S.B. Scalar Discrete Convolutions in Spaces of Sequences Summed with Exponential Weights—Part 1: One-Sided Invertibility. Integr. Equ. Oper. Theory 82, 575–600 (2015). https://doi.org/10.1007/s00020-014-2207-0
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DOI: https://doi.org/10.1007/s00020-014-2207-0