The Wiener--Hopf equation in Nevanlinna and Smirnov algebras

V. S. Vladimirov

Izvestiya. Mathematics , Tome 31 (1988) no. 1, pp. 77-94.

Résumé

A generalized Wiener–Hopf equation on the semiaxis is considered in the class of analytic functionals which are the Fourier transform of Nevanlinna algebras $N^\pm$ or Smirnov algebras $N_*^\pm$. The problem, connected with this equation, of factoring measurable functions $\rho(x)$ on the axis in the algebras $N_*^\pm$ which satisfy the condition $(1+x^2)^{-1}\ln|\rho(x)|\in \mathscr L_1(-\infty,\infty)$ and also the problem of linear junction $\rho\varphi^+=\psi^-+F^+$ in the algebras $N^\pm$ and $N_*^\pm$ are also considered. Bibliography: 24 titles.

Bibliographie

[1] Wiener N., Hopf E., “Über eine Klasse singulärer Integralgleichungen”, Sitz. Berliner Akademi Wiss, 1931, 696–706 | Zbl

[2] Fok V. A., “O nekotorykh intervalnykh uravneniyakh matematicheskoi fiziki”, Matem. sb., 14(56):1,2 (1944), 3–50

[3] Muskhelishvili N. I., Vekua N. P., “Kraevaya zadacha Rimana dlya neskolkikh neizvestnykh funktsii i ee prilozhenie k sistemam singulyarnykh integralnykh uravnenii”, Tr. Tbilis. matem. in-ta, XII, 1943, 1–46

[4] Muskhelishvili N. I., Singulyarnye integralnye uravneniya, Fizmatgiz, M., 1962

[5] Vekua N. P., Sistemy singulyarnykh integralnykh uravnenii, Nauka, M., 1970 | MR | Zbl

[6] Ambartsumyan V. A., Nauchnye trudy, t. 1, AN ArmSSR, Erevan, 1960 | Zbl

[7] Gakhov F. D., Kraevye zadachi, Nauka, M., 1977 | MR

[8] Chandrasekar S., Perenos luchistoi energii, IL, M., 1953

[9] Sobolev V. V., Perenos luchistoi energii v atmosferakh zvezd i planet, Gostekhizdat, M., 1956 | MR

[10] Krein M. G., “Integralnye uravneniya na polupryamoi s yadrom, zavisyaschim ot raznosti argumentov”, Uspekhi matem. nauk, 13:5 (1958), 3–120 | MR | Zbl

[11] Gokhberg I. Ts., Krein M. G., “Sistemy integralnykh uravnenii na polupryamoi s yadrami, zavisyaschimi ot raznosti argumentov”, Uspekhi matem. nauk, 13:2 (1958), 3–72 | MR

[12] Maslennikov M. V., “Problema Milna s anizotropnym rasseyaniem”, Tr. matem. in-ta im. V. A. Steklova AN SSSR, 97, 1968, 3–134 | MR

[13] Arabadzhyan L. G., Engibaryan N. B., “Uravneniya v svertkakh i nelineinye funktsionalnye uravneniya”, Itogi nauki i tekhniki. Ser. matem. analiz, 22, VINITI, M., 1984, 175–244 | MR

[14] Vladimirov V. S., Obobschennye funktsii v matematicheskoi fizike, Nauka, M., 1979 | MR

[15] Vladimirov V. S., Drozhzhinov Yu. I., Zavyalov B. I., Mnogomernye tauberovy teoremy dlya obobschennykh funktsii, Nauka, M., 1986 | MR

[16] Goluzin G. M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1966 | MR

[17] Vladimirov V. S., Volovich I. V., “Ob odnoi modeli statisticheskoi fiziki”, TMF, 54:1 (1983), 8–22 | MR

[18] Vladimirov V. S., “Proizvedenie Blyashke v “obobschennom edinichnom kruge” i polnaya ortonormirovannaya sistema v trube buduschego”, Tr. matem. in-ta im. V. A. Steklova AN SSSR, 166, 1984, 44–51 | Zbl

[19] Shvedenko S. V., “Klassy Khardi i svyazannye s nimi prostranstva analiticheskikh funktsii v edinichnom kruge, polikruge i share”, Itogi nauki i tekhniki. Ser. matem. analiz, 23, VINITI, M., 1985, 3–124 | MR

[20] Komatsu H., “Ultradistributions. I: Structure theorems and a characterization”, J. Fac. Sci. Univ. Tokyo. Section IA, 20 (1973), 25–105 | MR | Zbl

[21] Komatsu H., “Ultradistributions. II: The kernel theorem and ultradistributions with support in a submanifold”, J. Fac. Sci. Univ. Tokyo. Section IA, 24 (1977), 607–628 | MR | Zbl

[22] Gelfand I. M., Shilov G. E., Prostranstva osnovnykh i obobschennykh funktsii. Obobschennye funktsii, 2, Fizmatgiz, M., 1958 | MR

[23] Privalov I. I., Granichnye svoistva analiticheskikh funktsii, GITTL, M., 1950

[24] Aleksandrov A. B., “Essays on non locally convex Hardy classes”, Complex Analysis and Spectral Theory, Lecture Notes in Mathem., 864, Springer, 1981, 1–89 | MR