A generalization of the Wiener–Hopf method for convolution equations on a finite interval with symbols having power-like asymptotics at infinity

B. V. Pal'tsev

Sbornik. Mathematics, Tome 41 (1982) no. 3, pp. 289-328

Résumé

A generalization of the Wiener–Hopf method is obtained for convolution equations on the finite interval $(-T,T)$ $$ (\mathbf Ku)(t)=f(t),\qquad|t|<T, $$ where $\mathbf K$ is the convolution operator $\mathbf Ku(t)=(r_{(-T,T)}k*u)(t)$, $u(t)\in\mathscr S'(\mathbf R^1)$, $u(t)\equiv0$ for $|t|>T$, $*$ is the convolution operation, $k=k(t)$ is a kernel belonging to $\mathscr S'(\mathbf R^1)$, $r_{(-T,T)}$ is the operator of restriction of a generalized function to the interval $(-T,T)$, and $f(t)\in\mathscr D'(-T,T)$. Here $\mathscr S(\mathbf R^1)$ and $\mathscr S'(\mathbf R^1)$ are the Schwartz spaces of rapidly decreasing test functions and generalized functions of slow growth on $\mathbf R^1$, respectively. Bibliogrpahy: 19 titles.

Bibliographie

[1] N. Winer, E. Hopf, Über eine Klasse singulärer Integralgleichungen, Sitzungsber. Acad. Wiss., 1931

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

[3] B. Nobl, Primenenie metoda Vinera–Khopfa dlya resheniya differentsialnykh uravnenii v chastnykh proizvodnykh, IL, Moskva, 1962

[4] M. G. Krein, “Integralnye uravneniya na polupryamoi s yadrom, zavisyaschim ot raznosti argumentov”, Uspekhi matem. nauk, XIII:5(83) (1958), 3–120 | MR

[5] I. Ts. Gokhberg, M. G. Krein, “Sistemy integralnykh uravnenii na polupryamoi s yadrami, zavisyaschimi ot raznosti argumentov”, Uspekhi matem. nauk, XIII:2(83) (1958), 3–72

[6] L. R. Volevich, S. G. Gindikin, “Uravnenie Vinera–Khopfa v obobschennykh funktsiyakh”, Trudy Mosk. matem. ob-va, 35 (1976), 165–214 | Zbl

[7] F. D. Gakhov, Yu. I. Cherskii, Uravneniya tipa svertki, izd-vo “Nauka”, Moskva, 1978 | MR

[8] Z. Prësdorf, Nekotorye klassy singulyarnykh uravnenii, izd-vo “Mir”, Moskva, 1979 | MR

[9] M. P. Ganin, “Ob integralnom uravnenii Fredgolma s yadrom, zavisyaschim ot raznosti argumentov”, Izv. VUZ'ov, Matematika, 1963, no. 2, 31–43 | MR | Zbl

[10] B. V. Paltsev, “Asimptotika spektra i sobstvennykh funktsii operatorov svertki na konechnom intervale s odnorodnym preobrazovaniem Fure yadra”, DAN SSSR, 218:1 (1974), 28–31

[11] B. V. Paltsev, “O zadache Dirikhle dlya odnogo psevdodifferentsialnogo uravneniya, vstrechayuschegosya v teorii sluchainykh protsessov”, Izv. AN SSSR, seriya matem., 41 (1977), 1348–1387

[12] B. V. Paltsev, “Uravneniya svertki na konechnom intervale dlya odnogo klassa simvolov, imeyuschikh stepennuyu asimptotiku na beskonechnosti”, Izv. AN SSSR, seriya matem., 44 (1980), 322–394 | MR

[13] V. Yu. Novokshenov, “Uravneniya v svertkakh na konechnom otrezke i faktorizatsiya ellipticheskikh matrits”, Matem. zametki, 27:6 (1980), 935–946 | MR | Zbl

[14] V. S. Vladimirov, Obobschennye funktsii v matematicheskoi fizike, izd-vo “Nauka”, Moskva, 1979 | MR

[15] L. R. Volevich, B. P. Paneyakh, “Nekotorye prostranstva obobschennykh funktsii i teoremy vlozheniya”, Uspekhi matem. nauk, XX:1(121) (1965), 3–74

[16] S. G. Krein, Lineinye uravneniya v banakhovom prostranstve, izd-vo “Nauka”, Moskva, 1971 | MR

[17] R. Hunt, B. Muckenhoupt, R. Wheeden, “Weighted norm inequalities for the conjugate function and Hilbert transform”, Trans. Amer. Math. Soc., 176 (1973), 227–251 | DOI | MR | Zbl

[18] G. F. Mandzhavidze, B. V. Khvedelidze, “O zadache lineinogo sopryazheniya i singulyarnykh integralnykh uravneniyakh s yadrom Koshi s nepreryvnymi koeffitsientami”, Trudy Tbil. matem. in-ta, 28 (1962), 85–105 | Zbl

[19] B. V. Khvedelidze, “Metod integralov tipa Koshi v razryvnykh granichnykh zadachakh teorii golomorfnykh funktsii odnoi kompleksnoi peremennoi”, Itogi nauki i tekhniki. Sovremennye problemy matematiki, 7, 1975, 5–162