Harmonic analysis of functions on semisimple Lie groups.~II

D. P. Zhelobenko

Izvestiya. Mathematics , Tome 3 (1969) no. 6, pp. 1183-1217.

Résumé

A theory of harmonic analysis is developed for the class of functions (fundamental and generalized) with compact support on an arbitrary semisimple complex connected Lie group. Duality theorems are proved for the linear topological spaces of finite functions most often encountered in analysis (infinitely differentiable finite functions, finite functions in $L^2$ , and finite generalized functions). All results are analogs of the standard theorems of Paley–Wiener type in harmonic analysis on the line.

Bibliographie

[1] Berezin F. A., “Operatory Laplasa na poluprostykh gruppakh Li”, Tr. Mosk. matem. ob-va, 6 (1957), 371–463 | MR | Zbl

[2] Burbaki N., Topologicheskie vektornye prostranstva, IL, M., 1959

[3] Viner N., Peli R., Preobrazovaniya Fure v kompleksnoi oblasti, Nauka, M., 1964 | MR

[4] Gelfand I. M., Graev M. I., “Preobrazovaniya Fure bystro ubyvayuschikh funktsii na kompleksnykh poluprostykh gruppakh Li”, Dokl. AN SSSR, 131:3 (1960), 496–499 | MR

[5] Gelfand I. M., Graev M. I., Vilenkin N. Ya., Integralnaya geometriya i svyazannye s nei voprosy teorii predstavlenii, Obobschennye funktsii, 5, Fizmatgiz, M., 1962

[6] Gelfand I. M., Naimark M. A., Unitarnye predstavleniya klassicheskikh grupp, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 36, 1950 | MR | Zbl

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

[8] Zhelobenko D. P., “Operatsionnoe ischislenie i teoremy tipa Peli–Vinera dlya poluprostoi kompleksnoi gruppy Li”, Dokl. AN SSSR, 170:6 (1966), 1243–1246

[9] Zhelobenko D. P., “O garmonicheskom analize funktsii na poluprostykh kompleksnykh gruppakh Li. I”, Izv. AN SSSR. Ser. matem., 27:6 (1963), 1343–1394 | MR | Zbl

[10] Zhelobenko D. P., “K teorii lineinykh predstavlenii kompleksnykh i veschestvennykh grupp Li”, Tr. Mosk. matem. ob-va, 12 (1963), 53–98

[11] Zhelobenko D. P., “Simmetriya v klasse elementarnykh predstavlenii poluprostoi kompleksnoi gruppy Li”, Funktsionalnyi analiz i ego prilozheniya, 2 (1967), 15–38

[12] Zhelobenko D. P., “Analiz neprivodimosti v klasse elementarnykh predstavlenii poluprostoi kompleksnoi gruppy Li”, Izv. AN SSSR. Ser. matem., 32 (1968), 108–133

[13] Zhelobenko D. P., “Operatsionnoe ischislenie na poluprostoi kompleksnoi gruppe Li”, Izv. AN SSSR. Ser. matem., 33 (1969), 931–973

[14] Zhelobenko D. P., Naimark M. A., “Opisanie vsekh vpolne neprivodimykh predstavlenii poluprostoi kompleksnoi gruppy Li”, Dokl. AN SSSR, 171:1 (1966), 25–28 | Zbl

[15] Zhelobenko D. P., “Analog teorii Kartana–Veilya dlya beskonechnomernykh predstavlenii poluprostoi kompleksnoi gruppy Li”, Dokl. AN SSSR, 175:1 (1967), 24–27

[16] Zhelobenko D. P., “O beskonechno differentsiruemykh vektorakh v teorii predstavlenii”, Vestnik MGU {, seriya 1}, 1 (1965), 3–10

[17] Mityagin B. S., “Approksimativnaya razmernost i bazisy v yadernykh prostranstvakh”, Uspekhi matem. nauk, 16:4 (1961), 63–132 | MR | Zbl

[18] Naimark M. A., “Ob opisanii vsekh unitarnykh predstavlenii kompleksnykh klassicheskikh grupp. I”, Matem. sb., 35(77) (1954), 317–356 | MR

[19] Naimark M. A., Lineinye predstavleniya gruppy Lorentsa, Fizmatgiz, M., 1958 | MR

[20] Harish-Chandra, “The Plancherel formula for complex semisimple groups”, Trans. Amer. Math. Soc., 76:3 (1954), 485–528 | DOI | MR | Zbl

[21] Shilov G. E., Matematicheskii analiz, vtoroi spetsialnyi kurs, Nauka, M., 1965 | MR