Geodesic
Harmonic analysis of functions on semisimple Lie groups.~II
Izvestiya. Mathematics , Tome 3 (1969) no. 6, pp. 1183-1217

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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. 
@article{IM2_1969_3_6_a2,
     author = {D. P. Zhelobenko},
     title = {Harmonic analysis of functions on semisimple {Lie} {groups.~II}},
     journal = {Izvestiya. Mathematics },
     pages = {1183--1217},
     publisher = {mathdoc},
     volume = {3},
     number = {6},
     year = {1969},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1969_3_6_a2/}
}
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D. P. Zhelobenko. Harmonic analysis of functions on semisimple Lie groups.~II. Izvestiya. Mathematics , Tome 3 (1969) no. 6, pp. 1183-1217. http://geodesic.mathdoc.fr/item/IM2_1969_3_6_a2/