Geodesic
Complex Tauberian theorems for the one-sided and two-sided Stieltjes transform
Izvestiya. Mathematics, Tome 8 (1974) no. 1, pp. 145-176 
M. A. Subkhankulov; F. I. An. Complex Tauberian theorems for the one-sided and two-sided Stieltjes transform. Izvestiya. Mathematics, Tome 8 (1974) no. 1, pp. 145-176. http://geodesic.mathdoc.fr/item/IM2_1974_8_1_a8/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In the paper we prove one-sided and two-sided complex Tauberian theorems for the Stieltjes transform. Tauberian theorems of the type of Keldysh theorem [1] with a remainder for the one-sided Stieltjes transform in the real domain were studied in [2]–[5]. Theorems of such type in the complex domain have not been studied in detail. Here only one result [6], [7] for the one-sided case is known.

[1] Keldysh M. V., “Ob odnoi tauberovoi teoreme”, Tr. Matem. in-ta im. V. A.Steklova AN SSSR, 38, 1951, 77–86 | Zbl

[2] Vuckovic V., “Quelques theorems relativs a la transformation de Stieltjes”, Publ. de l'Institut Math. Acad. Serbe, 6 (1954), 63–74 | MR | Zbl

[3] Subkhankulov M. A., “Ostatochnyi chlen v tauberovoi teoreme Khardi–Littlvuda–Karlemana”, Izv. AN SSSR. Ser. mat., 25:6 (1961), 925–934 | MR | Zbl

[4] Subkhankulov M. A., “Nekotorye obschie tauberovy teoremy s ostatochnym chlenom”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 64, 1961, 239–266

[5] Ganelius T., “Tauberian theorems for the Stiltjes transform”, Math. Scand., 14 (1964), 213–219 | MR | Zbl

[6] Malliavin P., “Un theoreme tauberian avec reste la transformee de Stieltijes”, Comptes Rendus de l'Academ. Sciences, Paris, 255 (1962), 2351–2352 | MR | Zbl

[7] Pleijel A., “On a theorem by P. Malliavin”, Israel J. Math., 1 (1963), 166–168 | DOI | MR | Zbl

[8] Gradshtein I. S., Ryzhik I. M., Tablitsa integralov, summ ryadov i proizvedenii, Fizmatgiz, M., 1962