Geodesic
Singular numbers of a weighted convolution operator
Sbornik. Mathematics, Tome 61 (1988) no. 2, pp. 309-319 
O. G. Parfenov. Singular numbers of a weighted convolution operator. Sbornik. Mathematics, Tome 61 (1988) no. 2, pp. 309-319. http://geodesic.mathdoc.fr/item/SM_1988_61_2_a2/
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     author = {O. G. Parfenov},
     title = {Singular numbers of a~weighted convolution operator},
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     year = {1988},
     volume = {61},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1988_61_2_a2/}
}
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This article studies a convolution operator on $[0,2\pi]$ with $2\pi$-periodic kernel. The operator acts from $L_2(0,2\pi)$ to $L_2(0,2\pi;\mu)$ where $\mu$ is a Borel measure. An asymptotic formula is obtained for the singular numbers in the case of kernels satisfying a certain regularity condition and of measures with continuous densities. Examples and counterexamples are given. Bibliography: 7 titles.

[1] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov, Nauka, M., 1965

[2] Parfenov O. G., “Poperechniki odnogo klassa analiticheskikh funktsii”, Matem. sb., 117(159) (1982), 279–285 | MR | Zbl

[3] Rozenblyum G. V., “Pochti podobie operatorov i spektralnaya asimptotika psevdodifferentsialnykh operatorov na okruzhnosti”, Tr. MMO, 36 (1978), 59–84 | MR | Zbl

[4] Grenander U., Segë G., Teplitsevy formy i ikh prilozheniya, IL, M., 1961 | MR

[5] Suetin P. K., “Osnovnye svoistva mnogochlenov, ortogonalnykh po konturu”, UMN, XXI:2 (1966), 41–88 | MR

[6] Parfenov O. G., “Asimptotika singulyarnykh chisel operatorov vlozheniya nekotorykh klassov analiticheskikh funktsii”, Matem. sb., 115(157) (1981), 632–641 | MR | Zbl

[7] Parfenov O. G., “Singulyarnye chisla operatorov vlozhenii nekotorykh klassov analiticheskikh i garmonicheskikh funktsii”, Lineinye i nelineinye uravneniya v chastnykh proizvodnykh. Spektralnye asimptotiki, Izd-vo LGU, L., 1984, 56–66 | MR