Geodesic
Estimates of the deviations of partial Fourier sums on classes of continuous periodic functions of several variables
Izvestiya. Mathematics , Tome 17 (1981) no. 2, pp. 369-403.

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The author studies the magnitudes of the suprema of the deviations of rectangular Fourier sums on classes of continuous periodic functions determined by arbitrary partial moduli of continuity. Order precise upper estimates for these magnitudes are found, and in the case where the moduli of continuity are convex asymptotic equalities are obtained that are full analogues of the corresponding one-dimensional equalities of A. N. Kolmogorov and S. M. Nikol'skii. Bibliography: 12 titles. 
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A. I. Stepanets. Estimates of the deviations of partial Fourier sums on classes of continuous periodic functions of several variables. Izvestiya. Mathematics , Tome 17 (1981) no. 2, pp. 369-403. http://geodesic.mathdoc.fr/item/IM2_1981_17_2_a7/

[1] Stepanets A. I., Issledovaniya po ekstremalnym zadacham teorii summirovaniya ryadov Fure, Avtoreferat dokt. dissertatsii, Kiev, 1974

[2] Kolmogorov A. N., “Zur Grössenordnung des Restliedes Fourierschen Reihen differenrierbaren Functionen”, Ann. Math., 36 (1935), 521–526 | DOI | MR | Zbl

[3] Nikolskii S. M., “Priblizhenie periodicheskikh funktsii trigonometricheskimi mnogochlenami”, Tr. Matem. instituta im. V. A. Steklova AN SSSR, 15, 1945, 1–76 | MR | Zbl

[4] Bugaets P. T., “Asimptoticheskaya otsenka ostatka pri priblizhenii funktsii dvukh peremennykh summami”, Dokl. AN SSSR, 79:4 (1951), 203–205

[5] Stepanets A. I., “Otklonenie chastnykh summ ryada Fure na klassakh funktsii Gëldera dvukh peremennykh”, Dokl. AN SSSR, 206:3 (1972), 549–551 | Zbl

[6] Stepanets A. I., “Priblizhenie funktsii, udovletvoryayuschikh usloviyam Lipshitsa, summami Fure”, UMZh, 24:6 (1972), 599–609

[7] Stepanets A. I., Issledovaniya po ekstremalnym zadacham teorii summirovaniya ryadov Fure, Doktorskaya dissertatsiya, Kiev, 1974

[8] Podkorytov A. N., “Priblizhenie odnogo klassa funktsii dvukh peremennykh summami Fure”, Vestnik Leningradskogo universiteta, 13:3, 170–187

[9] Stepanets A. I.,, Priblizhenie summami Fure nepreryvnykh periodicheskikh funktsii mnogikh peremennykh, preprint IM–77–2, Kiev, 1977 | MR

[10] Korneichuk N. P., “Ekstremalnye znacheniya funktsionalov i nailuchshee priblizhenie na klassakh periodicheskikh funktsii”, Izv. AN SSSR. Ser. matem., 35 (1971), 93–124

[11] Stepanets A. I., “K odnoi zadache A. N. Kolmogorova v sluchae funktsii dvukh peremennykh”, UMZh, 24:5 (1972), 653–665 | Zbl

[12] Stepanets A. I., “Ob odnoi ekstremalnoi zadache v prostranstve nepreryvnykh periodicheskikh funktsii mnogikh peremennykh”, Voprosy teorii priblizhenii funktsii i ee prilozhenii, IM AN USSR, 1976, 160–178