Geodesic
Difference properties of functions of several variables
Izvestiya. Mathematics, Tome 3 (1969) no. 3, pp. 633-642 
M. F. Timan. Difference properties of functions of several variables. Izvestiya. Mathematics, Tome 3 (1969) no. 3, pp. 633-642. http://geodesic.mathdoc.fr/item/IM2_1969_3_3_a7/
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     author = {M. F. Timan},
     title = {Difference properties of functions of several variables},
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     url = {http://geodesic.mathdoc.fr/item/IM2_1969_3_3_a7/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this article, new identities are established for the so-called total, mixed, and partial differences of functions of several variables. As applications of these identities, expressions are given for mixed derivatives in terms of directional derivatives, as well as inequalities for total and mixed moduli of smoothness of continuous functions of $n$ variables. Moreover, it is shown that in the spaces $L_p$ $(1$, the orders of decrease of the total modulus of smoothness and of the sum of the moduli of smoothness in the coordinate directions only are the same.

[1] Timan M. F., “Nekotorye voprosy konstruktivnoi teorii funktsii mnogikh peremennykh”, Issledovaniya po sovremennym problemam konstruktivnoi teorii funktsii, Fizmatgiz, M., 1961, 247–251 | MR

[2] Timan M. F., “O svyazi mezhdu polnymi i smeshannymi modulyami gladkosti funktsii dvukh peremennykh”, Tezisy dokladov nauchnoi konferentsii po itogam nauchno-issled. raboty s/kh instituta za 1963 g., Promin, Dnepropetrovsk, 1965, 66–69

[3] Timan M. F., Avtoreferat dissertatsii, Tbilisi, 1962

[4] Timan M. F., “Nekotorye tozhdestva dlya funktsii mnogikh peremennykh i ikh primenenie”, Dokl. AN SSSR, 178:1 (1968), 40–42 | MR | Zbl

[5] Brudnyi Yu. A., Avtoreferat dissertatsii, Rostov-na-Donu, 1966

[6] Timan A. F., Teoriya priblizheniya funktsii deistvitelnogo peremennogo, Fizmatgiz, M., 1960

[7] Marcinkiewiez J., “Sur les multiplicateurs des series de Fourier”, Fundam. math., 32 (1938), 78–91

[8] Marchoud A., “Sur les dérivées et sur les differences des fonctions de variables reelles”, J. math. pures et appl., 6 (1927), 337–425