Geodesic
On the total variation for functions of several variables and a multidimensional analog of Helly's selection principle
Matematičeskie zametki, Tome 63 (1998) no. 1, pp. 69-80 
A. S. Leonov. On the total variation for functions of several variables and a multidimensional analog of Helly's selection principle. Matematičeskie zametki, Tome 63 (1998) no. 1, pp. 69-80. http://geodesic.mathdoc.fr/item/MZM_1998_63_1_a7/
@article{MZM_1998_63_1_a7,
     author = {A. S. Leonov},
     title = {On the total variation for functions of several variables and a multidimensional analog of {Helly's} selection principle},
     journal = {Matemati\v{c}eskie zametki},
     pages = {69--80},
     year = {1998},
     volume = {63},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1998_63_1_a7/}
}
TY  - JOUR
AU  - A. S. Leonov
TI  - On the total variation for functions of several variables and a multidimensional analog of Helly's selection principle
JO  - Matematičeskie zametki
PY  - 1998
SP  - 69
EP  - 80
VL  - 63
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MZM_1998_63_1_a7/
LA  - ru
ID  - MZM_1998_63_1_a7
ER  - 
%0 Journal Article
%A A. S. Leonov
%T On the total variation for functions of several variables and a multidimensional analog of Helly's selection principle
%J Matematičeskie zametki
%D 1998
%P 69-80
%V 63
%N 1
%U http://geodesic.mathdoc.fr/item/MZM_1998_63_1_a7/
%G ru
%F MZM_1998_63_1_a7

Voir la notice de l'article provenant de la source Math-Net.Ru

We introduce the new notion of total variation for the Hardy class of functions of several variables and state various properties, similar to those in the one-dimensional case, for functions belonging to this class. In particular, we prove a precise version of Helly's selection principle for this class.

[1] Hahn H., Theorie der reellen Funktionen, V. I, Springer, Berlin, 1921

[2] Hobson E. W., The theory of functions of a real variable and the theory of Fourier's series, V. I, 3rd ed., Cambridge Univ. Press, Cambridge, 1927

[3] Clarkson J. A., Adams C. R., “On the definition of bounded variation for functions of two variables”, Trans. Amer. Math. Soc., 35:4 (1933), 824–854 | DOI | MR | Zbl

[4] Vitushkin A. G., O mnogomernykh variatsiyakh, Gostekhizdat, M., 1955

[5] Natanson I. P., Teoriya funktsii veschestvennoi peremennoi, Nauka, M., 1974

[6] Hardy G. H., “On double Fourier series, and especially those which represent the double zeta-functions with real and incommensurable parameters”, Quart. J. Math. Oxford, 37 (1905), 53–79

[7] Bokhner S., Lektsii ob integralakh Fure (s dobavleniem avtora o monotonnykh funktsiyakh, integralakh Stiltesa i garmonicheskom analize), Fizmatgiz, M., 1962

[8] Billingsli P., Skhodimost veroyatnostnykh mer, Nauka, M., 1977

[9] Ivanov L. D., Variatsii mnozhestv i funktsii, Nauka, M., 1975

[10] Tikhonov A. N., Leonov A. S., Yagola A. G., Nelineinye nekorrektnye zadachi, Nauka, M., 1995

[11] Kamke E., Integral Lebega–Stiltesa, Fizmatgiz, M., 1959

[12] Shilov G. E., Gurevich B. L., Integral, mera, proizvodnaya, Nauka, M., 1967

[13] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, Nauka, M., 1977 | Zbl

[14] Young W. H., Young G. C., “On the discontinuities of monotone functions of several variables”, Proc. London Math. Soc. (2), 22 (1924), 124–142 | DOI

[15] Adams C. R., Clarkson J. A., “Properties of functions $f(x,y)$ of bounded variation”, Trans. Amer. Math. Soc., 36:4 (1934), 711–730 | DOI | MR | Zbl