Geodesic
Approximation of periodic functions of several variables by bilinear forms
Izvestiya. Mathematics , Tome 28 (1987) no. 1, pp. 133-150.

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

The orders of the quantities $$ \tau_M(F)_{p_1,p_2}=\sup_{f\in F}\inf_{\substack{u_i(\mathbf x),v_i(\mathbf y)\\i=1,\dots,M}}\biggl\|f(\mathbf x-\mathbf y)-\sum_{i=1}^Mu_i(\mathbf x)v_i(\mathbf y)\biggr\|_{p_1,p_2} $$ are obtained, where $F$ is a class of functions with mixed derivative, or the corresponding prelimiting difference, bounded in $L_q$. In the process some results of independent interest are obtained: a generalization of the Hardy–Littlewood theorem, and the orders of the best $M$-term trigonometric approximations. Bibliography: 16 titles. 
@article{IM2_1987_28_1_a6,
     author = {V. N. Temlyakov},
     title = {Approximation of periodic functions of several variables by bilinear forms},
     journal = {Izvestiya. Mathematics },
     pages = {133--150},
     publisher = {mathdoc},
     volume = {28},
     number = {1},
     year = {1987},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1987_28_1_a6/}
}
TY  - JOUR
AU  - V. N. Temlyakov
TI  - Approximation of periodic functions of several variables by bilinear forms
JO  - Izvestiya. Mathematics 
PY  - 1987
SP  - 133
EP  - 150
VL  - 28
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1987_28_1_a6/
LA  - en
ID  - IM2_1987_28_1_a6
ER  - 
%0 Journal Article
%A V. N. Temlyakov
%T Approximation of periodic functions of several variables by bilinear forms
%J Izvestiya. Mathematics 
%D 1987
%P 133-150
%V 28
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1987_28_1_a6/
%G en
%F IM2_1987_28_1_a6
V. N. Temlyakov. Approximation of periodic functions of several variables by bilinear forms. Izvestiya. Mathematics , Tome 28 (1987) no. 1, pp. 133-150. http://geodesic.mathdoc.fr/item/IM2_1987_28_1_a6/

[1] Schmidt E., “Zur Theorie der Linearen und nichtlinearen Integralgleichungen”, J. Math. Ann., 63 (1907), 433–476 | DOI | MR | Zbl

[2] Michelli C. A., Pinkus A., “Some Problems in the Approximation of Functions of Two Variables and $n$-Widths of Integral Operators”, J. Approxim. Th., 24 (1978), 51–77 | DOI | MR

[3] Miroshin N. V., Khromov V. V., “Ob odnoi zadache nailuchshei approksimatsii funktsii mnogikh peremennykh”, Matem. zametki, 32:5 (1982), 721–727 | MR | Zbl

[4] Kashin B. S., “Poperechniki nekotorykh konechnomernykh mnozhestv i klassov gladkikh funktsii”, Izv. AN SSSR. Ser. matem., 41:2 (1977), 334–351 | MR | Zbl

[5] Temlyakov V. N., “Poperechniki nekotorykh klassov funktsii neskolkikh peremennykh”, Dokl. AN SSSR, 267:2 (1982), 314–317 | MR

[6] Nikolskii S. M, Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1969 | MR

[7] Zigmund A., Trigonometricheskie ryady, Mir, M., 1965 | MR

[8] Timan M. F., “O vlozhenii klassov funktsii”, Izv. vuzov. Matematika, 1974, no. 10, 61–74 | MR | Zbl

[9] Potapov M. K., “Teoremy vlozheniya v smeshannoi metrike”, Trudy Matem. in-ta im. V. A. Steklova AN SSSR, 156, 1980, 143–156 | MR | Zbl

[10] Temlyakov V. N., “O priblizhenii periodicheskikh funktsii neskolkikh peremennykh s ogranichennoi smeshannoi raznostyu”, Dokl. AN SSSR, 253:3 (1980), 544–548 | MR | Zbl

[11] Temlyakov V. N., “Priblizhenie funktsii s ogranichennoi smeshannoi raznostyu trigonometricheskimi polinomami i poperechniki nekotorykh klassov funktsii”, Izv. AN SSSR. Ser. matem., 46:1 (1982), 171–186 | MR | Zbl

[12] Temlyakov V. N., “Priblizhenie periodicheskikh funktsii neskolkikh peremennykh s ogranichennoi smeshannoi raznostyu”, Matem. sb., 113(155):1 (1980), 65–80 | MR | Zbl

[13] Temlyakov V. N., “Priblizhenie periodicheskikh funktsii neskolkikh peremennykh s ogranichennoi smeshannoi proizvodnoi”, Dokl. AN SSSR, 248:3 (1979), 527–531 | MR | Zbl

[14] Kakhan Zh.-P., Absolyutno skhodyaschiesya ryady Fure, Mir, M., 1976

[15] Stechkin S. B., “Ob absolyutnoi skhodimosti ortogonalnykh ryadov”, Dokl. AN SSSR, 102:1 (1955), 37–40 | Zbl

[16] Pich A., Yadernye, lokalno vypuklye prostranstva, Mir, M., 1967 | MR