Geodesic
Diameters of a class of smooth functions in the space $L_2$
Matematičeskie zametki, Tome 22 (1977) no. 5, pp. 671-678 
R. S. Ismagilov; Kh. Nasyrova. Diameters of a class of smooth functions in the space $L_2$. Matematičeskie zametki, Tome 22 (1977) no. 5, pp. 671-678. http://geodesic.mathdoc.fr/item/MZM_1977_22_5_a7/
@article{MZM_1977_22_5_a7,
     author = {R. S. Ismagilov and Kh. Nasyrova},
     title = {Diameters of a~class of smooth functions in the space $L_2$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {671--678},
     year = {1977},
     volume = {22},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_22_5_a7/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The class $V_\psi$, consisting of the smooth functions $f(t)$, $0\le t\le1$, satisfying the condition $\int_0^1\psi[f^{(r)}(t)]\,dt\le1$, where the function $\psi(t)$ is nonnegative and $r$ is a natural number, is studied. Under certain restrictions on the function $\psi(t)$ ensuring the compactness of the class $V_\psi$, the order of decrease of the Kolmogorov diameters $d_n(V_\psi)$ is computed. The analogous problem for the case $r=1$ is solved also for functions of several variables.