Geodesic
On the divergence of Lagrange interpolation processes on sets of the second category
Sbornik. Mathematics, Tome 55 (1986) no. 2, pp. 511-528 
Al. A. Privalov. On the divergence of Lagrange interpolation processes on sets of the second category. Sbornik. Mathematics, Tome 55 (1986) no. 2, pp. 511-528. http://geodesic.mathdoc.fr/item/SM_1986_55_2_a12/
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     author = {Al. A. Privalov},
     title = {On~the divergence of {Lagrange} interpolation processes on sets of the second category},
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     year = {1986},
     volume = {55},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1986_55_2_a12/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

If $\omega$ is a real nondecreasing semiadditive function, continuous on $[0;1]$, such that $\omega(0)=0$ and $\varlimsup_{n\to\infty}\omega\bigl(\frac1n\bigr)\ln n>0$, then for every matrix of interpolation knots on $[0;1]$ there are a function $f$, continuous on $[0;1]$, whose modulus of continuity $\omega(f,\delta)=O\{\omega(\delta)\}$, and a set $\mathscr E$ of second category on $[0;1]$ such that the Lagrange interpolation process for $f$ diverges everywhere on $\mathscr E$. Bibliography: 10 titles.

[1] Privalov A. A., “O raskhodimosti interpolyatsionnykh protsessov v fiksirovannoi tochke”, Matem. sb., 66(108) (1965), 272–286 | MR | Zbl

[2] Privalov A. A., “O raskhodimosti interpolyatsionnykh protsessov na mnozhestvakh vtoroi kategorii”, Matem. zametki, 18:2 (1975), 179–183 | MR | Zbl

[3] Bernshtein S. N., “Neskolko zamechanii ob interpolirovanii”, Sobranie sochinenii, 1, Izd-vo AN SSSR, M., 1952, 253–263

[4] Stechkin S. B., “Obobschenie nekotorykh neravenstv S. N. Bernshteina”, DAN SSSR, 60:9 (1948), 1511–1514 | Zbl

[5] Mills T. M., Varma A. K., “A new proof of S. A. Teljakovskii's approximation theorem”, Stud. sci. math. hung., 14:1–3 (1979), 241–246 | MR | Zbl

[6] Bernshtein S. N., “Ob ogranichenii mnogochlena $P_n(x)$ stepeni $n$ na vsem otrezke po ego znacheniyam v $n+1$ tochkakh”, Sobranie sochinenii, 2, Izd-vo AN SSSR, M., 1954, 107–126

[7] Bernshtein S. N., Ekstremalnye svoistva polinomov, ONTI, M.-L., 1937

[8] Lozinskii S. M., “Prostranstva $\widetilde C_{\omega}$ i $\widetilde C_{\omega}^*$ i skhodimost interpolyatsionnykh protsessov v nikh”, DAN SSSR, 59:8 (1948), 1389–1392 | MR | Zbl

[9] Korneichuk N. P., Ekstremalnye zadachi teorii priblizheniya, Nauka, M., 1976 | MR

[10] Sege G., Ortogonalnye mnogochleny, Fizmatgiz, M., 1962