Geodesic
Sharp estimates for derivatives of functions in the Sobolev classes $\mathring W_2^r(-1,1)$
Informatics and Automation, Function theory and differential equations, Tome 269 (2010), pp. 143-149 
G. A. Kalyabin. Sharp estimates for derivatives of functions in the Sobolev classes $\mathring W_2^r(-1,1)$. Informatics and Automation, Function theory and differential equations, Tome 269 (2010), pp. 143-149. http://geodesic.mathdoc.fr/item/TRSPY_2010_269_a10/
@article{TRSPY_2010_269_a10,
     author = {G. A. Kalyabin},
     title = {Sharp estimates for derivatives of functions in the {Sobolev} classes $\mathring W_2^r(-1,1)$},
     journal = {Informatics and Automation},
     pages = {143--149},
     year = {2010},
     volume = {269},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2010_269_a10/}
}
TY  - JOUR
AU  - G. A. Kalyabin
TI  - Sharp estimates for derivatives of functions in the Sobolev classes $\mathring W_2^r(-1,1)$
JO  - Informatics and Automation
PY  - 2010
SP  - 143
EP  - 149
VL  - 269
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2010_269_a10/
LA  - ru
ID  - TRSPY_2010_269_a10
ER  - 
%0 Journal Article
%A G. A. Kalyabin
%T Sharp estimates for derivatives of functions in the Sobolev classes $\mathring W_2^r(-1,1)$
%J Informatics and Automation
%D 2010
%P 143-149
%V 269
%U http://geodesic.mathdoc.fr/item/TRSPY_2010_269_a10/
%G ru
%F TRSPY_2010_269_a10

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

Explicit formulas are obtained for the maximum possible values of the derivatives $f^{(k)}(x)$, $x\in(-1,1)$, $k\in\{0,1,\dots,r-1\}$, for functions $f$ that vanish together with their (absolutely continuous) derivatives of order up to $\le r-1$ at the points $\pm1$ and are such that $\|f^{(r)}\|_{L_2(-1,1)}\le1$. As a corollary, it is shown that the first eigenvalue $\lambda_{1,r}$ of the operator $(-D^2)^r$ with these boundary conditions is $\sqrt2(2r)!(1+O(1/r))$, $r\to\infty$.

[1] Tikhomirov V.M., Nekotorye voprosy teorii priblizhenii, Izd-vo MGU, M., 1976 | MR

[2] Magaril-Ilyaev G.G., Tikhomirov V.M., Vypuklyi analiz i ego prilozheniya, Editorial URSS, M., 2000

[3] Kalyabin G.A., “Nekotorye zadachi dlya prostranstv Soboleva na polupryamoi”, Tr. MIAN, 255, 2006, 161–169 | MR

[4] Akhiezer N.I., Lektsii po teorii approksimatsii, Nauka, M., 1965 | MR | Zbl

[5] Böttcher A., Widom H., “From Toeplitz eigenvalues through Green's kernels to higher-order Wirtinger–Sobolev inequalities”, The extended field of operator theory, Oper. Theory Adv. and Appl., 171, Birkhäuser, Basel, 2007, 73–87 | DOI | MR

[6] Prudnikov A.P., Brychkov Yu.A., Marichev O.I., Integraly i ryady: Elementarnye funktsii, Nauka, M., 1981 | MR | Zbl