Geodesic
Asymptotic behaviour of the Lebesgue constants of periodic interpolation splines with equidistant nodes
Sbornik. Mathematics, Tome 191 (2000) no. 8, pp. 1233-1242 
Yu. N. Subbotin; S. A. Telyakovskii. Asymptotic behaviour of the Lebesgue constants of periodic interpolation splines with equidistant nodes. Sbornik. Mathematics, Tome 191 (2000) no. 8, pp. 1233-1242. http://geodesic.mathdoc.fr/item/SM_2000_191_8_a5/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Associated with each continuous function $f$ of period 1 is the periodic spline $s_{r,n}(f)$ that has degree $r$, defect 1, nodes at the points $x_i=i/n$, $i=0,1,\dots,n-1$ and that interpolates $f$ at these points for $r$ odd and at the mid-points of the intervals $[x_i,x_{i+1}]$ for $r$ even. For the corresponding Lebesgue constants $L_{r,n}$, that is the norms of the operators $f(x)\to s_{r,n}(f)$ from $C$ to $C$, the asymptotic formula $$ L_{r,n}=\frac2\pi\log\min(r,n)+O(1), $$ is established, which holds uniformly in $r$ and $n$.

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