Geodesic
Asymptotic behaviour of the~Lebesgue constants of periodic interpolation splines with equidistant nodes
Sbornik. Mathematics, Tome 191 (2000) no. 8, pp. 1233-1242

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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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     author = {Yu. N. Subbotin and S. A. Telyakovskii},
     title = {Asymptotic behaviour of {the~Lebesgue} constants of periodic interpolation splines with equidistant nodes},
     journal = {Sbornik. Mathematics},
     pages = {1233--1242},
     publisher = {mathdoc},
     volume = {191},
     number = {8},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2000_191_8_a5/}
}
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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/