Geodesic
Asymptotic behaviour of the Lebesgue constants of periodic interpolation splines with equidistant nodes
Sbornik. Mathematics, Tome 191 (2000) no. 8, pp. 1233-1242 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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     title = {Asymptotic behaviour of {the~Lebesgue} constants of periodic interpolation splines with equidistant nodes},
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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/

[1] Subbotin Yu. N., “O svyazi mezhdu konechnymi raznostyami i sootvetstvuyuschimi proizvodnymi”, Tr. MIAN, 78, Nauka, M., 1965, 24–42 | MR | Zbl

[2] Zhensykbaev A. A., “Tochnye otsenki ravnomernogo priblizheniya nepreryvnykh periodicheskikh funktsii splainami $r$-go poryadka”, Matem. zametki, 13:2 (1973), 217–228 | MR | Zbl

[3] Richards F. B., “Best bounds for the uniform periodic spline interpolation operator”, J. Approx. Theory, 7 (1973), 302–317 | DOI | MR | Zbl

[4] Zhensykbaev A. A., “Zamechanie o konstantakh Lebega operatora splain-interpolyatsii”, Issledovaniya po sovremennym problemam summirovaniya i priblizheniya funktsii i ikh prilozheniyam, no. 5, Dnepropetr. gos. un-t, Dnepropetrovsk, 1974, 50–52

[5] Richards F., “The Lebesque constants for cardinal spline interpolation”, J. Approx. Theory, 14 (1975), 83–92 | DOI | MR | Zbl

[6] Meinardus G., “Über die Norm des Operators der Kardinalen Spline-Interpolation”, J. Approx. Theory, 16 (1976), 289–298 | DOI | MR | Zbl