Geodesic
Norms on $L$ of Periodic Interpolation Splines with Equidistant Nodes
Matematičeskie zametki, Tome 74 (2003) no. 1, pp. 108-117 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the set $S_{r,n}$ of periodic (with period 1) splines of degree $r$ with deficiency 1 whose nodes are at $n$ equidistant points $x_i=i / n$. For $n$-tuples $\mathbf y=(y_0, y_1, \dots,y_{n-1})$, we take splines $s_{r,n}(\mathbf y, x)$ from $S_{r,n}$ solving the interpolation problem $$ s_{r, n} (\mathbf y, t_i)=y_i, $$ where $t_i = x_i$ if $r$ is odd and $t_i$ is the middle of the closed interval $[x_i , x_{i+1}]$ if $r$ is even. For the norms $L_{r,n}^*$ of the operator $\mathbf y\to s_{r,n} (\mathbf y, x)$ treated as an operator from $l^1$ to $L^1 [0,1]$ we establish the estimate $$ L_{r, n}^*=\frac{4}{\pi^2 n} \log \min (r, n)+O\biggl(\frac{1}{n} \biggr) $$ with an absolute constant in the remainder. We study the relationship between the norms $L_{r,n}^*$ and the norms of similar operators for nonperiodic splines. 
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Yu. N. Subbotin; S. A. Telyakovskii. Norms on $L$ of Periodic Interpolation Splines with Equidistant Nodes. Matematičeskie zametki, Tome 74 (2003) no. 1, pp. 108-117. http://geodesic.mathdoc.fr/item/MZM_2003_74_1_a11/

[1] Subbotin Yu. N., Telyakovskii S. A., “Asimptotika konstant Lebega periodicheskikh interpolyatsionnykh splainov s ravnootstoyaschimi uzlami”, Matem. sb., 191:8 (2000), 131–140 | MR | Zbl

[2] Stechkin S. B., Subbotin Yu. N., Splainy v vychislitelnoi matematike, Fizmatgiz, M., 1976

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

[4] Gradshtein I. S., Ryzhik I. M., Tablitsy integralov, summ, ryadov i proizvedenii, Fizmatgiz, M., 1962

[5] Marsden M. J., Richards F. B., Riemenschneider S. D., “Cardinal spline interpolation operators on $l^p$ data”, Indiana University Math. J., 24 (1975), 677–689 | DOI | MR | Zbl

[6] Schoenberg I. J., “Cardinal interpolation and spline functions”, J. Approx. Theory, 2 (1969), 167–206 | DOI | MR | Zbl

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