Geodesic
Orthogonal projection and minimal linear interpolation on a $n$-dimensional cube
Modelirovanie i analiz informacionnyh sistem, Tome 14 (2007) no. 3, pp. 8-28

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Let $H$ be the orthogonal projection onto polynomials of $n$ variables of degree $\le 1$ and $\|\cdot\|$ be the norm of an operator from $C([0,1]^n)$ to $C([0,1]^n)$. In this paper we show that $C_1\theta_n\le\|H\|\le C_2\theta_n$, $n\in\mathrm{N}$. Here $\theta_n$ denotes the minimal norm of a projection dealing with the linear interpolation on the cube $[0,1]^n$. The proofs make use of certain properties of the Eulerian numbers and the central $B$-splines and also some previous results of the author. 
@article{MAIS_2007_14_3_a1,
     author = {M. V. Nevskij},
     title = {Orthogonal projection and minimal linear interpolation on a $n$-dimensional cube},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {8--28},
     publisher = {mathdoc},
     volume = {14},
     number = {3},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2007_14_3_a1/}
}
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M. V. Nevskij. Orthogonal projection and minimal linear interpolation on a $n$-dimensional cube. Modelirovanie i analiz informacionnyh sistem, Tome 14 (2007) no. 3, pp. 8-28. http://geodesic.mathdoc.fr/item/MAIS_2007_14_3_a1/