Geodesic
On interpolation theory in the complex domain
Izvestiya. Mathematics, Tome 6 (1972) no. 4, pp. 782-787 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is shown that for the nodes $z_k^{(n)}=e^{i\theta_k^{(n)}}$, where $\theta_k^{(n)}=\frac{(2k+1)\pi}n$, $k=1,\dots,n$; $n=1,2,\dots$, the following statements hold: 1) The Hermite–Fejér interpolation process for an arbitrary polynomial converges in $|z|\leqslant1$ with rapidity $O\bigl(\frac1n\bigr)$. 2) The process $R_n(f,z)=\sum_{k=1}^nf\bigl(z_k^{(n)}\bigl)\bigl[l_k^{(n)}(z)\bigr]^2$, where $\bigl\{l_k^{(n)}(z)\bigr\}$ are Lagrange fundamental polynomials with nodes $\bigl\{z_k^{(n)}\bigr\}$, diverges at all points $z\ne0$ of $|z|\leqslant1$ for every function $f(z)=z^s$, $s=0,1,2,\dots$ . 
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     author = {D. L. Berman},
     title = {On interpolation theory in the complex domain},
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     url = {http://geodesic.mathdoc.fr/item/IM2_1972_6_4_a5/}
}
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D. L. Berman. On interpolation theory in the complex domain. Izvestiya. Mathematics, Tome 6 (1972) no. 4, pp. 782-787. http://geodesic.mathdoc.fr/item/IM2_1972_6_4_a5/

[1] Bernshtein S. N., Neskolko zamechanii ob interpolirovanii. Sobr. soch., t. I, AN SSSR, M., 1952

[2] Faber G., “Über die interpolatorische Darstellung stetiger Funktionen”, Jahresber. DMV, 23 (1914), 192–210 | Zbl

[3] Grünwald G., “Über Divergenzerscheinungen”, Ann. Math., 37 (1936), 908–918 | DOI | MR | Zbl

[4] Marcinkiewicz J., “Sur la divergence des polynômes d'interpolation”, Acta Litt. as Sc. Szeged, 8 (1937), 131–135 | Zbl

[5] Natanson I. P., Konstruktivnaya teoriya funktsii, GITTL, M., L., 1949

[6] Fejér L., “Lagrangesche Interpolation und die zugehörigen konjngierten Punkte”, Math. Ann., 106 (1932), 1–55 | DOI | MR | Zbl

[7] Grünwald G., “On the theory of interpolation”, Acta Mathem., 75 (1943), 219–245 | DOI | MR | Zbl

[8] Fejér L., “Bestimmung derjenigen abszissen eines intervalles für welche die Quadratsumme der Gründfunctionen ein möglichts kleines maximum besitzt”, Ann. Scuola Norm. Sup. Pisa Cl. Sci.(2), 1:3 (1932), 263–276 | MR | Zbl

[9] Smirnov V. I., Lebedev N. A., Konstruktivnaya teoriya funktsii, kompleksnogo peremennogo, Nauka, M., 1964 | MR