On the theory of interpolation. (English) Zbl 0028.05001

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[1] S. Bernstein, Quelques remarques sur l’interpolation,Comm. Soc. Math. Charkow, 14 (1914).G. Faber, Über die interpolatorische Darstellung stetiger Funktionen,Jahresbericht d. D. Math. Vereinigung, 23 (1914), p. 102–210. A very simple proof is given inL. Fejér, Die Abschätzung eines Polynoms ...,Math. Zeitschrift, 32 (1930), p. 426–457.
[2] S. Bernstein, Sur la limitation des valeurs d’un polynome etc.,Bull. de l’Acad. des Science de l’U.R.S.S., (1931), p. 1025–1050. · Zbl 0004.00601
[3] G. Grünwald, Über Divergenzerscheinungen der Lagrangeschen Interpolationspolynome stetiger Funktionen,Annals of Mathematics, 37 (1936), p. 908–918. See alsoG. Grünwald, Über Divergenzerscheinungen der Lagrangeschen Interpolationspolynome,Acta Szeged, 7 (1935), p. 207–211;J. Marcinkiewicz, Sur la divergence des polynomes d’interpolation,Acta Szeged, 8 (1937), p. 131–135. · Zbl 0015.25202
[4] If then th interpolation polynomial can have any arbitrary degree, then for an arbitrary every where dense pointsystem we can find a sequence of interpolation polynomials which is convergent uniformly for all continuous functions.G. Grünwald,On Interpolation,Bull. of Am. Math. Soc., 47 (1941), p. 257–260. · Zbl 0025.04001
[5] De La Vallée Poussin, Sur la convergence des formules d’interpolation entre coordonées equidistantes,Bulletin de l’Académie Belgique, 1908.S. Bernstein, Sur nne formule d’Interpolation de M. de la Vallée Poussin,Comm. Soc. Math. Charkow, (4) 5. (1932), p. 61–64. The degree ofde la Vallée Poussins interpolation polynomials is 6n; that ofBernsteins is < 3 n .
[6] Fejérs first note on hermite interpolation:L. Fejér, Über Interpolation,Nachrichten d. K. Gesellschaft zu Göttingen (1916), p. 66–91. For the further investigation see the papers ofFejér cited later.
[7] In the Tchebycheff case the degree of then th interpolation polynomial must ben+cn (c>o) at least, if we desire convergence for all continuous functions.G. Grünwald, On a theorem of S. Bernstein,Acta Szeged, in the Press. It is very likely that this is true in the general case, too.
[8] See the note ofL. Fejér cited in note 1, p. 220.
[9] See e. g.L. Fejér, Lagrangesche Interpolation und die zugehörigen konjugierten Punkte,Mathematische Annalen, 106 (1932), p. 1–55.L. Fejér, On the characterisation of some remarkable systems of points of interpolation by means of conjugate points,American Math. Monthly, 41 (1934), p. 1–14.P. Erdos.P. Turán, On Interpolation II,Annals of Mathematics, 39 (1938), p. 703–724. · Zbl 0003.25003
[10] We note that the results are – with few exceptions – the best possibles.
[11] The notations used are introduced – with few exceptions – byL Fejér. The results of this §. are due toL. Fejér.
[12] SeeFejérs first paper cited in note 2, p. 222.
[13] SeeFejérs first paper cited in note 2, p. 222.
[14] We mention the investigations ofde la Vallée Poussin, Lebesgue, S. Bernstein, D. Jackson.
[15] SeeFejérs first paper cited in note 2, p. 222.
[16] The idea of the proof due toLebesgue, Haar, Faber. SeeFejérs first cited paper in note 2, p. 222.
[17] See the paper of.P. Erdos-P. Turán cited in note 2, p. 222.
[18] L. Fejér, Bestimmung derjenigen Abscissen eines Intervalles, für welche die Quadratsumme der Grundfunktionen der Lagrangeschen Interpolation im Intervalle ein möglichst kleines Maximum besitzt,Annali della R. Scuola Normale Superiore di Pisa (1932), p. 3–16. · JFM 58.0373.03
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