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Notes on a paper of G. Grünwald

Notes on a paper of G. Grünwald Acta h.lathematicae Academia Scientiarum Hungaricae Tomus 26 (1--2), (1975), 191--197. By P. O. H. VIERTESI (Budapest) ][. Notations and results Let us consider the Lagrange-interpolation process for a continuous function. f(x) on the interval [-l, 1J, i.e. let (1.1) L.(f; x) = Zf(xx.n)lk,.(x) (--1 <: x ~ 1), k=l where co(x) with co (x) = e(x- xi,,) (x- x~,~)... (x - x,,,). (1.2) l~,,,(x)= For the Tchebyscheff abscissas 2k- 1 (1.3) xk,, = cos 0k,,, = cos --ffn---n ~r (k = 1,2, ...,n; n = 1,2, ...) we have with x-=cos 0 (1.4) co (x) = T. (x) = cos (n arccos x) = cos nO ( 1 < <1 0<0<~ 0 cos nO sin 0k (-1 =~x<= 1, 0 < 0< 7r) (1.5) /~,.(x) =/k,.[0J = (-1) ~+~ n(cos 0 - cos 0k) According to a result of G. GRUNWALD and J. MARCINKIEWICZ we may find a con- tinuous function f~(x) such that the sequence L,(fi; x) is divergent for all points of the interval - 1 =<x<= l (see [2] and [3]). On the other hand GR~ONWALD [1] proved the following THEORE~ 1.1 (G. Grfinwald). Let f(x) be a continuous fimction in the interval [-1, 1]; then http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

Notes on a paper of G. Grünwald

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Springer Journals
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Copyright
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Mathematics; Mathematics, general
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0001-5954
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1588-2632
DOI
10.1007/BF01895961
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Abstract

Acta h.lathematicae Academia Scientiarum Hungaricae Tomus 26 (1--2), (1975), 191--197. By P. O. H. VIERTESI (Budapest) ][. Notations and results Let us consider the Lagrange-interpolation process for a continuous function. f(x) on the interval [-l, 1J, i.e. let (1.1) L.(f; x) = Zf(xx.n)lk,.(x) (--1 <: x ~ 1), k=l where co(x) with co (x) = e(x- xi,,) (x- x~,~)... (x - x,,,). (1.2) l~,,,(x)= For the Tchebyscheff abscissas 2k- 1 (1.3) xk,, = cos 0k,,, = cos --ffn---n ~r (k = 1,2, ...,n; n = 1,2, ...) we have with x-=cos 0 (1.4) co (x) = T. (x) = cos (n arccos x) = cos nO ( 1 < <1 0<0<~ 0 cos nO sin 0k (-1 =~x<= 1, 0 < 0< 7r) (1.5) /~,.(x) =/k,.[0J = (-1) ~+~ n(cos 0 - cos 0k) According to a result of G. GRUNWALD and J. MARCINKIEWICZ we may find a con- tinuous function f~(x) such that the sequence L,(fi; x) is divergent for all points of the interval - 1 =<x<= l (see [2] and [3]). On the other hand GR~ONWALD [1] proved the following THEORE~ 1.1 (G. Grfinwald). Let f(x) be a continuous fimction in the interval [-1, 1]; then

Journal

Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals

Published: May 21, 2016

References

  • On a convergence theorem for the Lagrange interpolation polynomials
    Grünwald, G.
  • Über Divergenzerscheinungen der Lagrangeschen Interpolationspolynome stetiger Funktionen
    Grünwald, G.
  • Sur la divergence des polynomes d'interpolation
    Marcinkiewicz, J.
  • A convergence theorem for certain Lagrange interpolation polynomials
    Webster, M. W.
  • Orthogonal Polynomials
    Szegő, G.
  • On certain linear operators. VII
    Vértesi, P. O. H.
  • Investigation of an interpolatory process. II
    Kis, O.
  • Theory of Approximations of Functions of a Real Variable
    Timan, A. F.