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Acta Matkematica Academiae Scientiarum Hungaricae Tomus 26 (1--2), (1975), 199--204. HERMITE--FEJI~R INTERPOLATION OMITTING SOME DERIVATIVES By P. O. H. VI~RTESI (Budapest) 1. Introduction. Let us consider the Hermite--Fej6r process (1.1) H,, (f; x) = Zf(xk, n)hk,, (X) k=J_ on the interval [-1, 1] for a continuousf(x) where (1.2) -~ 1 (n = 1,2, ...), -- ~ ~ Xn,n <: Xn_l, n -< ... <z X1, n = (1.3) c~,(x) = 1[ (x-xk.,). k=l o~,(x) (1.4) ~,,,(x) = ~(xk.,)(x- xk,,) (k = 1, 2, ..., n), o);'(xk.,) ] (1.5) hk,,(x) = ]1 o~; (x~.,) (x- xk, ,)j IL, (x) (k = 1, 2, ..., n). It is well known that (1.6) H,(f; xk,,) =f(xk,,), H~(f; Xk,,, ) = 0 (k = 1, 2, ..., n). For the Chebysheff nodes 2k - 1 (1.7) xl,,, = cos ~k,, = COS ~ ~ (k = 1, 2, ..., n) according to a classical result of L. FEJ~R [1], H,(f; x) converges uniformly to f(x). On the other hand, the famous FABER theorem ([8]) states that for all systems of nodes (1.1) there is a continuous function f~(x) on [-1, 1] such that g-m J[L,(fl; x)t]=~ where n ~ ?/ (1.8) Z,(f; x)
Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals
Published: May 21, 2016
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