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What is the condition for a real valued function of a real variable to have a Newton series which converges to that function pointwise?

It feels like there should be a condition similar to that for the Taylor series.

user157872
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    Could you be a little more specific about what you mean by the Newton series? I can find references to the "Newton polynomial" of a function, but it's not uniquely defined. – Jack M Aug 10 '14 at 13:53
  • The Newton series being the limit of the sequence of Newton polynomials where the forward difference in x is some arbitrary finite real number, – user157872 Aug 10 '14 at 14:07
  • Doesn't it depend on which sample points you're using? – Jack M Aug 10 '14 at 14:26
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    @Jack M since he is speaking about forward difference, it can be assumed the points are equidistant, 1-separated unless one mentions time scales. – Anixx Dec 09 '14 at 00:49

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I have been pointed recently that convergence of Newton series is covered extensively in Gelfond, A. O. Calculus of finite differences. Translated from the Russian. International Monographs on Advanced Mathematics and Physics. Hindustan Publishing Corp., Delhi, 1971

The Russian original is available online:

http://inis.jinr.ru/sl/vol1/UH/_Ready/Mathematics/Gel%27fond%20A.%20Ischislenie%20konechnyh%20raznostej%20%281959%29%28ru%29%28L%29%28201s%29.pdf

Page 141 onwards.

Anixx
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  • Hi - I've been looking for these results for a while and I cannot seem to find Gelfond's book online in english. As I do not understand Russian would it be possible to summarise the results? – Jpk Mar 20 '20 at 11:27