Abstract
It is a well-known fact that the classical (i.e. polynomial) divided difference of orderm, when applied to a functiong, converges to themth-derivative of this function, if the evaluation points all collapse to a single one.
In the first part of this paper we shall sharpen this result in the sense that we prove the existence of an asymptotic expansion with limitg (m) /m!. This result allows the application of extrapolation methods for the numerical differentiation of funtions.
Moreover, in the second and main part of the paper we study generalized divided differences, which were introduced by Popoviciu [10] and further investigated for example by Karlin [2], Walz [15] and, mainly, Mühlbach [6–8]; we prove the existence of an asymptotic expansion also for these generalized divided differences, if the underlying function space is a Polya space. As a by-product, our results show that the generalized divided difference of orderm converges to the value of a certainmth order differential operator.
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Communicated by G. Mühlbach
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Walz, G. Asymptotic expansions for classical and generalized divided differences including applications. Numer Algor 7, 161–171 (1994). https://doi.org/10.1007/BF02140680
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DOI: https://doi.org/10.1007/BF02140680