Abstract
We will analyse the asymptotic behavior ofr n(x) (0≤x≤1), the remainder term of the Lagrange interpolation withn+1 equidistant nodes, in the case, as treated by C. Runge, where the interpolated function is an analytic, function which is regular in a certain complex domain including the real line segment [0, 1]. Specifically, it is proved that there exists a setS⊆[0, 1] such that i) the power (or potency) ofS is that of the continuum; ii) for anyx∈S,\(\frac{{\lim }}{{x \to \infty }}r_n \left( x \right) = 0\) irrespective off(z). We will also investigate the divergence property ofr n(x). Finally, we will introduce a new concept of essential convergence, in terms of which divergence property is discussed.
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Sugihara, M. Some remarks on the Lagrange interpolation with equidistant nodes. Japan J. Appl. Math. 2, 273–284 (1985). https://doi.org/10.1007/BF03167077
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DOI: https://doi.org/10.1007/BF03167077