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A necessary condition for convergence of interpolating parabolic and cubic splines

Authors:	N L Zmatrakov

Institution:	(1) Institute of Mathematics and Mechanics, UNTs, Academy of Sciences of the USSR, USSR

Abstract:	Let the sequence of nets _n be such that $\mathop {\lim }\limits_{n \to \infty } \mathop {\max }\limits_i h_i^{(n)} = 0$ , where h_i ⁽ⁿ⁾ are the lengths of the segments of a net. The bound $\mathop {\max }\limits_{\left\| {i - j} \right\| = 1} \frac{{h_i^{(n)} }}{{h_j^{(n)1 - \alpha } }} \leqslant R< \infty$ is necessary in order that interpolating parabolic and cubic splines converge for any function in C ( = 0) or C(0 < < 1), where C is the class of functions satisfying a Lipschitz condition of order. It is also shown that this bound cannot essentially be weakened.Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 165–178, February, 1976.The author thanks Yu. N. Subbotin for a useful discussion of the results obtained.

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