Abstract
A cardinal spline analog of the Markov theorem is given. It is applied to derive the necessary conditions for a function to be the limit of its cardinal spline interpolents as their degree trends to infinity. Sufficient conditions for this to happen are given in [8].
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R. P. Boas, Jr.,Entire Functions, Academic Press, Inc., New York, 1954.
A. Kolmogorov,On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval, Amer. Math. Soc. Trans. (1)2 (1962), 233–243.
A. C. Schaeffer and R. J. Duffin,On some inequalities of S. Bernstein and W. Markoff for derivatives of polynomials, Bull. Amer. Math. Soc.44 (1938), 289–297.
I. J. Schoenberg,Cardinal interpolation and spline functions, J. Approximation Theory,2 (1969), 167–206.
I. J. Schoenberg,Cardinal interpolation and spline functions II. Interpolation of data of power growth, J. Approximation Theory6 (1972), 404–420.
I. J. Schoenberg,The elementary cases of Landau’s problem of inequalities between derivatives, Amer. Math. Monthly80 (1973), 121–158.
I. J. SchoenbergCardinal spline interpolation, (to be published by SIAM as the 12th Regional Conference Monograph).
I. J. Schoenberg,Notes on spline functions III: On the convergence of the interpolating cardinal splines as their degree tends to infinity, Israel J. Math.16 (1973), 87–93, MRC T. S. R. #1326, February 1973.
J. N. Subbotin,On the relation between finite differences and the corresponding derivatives, Proc. Steklov Inst. Math.78 (1965), 24–42. Amer. Math. Soc. Translations (1967), 23–42.
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Sponsored by the United States Army under Contract No. DA-31-124-ARO-D-462.
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Richards, F.B., Schoenberg, I.J. Notes on spline functions IV: A cardinal spline analogue of the theorem of the brothers Markov. Israel J. Math. 16, 94–102 (1973). https://doi.org/10.1007/BF02761974
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DOI: https://doi.org/10.1007/BF02761974