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Transactions of the American Mathematical Society

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Necessary conditions for the convergence of cardinal Hermite splines as their degree tends to infinity


Author: T. N. T. Goodman
Journal: Trans. Amer. Math. Soc. 255 (1979), 231-241
MSC: Primary 41A15; Secondary 41A05
DOI: https://doi.org/10.1090/S0002-9947-1979-0542878-0
MathSciNet review: 542878
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Abstract: Let $ {\mathcal{S}_{n,s}}$ denote the class of cardinal Hermite splines of degree n having knots of multiplicity S at the integers. In this paper we show that if $ {f_n}\, \to \,f$ uniformly on R, where $ {f_n}\, \in \,{\mathcal{S}_{{i_{n,s}}}}\,{i_n}\, \to \,\infty $ as $ n\, \to \,\infty $, and f is bounded, then f is the restriction to R of an entire function of exponential type $ \leqslant \,S$. In proving this result, we need to derive some extremal properties of certain splines $ {\mathcal{E}_{n,s}}\, \in \,{\mathcal{S}_{n,s}}$, in particular that $ \vert\vert{\mathcal{E}_{n,s}}\vert{\vert _\infty }$ minimises $ \vert\vert S\vert{\vert _\infty }$ over $ S\, \in \,{\mathcal{S}_{n,s}}$ with $ \vert\vert{S^{(n)}}\vert{\vert _\infty }\, = \,\vert\vert\mathcal{E}_{n,s}^{(n)}\vert{\vert _\infty }$.


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  • [1] A. S. Cavaretta, Jr., On cardinal perfect splines of least sup-norm on the real axis, J. Approximation Theory 8 (1973), 285-303. MR 0350263 (50:2756)
  • [2] -, An elementary proof of Kolmogorov's theorem, Amer. Math. Monthly 82 (1974), 480-486.
  • [3] J. W. Jerome and L. L. Schumaker, On Lg-splines, J. Approximation Theory 2 (1969), 29-49. MR 0241864 (39:3201)
  • [4] S. Karlin, Total positivity, vol. 1, Stanford Univ. Press, Stanford, California, 1968. MR 0230102 (37:5667)
  • [5] S. L. Lee and A. Sharma, Cardinal lacunary interpolation by g-splines. I, The characteristic polynomials, J. Approximation Theory 16 (1976), 85-96. MR 0415141 (54:3232)
  • [6] P. R. Lipow and I. J. Schoenberg, Cardinal interpolation and spline functions. III, Cardinal Hermite interpolation, Linear Algebra and Appl. 6 (1973), 273-304. MR 0477565 (57:17084)
  • [7] G. G. Lorentz, Zeros of splines and Birkhoff's kernel, Math. Z. 142 (1975), 173-180. MR 0393950 (52:14757)
  • [8] M. J. Marsden and S. D. Riemenschneider, Cardinal Hermite spline interpolation. Convergence as the degree tends to infinity, Trans. Amer. Math. Soc. 235 (1973), 221-244. MR 0463752 (57:3692)
  • [9] C. A. Micchelli, Oscillation matrices and cardinal spline interpolation, Splines and Approximation Theory, Academic Press, New York, 1976. MR 0481735 (58:1834)
  • [10] 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. MR 0425438 (54:13393)
  • [11] -, Notes on spline functions. IV, A cardinal spline analogue of the theorem of the brothers Markov, Israel J. Math. 16 (1973), 94-102. MR 0425439 (54:13394)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0542878-0
Keywords: Cardinal spline interpolation, cardinal Hermite splines, Euler splines, Chebyshev polynomials
Article copyright: © Copyright 1979 American Mathematical Society

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