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A remainder formula and limits of cardinal spline interpolants


Authors: T. N. T. Goodman and S. L. Lee
Journal: Trans. Amer. Math. Soc. 271 (1982), 469-483
MSC: Primary 41A15
DOI: https://doi.org/10.1090/S0002-9947-1982-0654845-7
MathSciNet review: 654845
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Abstract: A Peano-type remainder formula

$\displaystyle f(x) - {S_n}(f;\,x) = \int_{ - \infty }^\infty {{K_n}(x,\,t){f^{(n + 1)}}(t)\,dt} $

for a class of symmetric cardinal interpolation problems C.I.P. $ (E,\,F,\,{\mathbf{x}})$ is obtained, from which we deduce the estimate $ \vert\vert f - {S_{n,r}}(f;\,)\vert{\vert _\infty } \leqslant K\vert\vert{f^{(n + 1)}}\vert{\vert _\infty }$. It is found that the best constant $ K$ is obtained when $ {\mathbf{x}}$ comprises the zeros of the Euler-Chebyshev spline function. The remainder formula is also used to study the convergence of spline interpolants for a class of entire functions of exponential type and a class of almost periodic functions.

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DOI: https://doi.org/10.1090/S0002-9947-1982-0654845-7
Article copyright: © Copyright 1982 American Mathematical Society

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