A remainder formula and limits of cardinal spline interpolants

Goodman, T. N. T.; Lee, S. L.

doi:10.1090/S0002-9947-1982-0654845-7

A remainder formula and limits of cardinal spline interpolants
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by T. N. T. Goodman and S. L. Lee PDF

Trans. Amer. Math. Soc. 271 (1982), 469-483 Request permission

Abstract:

A Peano-type remainder formula \[ f(x) - {S_n}(f; x) = \int _{ - \infty }^\infty {{K_n}(x, t){f^{(n + 1)}}(t) dt} \] for a class of symmetric cardinal interpolation problems C.I.P. $(E, F, {\mathbf {x}})$ is obtained, from which we deduce the estimate $||f - {S_{n,r}}(f; )|{|_\infty } \leqslant K||{f^{(n + 1)}}|{|_\infty }$. 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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