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Computing reproducing kernels with arbitrary boundary constraints. (English) Zbl 0777.65003

The space of all functions with \(m-1\) absolutely continuous derivatives and a square-integrable \(m\)th derivative is denoted by \(H^ m\), \(m\geq 1\). Smoothing with \(L\)-splines involves the use of the penalty term \(\| x\|^ 2_ 2=\int(Lx)^ 2(t)dt\), where \(L\) is an arbitrary linear differential operator of order \(m\). This also involves the choice of \(m\) constraint functionals to define two orthogonal subspaces \(H_ 1=\text{ker }L\) and a complementary subspace \(H_ 2\) which partition \(H^ m\).
Computing the reproducing kernel for \(H_ 2\) is generally difficult. This problem is mainly addressed in this paper and techniques for computing the kernels are developed and illustrated [cf. G. Wahba, Spline models for observational data, SIAM, Philadelphia, PA (1990; M.R. 91g:62028); and the authors, J. R. Stat. Soc., Ser. B, 53, No. 3, 539-572 (1991; M.R. 93e:62161)].

MSC:

65D10 Numerical smoothing, curve fitting
65D07 Numerical computation using splines
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