## 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

### Keywords:

smoothing; $$L$$-splines; reproducing kernel
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