Natural splines of Birkhoff type and optimal approximation (Q2751658)
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scientific article; zbMATH DE number 1665007
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Natural splines of Birkhoff type and optimal approximation |
scientific article; zbMATH DE number 1665007 |
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16 September 2002
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natural splines
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Birkhoff-type interpolation
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linear functionals
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remainder operator
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Sard-type formula
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0.91965294
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0.9046573
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Natural splines of Birkhoff type and optimal approximation (English)
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For a given finite interval \([a,b]\) and \(r\) distinct knots on this interval, the problem of finding functions \(s\) in the Birkoff-type interpolatory set \(U(y)\) associated with an \(n\)-dimensional vector \(y=(y(1), \dots, y(n))\), \(n\) being an appropriate natural number, and minimizing the usual \(L(2)\) Lebesgue norm in \(U(y)\), is called ``a Birkhoff-type natural spline interpolation problem''. In this note, the existence and uniqueness of the solution to such a problem is analyzed (Theorem 1) in terms of the so-called ``optimal approximation formula of Sard-type''. In this respect, the standard theory on Peano kernels represents the essential tool in the proof of this theorem.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00041].
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