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A variational approach to spline functions theory. (English) Zbl 1121.41007

In this paper the author presents a variational approach to spline interpolation, i.e. interpolating splines are derived as unique solution of certain variational problems.
He proposes a survey by means of a sequence of theorems and results from the classical variational property of natural cubic interpolating splines J. C. Holladay [Math. Tables Aids Comput. 11, 233–243 (1957; Zbl 0084.34904)] up to some general theoretical developments of abstract splines M. Atteia [Hilbertian kernels and spline functions, Studies in Computational Mathematics. 4. Amsterdam: North-Holland (1992; Zbl 0767.41015)] F. I. Utreras [Rev. Mat. Apl. 9, No. 1, 87–95 (1987; Zbl 0679.41010)], passing through the \(D^m\)–splines C. de Boor [J. Math. Mech. 12, 747–749 (1963; Zbl 0116.27601)], the trigonometric splines I. J. Schoenberg [J. Math. Mech. 13, 795–825 (1964; Zbl 0147.32104)], the \(g\)–splines I. J Schoenberg [J. Math. Anal. Appl. 21, 207–231 (1968; Zbl 0159.08401)], the \(L\)–splines M. H. Schultz and R. S. Varga [Numer. Math. 10, 345–369 (1967; Zbl 0183.44402)], the \(Lg\)–splines J. W. Jerome and L. L. Schumaker [J. Approximation Theory 2, 29–49 (1969; Zbl 0172.34501)], the \(pLg\)–splines P. Copley and L. L. Schumaker [J. Approximation Theory 23, 1–28 (1978; Zbl 0409.41004)], the vector–valued \(Lg\)–splines G. S. Sidhu and H. L. Weinert [J. Math. Anal. Appl. 70, 505–529 (1979; Zbl 0435.65007)] and the thin plate splines J. Duchon [RAIRO, Anal. Numér. 10, 5–12 (1976)].
The variational problems become increasingly abstract within the paper and the same is for the concept of \`\` spline\'\'. The theory of variational splines leads to an extension in a natural and attractive way of the classical theory of interpolating splines to multivariate situations. Moreover it shows the power of functional analysis in yielding a unified approach to computational problems in interpolation.

MSC:

41A15 Spline approximation
65D07 Numerical computation using splines
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
65D05 Numerical interpolation
41A05 Interpolation in approximation theory
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