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.


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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