On Interpolation by Spline Functions and its Minimal Properties
Polynomials are wonderful even after they are cut into pieces, but the cutting must be done with care. One way of doing the cutting leads to the so-called spline functions. These were introduced by the writer in 1944 as a tool for the approximation of functions and were suggested by the work of T. N.E. Greville and other actuarial writers on the subject of osculatory interpolation (for the connections and references see , Part B). The other tool used was the law of the flow of heat in a homogeneous wire and with both tools combined extensive ballistic tables were approximated and thereby smoothed (see ). For recent largescale physical applications of cubic spline functions see . The spline functions used in  had equidistant knots. In 1945 H. B. Curry and the author wrote a paper on spline functions with arbitrarily spaced knots which for no good reason remained unpublished (see the Abstract ). For further work on spline functions and related topics see the list of references which claims no completeness.
KeywordsSpline Function Spline Interpolation Interpolation Problem Minimal Property Divided Difference
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- 2.Curry H.B. and I.J. Schoenberg: On Pólya frequency functions IV: The spline functions and their limits. Bull. Amer. Math. Soc. 53 (1947), Abstract 380t, 1114.Google Scholar
- 5.Kuntzmann J.: Méthodes numériques: Interpolation, dérivées (Dunod, Paris 1959).Google Scholar
- 6.Landis F. and E.N. Nilson: The determination of Thermodynamic properties by direct differentiation techniques. Progress in Internat. Research on Thermodynamic and Transport properties. Amer. Soc. Mech. Engineers, New York, 1962.Google Scholar
- 7.Nörlund, N.E.: Vorlesungen über Differenzenrechnung. (Berlin 1924).Google Scholar
- 8.Schoenberg, I. J.: Contributions to the problem of approximation of equidistant data by analytic functions. Quart. Appl. Math., 4 (1946), Part A 45–99, Part B 112-141.Google Scholar
- 9.Schoenberg, I.J. and Anne Whitney: On Pólya frequency functions III: The positivity of translation determinants with an application to the interpolation problem by spline curves. Trans. Amer. Math. Soc. 74 (1953), 246–259.Google Scholar
- 11.Steffensen, J.F.: Interpolation. Baltimore, 1927.Google Scholar