# Contributions to the Problem of Approximation of Equidistant Data by Analytic Functions

Part A.—On the Problem of Smoothing or Graduation. A First Class of Analytic Approximation Formulae
• I. J. Schoenberg
Chapter
Part of the Contemporary Mathematicians book series (CM)

## Abstract

Introduction. Let there be given a sequence of ordinates
$$\left\{ {{y_n}} \right\}\quad \left( {n = 0, \pm 1 \pm 2, \ldots } \right),$$
corresponding to all integral values of the variable x = n. If these ordinates are the values of a known analytic function F(x), then the problem of interpolation between these ordinates has an obvious and precise meaning: we are required to compute intermediate values F(x) to the same accuracy to which the ordinates are known. Undoubtedly, the most convenient tool for the solution of this problem is the polynomial central interpolation method. It uses the polynomial of degree k — 1, interpolating k successive ordinates, as an approximation to F(x) only within a unit interval in x, centrally located with respect to its k defining ordinates. Assuming k fixed, successive approximating arcs for F(x) are thus obtained which present discontinuities on passing from one arc to the next if k is odd, or discontinuities in their first derivatives if k is even (see section 2.121). Actually these discontinuities are irrelevant in our present case of an analytic function F(x). Indeed, if the interpolated values obtained are sufficiently accurate, these discontinuities will be apparent only if we force the computation beyond the intrinsic accuracy of the y n.

## Keywords

Basic Function Characteristic Function Interpolation Formula Spline Curve Polygonal Line
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## References

1. 1.
W. A. Jenkins, Osculatory interpolation: New derivation and formulae, Record of the American Institute of Actuaries, 15, 87 (1926).Google Scholar
2. Thomas N. E. Greville, The general theory of osculatory interpolation, Transactions of the Actuarial Society of America, 45, 202–265 (1944).Google Scholar
3. 2.
See J. M. Whittaker, Interpolator function theory, Cambridge Tracts in Mathematics, 1935, pp. 62-64, for a discussion of the relation between the cardinal series and Stirling’s interpolation series. The cardinal series was probably first investigated in an important mémoire by Ch. J. de la Vallée Poussin, Sur la convergence des formules d’interpolation entre ordonnées équidistantes, Bull. Acad. Roy. Belgique, 1908, 319-410.Google Scholar
4. 4.
Compare G. J. Lidstone, Note on the computation of terminal values in graduation by Jenkins 1 modified osculatory formula, Transactions of the Faculty of Actuaries (Scotland), 12, 277 (1930).Google Scholar
5. 8.
See e. g. S. Bochner, Vorlesungen über Fouriersche Integrale, Leipzig, 1932, Satz 11b on p. 42.Google Scholar
6. 12.
See H. S. Carslaw, Mathematical theory of the conduction of heat, Dover Publications, New York, 1945, Chapter III, Section 16.Google Scholar
7. Certain smoothing properties of heat flow were already noticed by Ch. Sturm in 1886. See in this connection G. Pólya, Qualitatives über Wärmeausgleich, Z. angew. Math. u. Mech. 13, 125–128 (1933). It should be mentioned here that Weierstrass derived his famous approximation theorem by means of the integral (33). Finally see E. Czuber, Wahrscheinlichkeitsrechnung, vol. I, Leipzig-Berlin, 1924, pp. 417-418, for a brief sketch of a method of using (33) to derive analytic approximations to given data.