Skip to main content
SpringerLink
Log in

On smooth interpolation by continuously connected piecewise polynomials

  • Published:
Rendiconti del Circolo Matematico di Palermo Aims and scope Submit manuscript

Summary

In this paper, a method is presented for interpolating to a given set of equally-spaced points a set of piecewise polynomials of degreen which agree in all derivatives throughn−1 at each of the points. Efficient computing algorithms and theorems, based on recursive difference equations and modern matrix techniques, are developed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. T. Ross,General Description of the APT System, Servo-Mechanisms Laboratory, Mass. Inst. of Tech., vol. 1.

  2. H. Stone,Approximation of Curves by Line Segments, Math. Comp., vol. 15, pp. 40–47, (1961).

    Article  MATH  MathSciNet  Google Scholar 

  3. T. J. Rivlin,Smooth Interpolation, SIAM Review, vol. 1, pp. 60–63, (1959).

    Article  MATH  MathSciNet  Google Scholar 

  4. T. J. Rivlin,A Note on Smooth Interpolation, SIAM Review, vol. 2, pp. 27–30, (1960).

    Article  MATH  MathSciNet  Google Scholar 

  5. F. B. Hildebrand,Introduction to Numerical Analysis, New York, McGraw-Hill Co., Inc., pp. 91–121, (1956).

    MATH  Google Scholar 

  6. H. Rutishauser,Notes on Smooth Interpolation, ZAMP, vol. 11, pp. 508–513, (1960).

    Article  MATH  MathSciNet  Google Scholar 

  7. T. N. E. Greville and I. J. Schoenberg,Smoothing by Generalized Spline Functions, SIAM National Meeting, New York, June 7, (1965).

  8. E. T. Whittaker and G. Robinson,The Calculus of Observations, London, Blackie and Son, 2nd Ed., (1937).

    Google Scholar 

  9. T. N. E. Greville and H. Vaughan,Polynomial Interpolation in Terms of Symbolic Operators, Soc. Actuar. Trans. 6, pp. 413–476, (1954).

    MathSciNet  Google Scholar 

  10. E. N. Nilson and J. H. Ahlberg,Convergence Properties of the Spline Fit, J. SIAM, vol. 11, pp. 95–104, (1963).

    MATH  MathSciNet  Google Scholar 

  11. J. L. Walsh, J. H. Ahlberg and E. N. Nilson,Best Approximation Properties of the Spline Fit, J. Math. Mech. II, pp. 225–234, (1962).

    MathSciNet  Google Scholar 

  12. Boris Podolsky and Harry M. Denman,Conditions on Minimization Criteria for Smoothing, Math. Comp. 18, pp. 441–448, (1964).

    Article  MATH  MathSciNet  Google Scholar 

  13. Boris Podolsky,Curve Fitting With Allowance for Errors in Given Data, Technical Information Series No. R59 FPD 442, General Electric Company, Lockland, Ohio, July, (1959).

    Google Scholar 

  14. J. F. Reilly,On Lidstone’s Demonstration of the Osculatory Interpolation Formula, The Record of the American Institute of Actuar., vol. XIV, Part. 1, Number 29, pp. 12–20, June, (1925).

    Google Scholar 

  15. I. J. Schoenberg,Contributions on 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 Formula, Quart. Appl. Math. 4, MF 15945, pp. 45–99, (1946).

    MathSciNet  Google Scholar 

  16. I. J. Schoenberg,Contributions on the Problem of Approximation of Equidistant Data by Analytic Functions, Part B—On the Problem of Osculation Interpolation. A second Class of Analytic Formulae, Quart. Appl. Math. 4, MF 16959, pp. 112–141, (1946).

    MathSciNet  Google Scholar 

  17. W. A. Jenkins,Osculatory Interpolation: New Derivation and Formulae, Record of the American Institute of Actuaries 15, pp. 87, (1926).

    Google Scholar 

  18. T. N. E. Greville,The General Theory of Osculatory Interpolation, Transactions of the Actuarial Society of America 45, pp. 202–265, (1944).

    MathSciNet  Google Scholar 

  19. I. J. Schoenberg,On Best Approximation of Linear Operators, Koninki. Nederl. Akad. Van Wetenschappen, Proceedings, Ser. A, Vol. 67, pp. 155–163, (1964).

    MathSciNet  MATH  Google Scholar 

  20. I. J. Schoenberg,Spline Interpolation and the Higher Derivatives, Proc. Nat. Acad. Sci. U.S.A., pp. 24–28, (1964).

  21. I. J. Schoenberg,On Trigonometric Spline Interpolation, J. Math. Mech. 13, pp. 795–825, (1964).

    MATH  MathSciNet  Google Scholar 

  22. I. J. Schoenberg,Spline Interpolation and Best Quadrature Formulae, Bull. Amer. Math. Soc. 70, pp. 143–148, (1964).

    Article  MATH  MathSciNet  Google Scholar 

  23. C. Lanczos,Applied Analysis, Englewood Cliffs, Prentice Hall, Inc., p. 235, (1956).

    Google Scholar 

  24. J. C. Holladay,Smoothest Curve Approximation, MTAC, vol. 11, pp. 233–243, (1957).

    MATH  MathSciNet  Google Scholar 

  25. J. F. Price and R. H. Simonson,Various Methods and Computer Routines for Approximations, Curve Fittings, and Interpolation, Boeing Scientific Research Laboratories Document D1-82-0151, pp. 12–16, February, (1962).

  26. G. Birkhoff and H. L. Garabedian,Smooth Surface Interpolation, J. Math. and Physics, vol. 39, pp. 258–268, (1960).

    MathSciNet  Google Scholar 

  27. M. H. Schultz,L 2-Approximation Theory of Even Order Multivariate Splines, SIAM Journal on Numerical Analysis, vol. 6, no. 3, pp. 467–475, (1969).

    Article  MATH  MathSciNet  Google Scholar 

  28. A. F. Cornock,The Numerical Solution of Poisson’s and the Bi-Harmonic Equations by Matrices, Proc. Cambridge Phil. Soc., vol. 50, (1954).

  29. O. Karlquist,Numerical Solution of Elliptic Differential Equations, Tellus 4, pp. 374–384, (1952).

    Article  MathSciNet  Google Scholar 

  30. R. E. von Holdt,Inversion of Triple Diagonal Compound Matrices, J. ACM, vol. 9, pp. 71–83, (1962).

    Article  MATH  Google Scholar 

  31. M. J. D. Powell,The Local Dependence of Least Squares Cubic Splines, SIAM Journal on Numerical Analysis, vol. 6, no. 3, pp. 398–413, (1969).

    MATH  MathSciNet  Google Scholar 

  32. E. Goursat,A Course in Mathematical Analysis, vol. I, New York, Dover Publications, Inc., pp. 443–451, (1959).

    MATH  Google Scholar 

  33. C. De Boor,Best Approximation Properties of Spline Functions of Odd Degree, J. Math. and Physics, vol. 41, pp. 212–218, (1962).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Seattle, U.S.A.

    A. C. Ahlin

Authors
  1. A. C. Ahlin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ahlin, A.C. On smooth interpolation by continuously connected piecewise polynomials. Rend. Circ. Mat. Palermo 20, 229–253 (1971). https://doi.org/10.1007/BF02844177

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02844177

Keywords

Search

Navigation