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L1 approximations of strictly convex functions by means of first degree splines

L1-Approximationen von strikt konvexen Funktionen mittels Splines ersten Grades

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Abstract

The L1 approximation of strictly convex functions by means of first degree splines with a fixed number of knots is studied. The main theoretical results are a system of equations for the knots, which solves the problem, and an estimate of the approximation error. The error estimation allows the determination of bounds for the number of knots needed so that the L1 approximation error does not exceed a given number. Finally, an algorithm is used, by means of which a solution to the system can be obtained.

Zusammenfassung

Es wird die L1-Approximation strikt konvexer Funktionen mittels Splines ersten Grades studiert. Die theoretischen Hauptergebnisse sind ein Gleichungssystem für die Knotenpunkte, welches das Problem löst, und eine Abschätzung des Approximationsfehlers. Diese Fehlerabschätzung erlaubt die Bestimmung von Schranken für die Anzahl der Knotenpunkte, die nötig sind, damit der L1-Approximationsfehler eine gegebene Zahl nicht übersteigt. Schließlich wird ein Algorithmus angegeben, mit dessen Hilfe eine Lösung des Gleichungssystems erhalten werden kann.

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References

  1. [1]

    Apostol, T. M.: Mathematical Analysis. p. 73. Reading, Mass.: Addison-Wesley 1973.

  2. [2]

    Blum, E. K.: Numerical Analysis and Computation: Theory and Practice, p. 208. Reading, Mass.: Addison-Wesley 1972.

  3. [3]

    Buck, R. C.: Advanced Calculus, 2nd ed., p. 246. Tokyo: McGraw-Hill Kogakusha Ltd. 1965.

  4. [4]

    Carrol, M. P., Braess, D.: On Uniqueness of L1 Approximation for certain families of Spline functions. J. Approx. Theory.12, 362 (1974).

  5. [5]

    Cox, M. G.: An algorithm for approximating convex functions by means of first degree splines. Comp. Jour.14, 272 (1971).

  6. [6]

    Davis, P. J.: Interpolation and Approximation. London: Blaisdell 1963.

  7. [7]

    Isaacson, E., Keller, H. B.: Analysis of Numerical Methods. New York: J. Wiley 1966.

  8. [8]

    Phillips, G. M.: Algorithms for piecewise straight line approximations. Comp. Jour.11, 211 (1968).

  9. [9]

    Rice, J. R.: The Approximation of Functions, Vol. 1, p. 106. Reading, Mass: Addison-Wesley 1964.

  10. [10]

    Stone, H.: Approximation of Curves by line segments. Math. Comp.15, 40 (1961).

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Kioustelidis, J.B., Spyropoulos, K.J. L1 approximations of strictly convex functions by means of first degree splines. Computing 20, 35–45 (1978). https://doi.org/10.1007/BF02241900

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Keywords

  • Approximation Error
  • Computational Mathematic
  • Theoretical Result
  • Fixed Number
  • Main Theoretical Result