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Journal of Optimization Theory and Applications

, Volume 29, Issue 2, pp 199–213 | Cite as

Nonlinear programming, approximation, and optimization on infinitely differentiable functions

  • V. A. Ubhaya
Contributed Papers

Abstract

A nonnegative, infinitely differentiable function φ defined on the real line is called a Friedrichs mollifier function if it has support in [0, 1] and ∫ 0 1 φ(t)dt=1. In this article, the following problem is considered. Determine Δ k =inf∫ 0 1 (k)(t)|dt,k=1, 2, ..., where φ(k) denotes thekth derivative of φ and the infimum is taken over the set of all mollifier functions φ, which is a convex set. This problem has applications to monotone polynomial approximation as shown by this author elsewhere. The problem is reducible to three equivalent problems, a nonlinear programming problem, a problem on the functions of bounded variation, and an approximation problem involving Tchebycheff polynomials. One of the results of this article shows that Δ k =k!22k−1,k=1, 2, .... The numerical values of the optimal solutions of the three problems are obtained as a function ofk. Some inequalities of independent interest are also derived.

Key Words

Nonlinear programming approximation of functions Tchebycheff polynomials infinitely differentiable functions 

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References

  1. 1.
    Morrey, C. B., Jr.,Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, Germany, 1966.Google Scholar
  2. 2.
    Ubhaya, V. A.,Moduli of Monotonicity with Applications to Monotone Polynomial Approximation, SIAM Journal on Mathematical Analysis, Vol. 7, pp. 117–130, 1976.Google Scholar
  3. 3.
    Ubhaya, V. A.,Nonlinear Programming and an Optimization Problem on Infinitely Differentiable Functions, Journal of Optimization Theory and Applications, Vol. 29, No. 3, 1979.Google Scholar
  4. 4.
    Lorentz, G. G.,Approximation of Functions, Holt, Rinehart, and Winston, New York, New York, 1966.Google Scholar
  5. 5.
    Ubhaya, V. A.,On the Degree of Approximation by Monotone Polynomials, Case Western Reserve University, Cleveland, Ohio, Department of Operations Research, Technical Memorandum No. 410, 1976.Google Scholar
  6. 6.
    Ubhaya, V. A.,On the Degree of Approximation by Monotone Polynomials, Notices of the American Mathematical Society, Vol. 23, p. A-588, 1976.Google Scholar
  7. 7.
    Ubhaya, V. A.,Isotone Optimization, I and II, Journal of Approximation Theory, Vol. 12, pp. 146–159 and 315–331, 1974.Google Scholar
  8. 8.
    Ubhaya, V. A.,An O(n)-Algorithm for Discrete n-Point Convex Approximation, with Application to the Continuous Case, Journal of Mathematical Analysis and Applications (to appear).Google Scholar
  9. 9.
    Dunford, N., andSchwartz, J. T.,Linear Operators, Part I, John Wiley and Sons (Interscience Publishers), New York, New York, 1958.Google Scholar
  10. 10.
    Royden, H. L.,Real Analysis, Second Edition, The Macmillan Company, New York, New York, 1968.Google Scholar
  11. 11.
    Meinardus, G.,Approximation of Functions: Theory and Numerical Methods, Springer-Verlag, Berlin, Germany, 1967.Google Scholar
  12. 12.
    Neustadt, L. W.,Optimization, a Moment Problem, and Nonlinear Programming, SIAM Journal on Control, Vol. 2, pp. 33–53, 1964.Google Scholar
  13. 13.
    Jolley, L. B. W.,Summation of Series, Second Revised Edition, Dover Publications, New York, New York, 1961.Google Scholar

Copyright information

© Plenum Publishing Corporation 1979

Authors and Affiliations

  • V. A. Ubhaya
    • 1
  1. 1.Department of Operations ResearchCase Western Reserve UniversityCleveland

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