Efficiency of Chebyshev Approximation on Finite Subsets

ABSTRACT

Chebyshev approximation on an interval and closed subsets by a Haar subspace are considered. The closeness of best approximations on subsets to the best approximation on the interval is examined. It is shown that under favorable conditions the difference is O((density of the subset)2), making it unnecessary to use very large finite subsets to get good approximations on the interval.

REFERENCES

• 1: DE BOOR, C. Review of {4}. Math. Reviews 34 (1967),/{4473, 869.
• 2: PE~TRE, J. Approximation of norms. J. Approx. Theory S (1970), 243-260.
• 3: RICE, j. The Approximation of Functions, Vol. 1. Addison-Wesley, Reading, Mass., 1964.
• 4: RIVLIN, T. J., AND CHENEY, E.W. A comparison of uniform approximations on an interval and a finite subset thereof. SIAM J. Numer. Anal. S (1966), 311-320.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.2 Approximation
          Subjects: Chebyshev approximation and theory

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.1 Numerical Algorithms and Problems
          Subjects: Computations in finite fields

General Terms:
Theory, Verification