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Approximation by Polynomials

  • Ker-I Ko
Part of the Progress in Theoretical Computer Science book series (PTCS)

Abstract

The classical Weierstrass approximation theorem states that for every continuous function f on [0, 1] and for any n ≥ 0, there is a real-valued polynomial function φ n such that |f(x) - φ n (x){ ≤ 2-n for all x ∈ [0, 1], In this chapter we investigate the polynomial-time version of the Weierstrass approximation theorem: Is the sequence |φ n } polynomial-time computable, if f is known to be polynomial-time computable? Pour-El and Caldwell [1975] proved that the recursive version of the Weierstrass approximation theorem holds. Shepherd-son [1976] has pointed out the importance of this recursive version of the Weierstrass approximation theorem: It provides a straight-line type finite program for function f. Also, some numerical operations on f, such as integration, can be computed by performing the operation on the approximation polynomials φ n . We show that the weak form of the polynomial-time version of the Weierstrass approximation theorem does hold and so a polynomial-time evaluable straight-line program for φ n can be found in polynomial time if f is itself polynomial-time computable. However, the strong form of the theorem that requires the output of the coefficients of φ n fails. Thus, the integrals of φ n may still be hard to compute.

Keywords

Polynomial Function Piecewise Linear Function Arithmetic Circuit CHEBYSHEV Approximation Binary Expansion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 1991

Authors and Affiliations

  • Ker-I Ko
    • 1
  1. 1.Department of Computer ScienceState University of New York at Stony BrookStony BrookUSA

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