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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© 1991 Birkhäuser Boston
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Ko, KI. (1991). Approximation by Polynomials. In: Complexity Theory of Real Functions. Progress in Theoretical Computer Science. Birkhäuser Boston. https://doi.org/10.1007/978-1-4684-6802-1_9
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DOI: https://doi.org/10.1007/978-1-4684-6802-1_9
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4684-6804-5
Online ISBN: 978-1-4684-6802-1
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