Complexity Theory of Real Functions pp 247-273 | Cite as

# Approximation by Polynomials

## 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## Preview

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