Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Suppose I want to construct a sequence of polynomials that converges to $|x|$ pointwise.

I am pretty good on proving that sequences of functions converge to things pointwise, but I am having trouble actually coming up with a sequence of polynomials that converges to $|x|$. Someone told me to try using the Taylor series expansion of $(1−x)^{1/2}$, but that just confused me, since I am very rusty on Taylor series. Would anyone mind explaining this to me a little?

Thanks!!

share|improve this question
    
Maybe with an orthogonal polynomial? –  Pragabhava Oct 12 '12 at 18:21
4  
Over what domain? –  copper.hat Oct 12 '12 at 18:23
    
The Bernstein polynomials are a nice way of approximating a continuous function over $[0,1]$. en.wikipedia.org/wiki/Bernstein_polynomial –  copper.hat Oct 12 '12 at 18:23

1 Answer 1

You can use Chebyshev polynomials of the first kind.

$$ P_{2n}(x)=\frac{2}{\pi}-\sum_{j=1}^{n}\frac{4(-1)^j}{\pi(4j^2-1)}T_{2j}(x)$$

converges uniformly to $|x|$ over the compact set $[-1,1]$, in virtue of the Fourier-Chebyshev series (the Chebyshev polynomials of the first kind are a complete base of orthonormal functions wrt the scalar product $<f,g>=\int_{-1}^{1}\frac{f(x)g(x)dx}{\sqrt{1-x^2}}$).

This choice is quite good since:

$$ \sup_{x\in(-1,1)}||x|-P_{2n}(x)|=O\left(\frac{1}{n}\right).$$

See also: Weierstrass approximation Theorem.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.