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    $\begingroup$ Substitute x = cos t and write p(x) as a linear combination of terms of the form cos nt. As t varies these terms will go in and out of phase with respect to each other, and the only constraint you have is on the coefficient of cos nt, so to keep these phases from constructively interfering you get rid of all the other terms. That's my intuition, anyway. $\endgroup$ Commented May 21, 2010 at 22:56
  • $\begingroup$ My answer to your title question would be "orthogonalization with semicircular weighting" but now I prefer KConrad's answer. $\endgroup$ Commented May 21, 2010 at 23:01
  • $\begingroup$ The Chebyshev polynomials form one family of orthogonal polynomials on $[-1,1]$, and are (with the Legendre polynomials) particular instances of Jacobi polynomials corresponding to the measure $dx/(1-x)^\alpha(1+x)^\beta$. The name "Chebyshev polynomials" is also used in the problem of minimizing sup norm by polynomials with integer coefficients; see math/0101166. $\endgroup$ Commented May 22, 2010 at 7:41
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    $\begingroup$ I feel as though none of the answers so far quite answers the question. There are many beautiful characterizations of the Chebyshev polynomials, but what does any of them have to do with the minimization problem in the question? But perhaps the answer to the OP is that the Chebyshev polynomials were thought of for a different reason, and it then became clear what else they could do. (I don't know whether that is the actual story -- does anyone know their history here?) $\endgroup$ Commented May 22, 2010 at 9:30