Minimizing an Integral (Chebyshev Polynomials)

Question

I'd like to minimize the integral $$\int_{-1}^{1}\sqrt{1-x^2}\left(f(x)-p_n(x)\right)^2dx$$ given $f(x)\in C^0\left([-1,1]\right)$ and where $p_n(x)$ is allowed to range over all polynomials of degree $\leq n$. The weight is a good indicator that $p_n(x)$ is related to the Chebyshev polynomials, but I'm not sure how I would show that they might be the minimizers of this functional without being able to take derivatives.

Answer 1

If we consider $L^2(-1,1)$ equipped with the following dot product $$\langle f(x), g(x)\rangle = \int_{-1}^{1} f(x)\,g(x)\,\sqrt{1-x^2}\,dx $$ we have that Chebyshev polynomials of the second kind are a complete base of orthogonal functions: $$ \langle U_m(x), U_n(x)\rangle = \frac{\pi}{2}\,\delta(m,n) $$ By Parseval's theorem, the best $L^2$-approximation of $f(x)$ in terms of polynomials having degree $\leq n$ is given by the projection of $f(x)$ on the subspace generated by $U_0,U_1,\ldots,U_n$:

$$ p_n(x) = \sum_{k=0}^{n} U_k(x)\cdot\frac{2}{\pi}\int_{-1}^{1}f(x)\,U_k(x)\,\sqrt{1-x^2}\,dx.$$