Chebyshev projection of a semicircle ($p_n(x)=\frac{2}{\pi }-\frac{4}{\pi }{\sum }_{m=1}^{\lfloor n/2\rfloor }\frac{T_{2m}(x)}{4m^2-1}$)

For weighted least-squares polynomial in an orthogonal-polynomial basis with weight $(1-x^2{)}^{-1/2}$ on $[-1,1]$, the projection of $\sqrt{1-x^2}$ onto polynomials of degree at most $n$ is the displayed sum. Put $x=\cos \theta$ to obtain $T_j(x)=\cos j\theta$ and $f(x)=\sin \theta$ on $[0,\pi ]$. The constant coefficient is $2/\pi$, odd coefficients vanish, and the coefficient of $T_{2m}$ is $4/[\pi (1-4m^2)]$. For odd $n$ the optimal polynomial can have degree $n-1$.