Approximating $\sin(\frac{x}{2})$ by trigonometric polynomials in the uniform norm

Question

Find a trygonometric polynomial of the form $$a_0+a_{1} \sin (x)+ a_2 \cos (x)$$ that best approximates the function $$\sin\left(\frac{x}{2}\right)$$ in the uniform norm on the interval $[-\pi,\pi]$. I can't think of a way to approach this problem. Should I be looking for alternating points or is there a way to transform this into polynomial approximation? I would appreciate a hint.

Answer 1

At $x=\pm \pi$ you get the residuals $f(\pi)-p(\pi)=1-a_0+a_2$ and $f(-\pi)-p(-\pi)=-1-a_0+a_2$. The sum of these errors is $$ |f(\pi)-p(\pi)|+|f(-\pi)-p(-\pi)|=|1+(a_2-a_0)|+|-1+(a_2-a_0)|=2\max(1,|a_2-a_0|). $$ The best result you can get in the sense of a uniform error is when both errors are $1$.

However, the uniform distance of the given function to the zero function $p(x)=0$ is $1$, so that this trivial approximation already realizes the best uniform approximation.