Let's define the n-th degree Chebyshev polynomials by 


$$ T\_{n} (x)=\cos(n\arccos(x)).$$


Find a polynomial $P$ such that


$$\mid y- P (x) \mid$$


is minimal, using the first three Chebyshev polynomials for the following data:


$$ \begin{bmatrix}
x & -1 & -0.5 & 0 & 0.5 & 1 \\
y & 0.6346 & 0.6565 & 1 & 1.5230 & 1.5756 
\end{bmatrix}. $$


How could we manage to approach such a problem?



Chebyshev interpolation
Uniform approximation by polynomials on a finite set of points is studied in Section 1.3 of



> 
> T.J. Rivlin, An introduction to the approximation of functions, Dover,
>  2003.
> 
> 
> 


An explicit solution to your (non trivial) question is given on p.36 of that book.


