Chebyshev interpolation [closed]

Question

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?

Answer 1

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.