(Solved): Problem 4. Pick three square-integrable functions defined on \( [-1,1] \) which are not polynomial ...

Problem 4. Pick three square-integrable functions defined on \( [-1,1] \) which are not polynomials. They do not need to be smooth or even continuous. Now, for \( n=5 \), compute the coefficients \( c_{0}, \ldots, c_{5} \) for each function \( f \) and form the approximation: \[ \hat{f}_{k}(x)=c_{0} p_{0}(x)+\cdots+c_{k} p_{k}(x), \quad 0 \leq k \leq 5, \] and the relative error: \[ e_{k}=\frac{\left\|\hat{f}_{k}-f\right\|}{\|f\|} \]