Present a concise formulation of the theoretical motivation of your answer, and then proceed to solve the following problems.

1. Consider

   ${\displaystyle S\left(x\right)=\{\begin{array}{ll}ax+b& x⩽0\\ \tan \left(x\right)& 0⩽x⩽\pi /4\\ cx+d& x⩾\pi /4\end{array}..}$

   Determine $a,b,c,d$ such that $S\left(x\right)$ is a spline function. With $S^{\left(k\right)}=d^{k}S/d{x}^k$ denoting the $k^{\mathrm{th}}$ derivative of $S\left(x\right)$, up to what $k$ is $S^{\left(k\right)}\left(x\right)$ continuous?

2. Find a bound for the maximum error in approximation of $f:[-1,1]\to ℝ$,

   ${\displaystyle f\left(x\right)=\cos (4\pi \sin \left(4\pi x\right))+\sin (4\pi \cos \left(4\pi x\right))}$

   by interpolation of the data $𝒟=\{(x_i,y_i),y_i=f\left(x_i\right),i=0,1,2\}$ with $x_i$ the roots of the Chebyshev polynomial $T_3\left(x\right)$.

3. Find $L_2\left(t\right)$, the quadratic polynomial within the set $\{L_0\left(t\right),L_1\left(t\right),L_2\left(t\right),..\}$ orthonormal with respect to the scalar product

   ${\displaystyle (f,g)=\underset{-1}{\overset{1}{\int }}f\left(t\right)g\left(t\right)dt.}$

4. Construct a numerical method to $f\left(x\right)=0$ using a quadratic polynomial approximation of $f$.