Solve the problems for your appropriate course track. Problems probe understanding of the definitions and results from the modules on real function approximation. Formulate your answers clearly and cogently. Sketch out an approach on scratch paper first. Then briefly transcribe the approach to the answer you turn in, followed by appropriate calculations and conclusions. Use concise, complete English sentences in the description of your approach.

Each question is meant to be completely answered and transcribed from proof to final copy within ten minutes. The test also contains 5 short Quiz 03 questions, meant to be answered in one minute each. Present a succint motivation for your quiz answers taking into account time constraints.

1Track 1

1. Construct the polynomial interpolant of data $𝒟=\{(1,3),(2,4),(3,-1),(4,-18)\}$ in Lagrange form.

2. Construct the Newton form of the polynomial interpolant of the above data set, presenting the table of divided differences.

3. Efficiently evaluate the Newton form of the polynomial interpolant determined above at $t=-1$, using Horner's scheme.

4. Replace the sampling points $x_i=1+i$, $i=0,..,3$ in the data set $𝒟$ so as to minimize the interpolation error over the interval $[1,4]$

5. Quiz questions:

   1. True or false: a basis exists for all vector spaces.

   2. True or false: a basis can be effectively constructed for all vector spaces.

   3. What is the projection of $f:ℝ\to ℝ$ along $b_k\left(t\right)$ in a Hilbert space spanned by $\{b_1\left(t\right),b_2\left(t\right),..,b_k\left(t\right),..\}$?

   4. What is the error of a polynomial interpolant $p\left(t\right)$ of $f:ℝ\to ℝ$, sampled by $𝒟=\{(x_i,y_i=f\left(x_i\right)),i=0,\dots ,n\}$ when $t=x_i$?

   5. What is the degree of the polynomial interpolant of data $𝒟=\{(x_i,y_i=f\left(x_i\right),y_i^{\text{'}}=f^{\text{'}}\left(x_i\right)),i=0,1\}$?

2Track 2

1. Construct the Hermite interpolant of data $𝒟=\{(x_i,y_i=f\left(x_i\right),y_i^{\text{'}}=f^{\text{'}}\left(x_i\right)),i=0,1\}=\{(1,3,2),(4,-18,-25)\}$ in Newton form.

2. Construct the Hermite interpolant of the above data in Lagrange form.

3. Present a spline interpolant $S$ of data set $𝒟=\{(x_i=ih,y_i=f\left(x_i\right)),i=0,..,n\}$, $h=\pi /n$, where the restriction of $S$ to interval $[x_{i-1},x_i]$ is of the form

   ${\displaystyle S_i\left(t\right)=a_i+b_i\cos t+c_i\sin t}$

4. Find the best inf-norm approximant of, $f:[0,\pi /2]\to ℝ$, $f\left(t\right)=\cos t$

5. Quiz questions:

   1. True or false: The monomials $ℳ\left(t\right)=\{1,t,t^2,..\}$ are a basis for $C^{\infty }\left(ℝ\right)$

   2. True or false: In a Hilbert space, Cauchy sequences converge to an element within the space

   3. What is the degree of the polynomial interpolant of data $𝒟=\{(x_i,y_i=f\left(x_i\right),y_i^{\text{'}}=f^{\text{'}}\left(x_i\right),y_i^{\text{'}\text{'}}=f^{\text{'}\text{'}}\left(x_i\right)),i=0,1,2\}$?

   4. True or false: An best inf-norm approximant exists in all Banach spaces.

   5. True or false: There are more polynomials than there are continuous functions.