You understand that you need to perform an orthogonal projection of f(t) onto the function space of your choice to find the best-possible approximation. To solve this exercise, you work out all of the following, step by step, and answer all the questions. You can get help with the involved calculations, but you need to be able to explain all steps of your solution in terms of what you are doing and how – “just” the eventual computation can be deferred to some “helping hand", for ex., from Wolfram Alpha: - 1. Start with d = 0: How does Bo look? What kind of spline so(t) can you get (on the basis of Bo)? How do all of them vary (among each other)? Perform the projection and find the optimal p0,0. Calculate also the approximation error, i.e., the Euclidean distance between fo(t) and f(t), given that Ĵo(t) is the best-possible approximation of ƒ (t) on the basis of Bo. 2. Next, consider d = 1: How does B₁ look? What kind of spline s₁ (t) can you get on the basis of B₁? Provide the formulæ for all b₁,k (t) in B₁ and sketch them. Are these polynomials b₁,k (t) in B₁ mutually orthogonal/orthonormal to each other? What's the formula for the projection of f(t) onto B₁? Do the necessary computation(s) and present all ô₁,k for the best-possible approximation f₁(t) wrt. B₁ . Compare f₁(t) to fo(t) regarding a possible improvement of the approximation. Explain your finding(s) from this comparison. Any conclusions for larger d? 3. Clearly, the case of d = 2 is next. Provide the formulæ for the b2,k (t) and sketch them, considering the particular values of them at the tk points. Are the basis functions of B2 mutually orthogo- nal/orthonormal to each other? Calculate the projection to get the optimal p2,k. Calculate also the approximation error. Discuss a possible improvement. Any additional, interesting observation(s)? Assume that you fancy polynomial splines, while you actually need ƒ(t) = e²/3 – 1 for t€ [−1, 1]. See the figure for a plot of f(t). Your goal is to approximate f(t) with an inter- polating polynomial spline of degree d that is given as sa(t) = • Σk=0 Pd,k bd,k(t) so that sd(tk) = = Pd,k for tk = −1 + 2 (given d > 0) with basis functions bd,k(t) = Σi±0 Cd,k,i = • The special case of d 0 is trivial: the only basis function b0,0 (t) is constant 1 and so(t) is thus constant po,0 for all t = [−1, 1]. ...9 The d+1 basis functions bd,k (t) form a ba- sis Bd {ba,o(t), ba,1(t), bd,d(t)} of the function space of all possible sα (t) functions. Clearly, you wish to find out, which of them given a particular maximal degree d is the best-possible approximation of f(t) in the least- squares sense. _ 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1 function f(t) = exp((2t)/3) - 1 to project -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t f(t) Even though you know yourself how to get the coefficients Cd,k,i for the basis functions bd,k (t) - it's recommended to give this a try, actually – we provide you with them for the degrees up to d = 2: 1 bo,o(t) = 1 b1,... (t) (1, t) b2....(t) -1 1 (1, t, t²) 020 -1 0 1 1-2 1