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The questions are attached below. The course is on discrete inverse problems.

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS

2. (a) Evaluate the function

f(t) = .05 sin(1000t)+.5 cos(лt) — .4 sin(10)

at the 101 points given by 0:.01:1. Plot the resulting broken line interpolant.

(b) In order to study the slow scale trend of this function, we wish to find a low degree polynomial (degree at most 6) that best approximates ƒ in the least squares norm at the above 101 data points. By studying the figure from part (a) find out the smallest n that would offer a good fit in this sense. (Try to do this without further computing.)

(c) Find the best approximating polynomial v of degree n and plot it together with ƒ. What are your observations?

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