Question 4 (15 marks, C3) (a) Let f: R→ R be a continuously differentiable periodic function of period 27. Given that th

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Question 4 15 Marks C3 A Let F R R Be A Continuously Differentiable Periodic Function Of Period 27 Given That Th 1 (48.72 KiB) Viewed 33 times

Question 4 (15 marks, C3) (a) Let f: R→ R be a continuously differentiable periodic function of period 27. Given that the Fourier series of f(x) is Moreover, (i) f(x) sin: 8 x dx. show that Σ k=1 Σ + (4k-3 Find each of the following with justification. • fo k=1 4k³ 1 - k)2 2 k sin ka. 57² 12 4. (ii) fo f'(x) cos(3x) dx. (b) Let f: R→ R and g: R → R be piecewise differentiable functions that are integrable. Given that the Fourier transform of f is f(w), and the Fourier transform of g is g(w) = f(w)f(w + 1), [5] g(t) = f(r)e-if(t-r)dr. [5] [5]