Fourier series, math homework help

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gragra

Mathematics

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Fourier Series

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In this question, we construct a function f :T + R which is continuous everywhere but differentiable nowhere. Define the function W(0) := Ï sin((n!)20) (3) n! n=0 and define m-1 Am(0) := Σ sin((n!)20) n! (4) n=0 Bm(0) := sin((m!)20) m! (5) and Cm(0) := sin((n.)20) (6) n! n=m+1 1. Prove that IM8 1 k! 2 n! (7) k=n for sufficiently large n E N. 2. Prove that for ER and any sufficiently large k E N that there exists O ER such that л 3л |θ – φε k' k and (8) (9) sin(ko) - sin(ko) > 1. 3. Explain why the sum in (3) converges to a continuous function on T. 4. Prove that for each sufficiently large n e N, there exists a One T such that n!?|0 – Onl € [T, 31). (10) and |Bn(0) – Bn(ºn) 2 (11) n! (12) n.n! 5. For the same on, prove that A |An(0) – An (on) for some constant A> 0 and all sufficiently large n e N. 6. For the same on, prove that с Cn(0) - Cn(on) (n + 1)! for some constant C > 0 and all sufficiently large n e N. 7. Using the estimates for An, Bn and Cn, deduce that W is not differentiable at any 0 € T. (13) Note: For O E T, 10:= minkez 10 + 2nk], where o E R is an arbitrary representative of the equivalence class O E T= R/2 Z.
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