Fourier series, math homework help
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gragra
Mathematics
Description
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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