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0 1 2 3 4 5 6
1
0
1
mesh[sin(x
2)]
x
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0 1 2 3 4 5 61
0
1
LFEM
[sin(x
2)]
0 1 2 3 4 5 6
0
1
2
e5 > f
5=0.947
x
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0
1
spline[sin(x
2)]
x
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0 2 4 6 8 10 12
1
0
1
FT3
0 2 4 6 8 10 12
1
0
1
FFT[square2
(x)]
FT6
0 2 4 6 8 10 12
1
0
1
FT12
x
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0 1 2 3 4 5 61
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1V5
0 1 2 3 4 5 61
0
1 V4
Haarwavelets
[sin(x
2)]
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0
1W4
x
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0 1 2 3 4 5 6
1
0
1 V5
0 1 2 3 4 5 6
1
0
1 V4
DAUB4wavelets[sin(x
2)]
0 1 2 3 4 5 6
1
0
1 W4
x
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t + t
0 0 0 0
0 0 0 0
0 0 0 0
1 1 1 1
1 1 1 1
1 1 1 1
0 0 0 0
0 0 0 0
0 0 0 0
1 1 1 1
1 1 1 1
1 1 1 1
t
x
t
j1 j j+1
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1
1 Im G
Re G
1
exp( )
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t + t
t t
0 0 0
0 0 0
0 0 0
0 0 0
1 1 1
1 1 1
1 1 1
1 1 1
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
x
t
j1 j j+1
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j+1j1 j0 0 0
0 0 0
0 0 0
0 0 0
1 1 1
1 1 1
1 1 1
1 1 1
0 0 0
0 0 0
0 0 0
0 0 0
1 1 1
1 1 1
1 1 1
1 1 1
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
0 0 0
0 0 0
0 0 0
0 0 0
1 1 1
1 1 1
1 1 1
1 1 1
0 0 0
0 0 0
0 0 0
0 0 0
1 1 1
1 1 1
1 1 1
1 1 1
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
t+ t
x
t
t
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t + t
0 0 0 0
0 0 0 0
0 0 0 0
1 1 1 1
1 1 1 1
1 1 1 1
0 0 0 0
0 0 0 0
0 0 0 0
1 1 1 1
1 1 1 1
1 1 1 1
0 0 0 0
0 0 0 0
0 0 0 0
1 1 1 1
1 1 1 1
1 1 1 1
0 0 0 0
0 0 0 0
0 0 0 0
1 1 1 1
1 1 1 1
1 1 1 1
t
x
t
j1 j j+1
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0
aliasing
aliased spectrumtrue spectrum
kkc c
A(k)
k
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