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Файл:Additional Chapters of Higher Mathematics for Masters in Civill and Geotechnical Engineering. Учебное пособие по дополнительным разделам высшей матема
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Chapter 2. Fourier series, wavelets, and integral transforms
f
f
f
f
fg
y
f
f
x
f
f
81
_____________________________________________________________________________
ˆ
The obtained expression reveals that
arbitrary function
Proposition 2.3.4
Let
A.
1
n
()
fL∈ is obvious.
1
,()
fg L∈ , then
n
()
χ ξ
gfg∗=⋅
^
satisfies the condition B. Generalization to an
()
Q
ˆ
ˆ
. (2.3.11)
If
B.
If
C.
Proof
Proof A flows out from the main definitions:
Proofs of conditions B and C are obvious.
Proposition 2.3.5
:() ( )ffτ→−
xxh
h
:() ()
l
flμ→xx
()^ ( )
^()()
fg f g e dxdy
∗= −
()
∫∫
n
=−
=⋅
()( )
∫∫
n
ˆ
ˆ
fg
is the shifting operator, then
2
i
−πξ⋅
h
ˆ
τ=
()^
h
ef
(2.3.12)
is the multiplying operator, then
n
−
μ=μ
l
lf
yxy
yxy
eedxd
ˆ
(2.3.13)
1
−
l
2
i
−πξ⋅
x
2( )2
ii
−πξ⋅ − −πξ⋅
xy y
. (2.3.14)
Proof
The proof flows out from integration by parts of the left-hand side of (2.3.15).
Corollary
Let
Let both
then
( ,..., )
Px x
1
and
is a polynomial of
n
(the latter denotes derivative with respect to
∂
k
∂=πξ
()
( ) ^ (2 ,...,2 )
PD f P i i f=πξ πξ
()
ˆ
^2
kk
n variables and
. (2.3.15)
i
PD P≡∂ ∂
1
ˆ
n
1
n
()
L ,
( ) ( ,..., )
1
n
) belong to
k
, then
. (2.3.16)
Chapter 2. Fourier series, wavelets, and integral transforms
L
f
g
g
g
L
g
L
g
f
f
82
_____________________________________________________________________________
Proposition 2.3.6
2
Let
()
fL∈ , then
n
2
ˆ
fL∈ and
n
()
Proof
At first, suppose that
1
fL∗∈ and due to (2.3.11)
ˆ
ˆ
since
f= . Expression (2.3.18) yields
12
() ()
fL L∈∩. Let () ( )gf=−xx, then in view of (1.1.15)
fgfff∗=⋅=⋅
()
∫
^
()
n
ˆ
. (2.3.17)
f=
2
L
nn
^
fd f∗ξ=
2
ˆˆˆ
ˆ
, (2.3.18)
2
ˆ
. (2.3.19)
2
On the other hand, applying (2.3.8) and Remark 2.3.3, yield
() ()
∫
n
^(0)
fd gf f∗ξ=∗ =
2
. (2.3.20)
2
Combination of (2.3.19) and (2.3.20) with the remark that
completes the proof.
1
n
()
L is dense in 2()
n
L
Scholium 2.3.1 (Plancherel Theorem)
The integral Fourier transform is unitary operator in
Proposition 2.3.7 (Parseval’s identity)
1
Let
,()
fg L∈ , then
a)
b)
n
ˆ
() () () ()
xx xx
∫∫
nn
() () () ()
xx xx
∫∫
nn
Proof
a) Applying Fubini’s theorem on interchange of the order of integration yields:
gdx f gdx=
gdx f gdx=
2
n
()
L .
ˆ
ˆ
ˆ
. (2.3.21)
. (2.3.22)
Chapter 2. Fourier series, wavelets, and integral transforms
f
f
83
_____________________________________________________________________________
2
i
−π ⋅
ˆ
() () () ()
f g dx f e dy g dx
xx y x
∫∫∫
nn
b) The proof is analogous to the preceding.
=
(
yx xx
() () ()()
==
()
∫∫ ∫
nn
g e dx dy f g dx
(
xy
2
i
−π ⋅
xy
)
. (2.3.23)
)
ˆ
2.3.3. Fourier transform of rapidly decreasing
functions and tempered distributions
Discussion of Fourier transform of (tempered) distributions can be found in Edward’s
comprehensive manuscript (1965), and works by Stein and Weiss (1971), Kierat and
Sztaba (2003); see also classical works by Schwartz (1951, 1957, 1963).
Proposition 2.3.8
Let
Proof
The proof flows out directly from Definition 1.5.4 and expressions (2.3.15), (2.3.16).
Definition 2.3.2 (Fourier transform of tempered distributions)
Let
ψ∈
at
Remark 2.3.4
Applying (2.3.24), (1.5.8) to the tempered distribution ∂ψ yields
()
Sϕ∈
′
()
S
n
()
Sϕ∈
n
be a rapidly decreasing function, then
n
be a tempered distribution, its Fourier transform is defined by
ˆˆ
,,< ϕ ψ >≡< ϕ ψ > (2.3.24)
.
ˆ
,^ ,< ϕ ∂ψ >= − < ∂ϕ ψ >
()
. (2.3.25)
ˆ
n
Sϕ∈
()
.
Proposition 2.3.9
Let
1
(), 2
fL n∈≥
n
be a radial function:
()ff=xx
, then
()
ˆ
is also radial and
Chapter 2. Fourier series, wavelets, and integral transforms
ff
ff
y
84
_____________________________________________________________________________
∞
2
ˆ
=π
yy
()
π
−
(2)/2
n
y
∫
0
/2
n
() (2 )
ss J sds
−
(2)/2
n
, (2.3.26)
where
(2)/2nJ−
is the corresponding Bessel function.
Proof
The proof is based on representing the improper integral of the Fourier transform in spherical
coordinates:
where
∞
ˆ
y
()
is the sphere of the unit radius, and
1nS−
()
=
ss e dx ds
∫∫
0
⎛⎞
1
n
−
⎜⎟
⎜⎟
⎝⎠
2()
−π ⋅
S
1
n
−
′′
is
yyx
′′
∈yx
′
, (2.3.27)
. It can be shown, that the internal
1,nS−
integral in (2.3.27) is
2
2()
−π ⋅
edx J s
∫
S
1
n
−
′′
is
yyx
′
=π
π
(2)/2
n
−
s
y
()
−
(2 )
(2)/2
n
. (2.3.28)
2.3.4. Examples
Example 2.3.1
Let
()tδ
1 p≤≤∞, but it belongs to
where in view of (2.3.8) the identity (2.3.29) should be interpreted as
for any
Remark 2.3.5
Expression (2.3.29) remains valid in the n-dimensional case.
be Dirak δ-function (1.5.5). Such a function does not belong to
′
. Applying (2.3.24), gives:
S
ˆ
, (2.3.29)
δ=
1
ˆ
,,(0)()
.
Sϕ∈
ˆˆ
(2.3.30)
dx< ϕ δ >=< ϕ δ >= ϕ = ϕ
x
∫
n
p
(,)
L −∞ ∞
at any
Chapter 2. Fourier series, wavelets, and integral transforms
H
L
H
s
s
s
f
85
_____________________________________________________________________________
Example 2.3.2
Let ()
Heaviside function does not belong to
∞
S
⊂
where c is a constant. The value of this constant can be determined by the following
consideration. Let
In view of Proposition 2.3.1 the Fourier transform of
constant c in (2.3.31) should be taken zero. Now, with account of (2.3.32), (2.3.29) we arrive at
Example 2.3.3
Let ()
t be defined by
t be Heaviside step-function (2.2.1). Again, as it was in the preceding example,
p
(,)
L −∞ ∞
′
. Application of (1.5.13), (2.3.25), and (2.3.29) yields
ˆ
ξ= −δ
()
be a skew-symmetric Heaviside function:
H
kew
HtHt=−
skew
ˆ
ξ= + δ
()
1
πξ
i
2
() ()
1
πξ
2Hi
c
1
at any
1 p≤<∞
, but it obviously belongs to
. (2.3.31)
1
. (2.3.32)
2
ˆ
should be odd. Thus, for
H
kew
. (2.3.33)
2
H
kew
the
where
⎧
⎪
()
ft
=
⎨
⎪
⎩
. The Fourier transform of this function is
0a >
2
ia
−π ξ
−πξ
1sin(2)
ˆ
ξ= = +
()
fi
ea
πξ πξ πξ
22 2
i
[]
0, 0,
ta
∉
[]
, (2.3.34)
cos(2 ) 1
()
a
πξ−
. (2.3.35)
1, 0,
ta
∈
Function (2.3.34) is of great importance for the vibration analyses, as it specifies the typical
asymmetric step impulse. Slightly modifying function (2.3.34) by shifting it along the time axis,
gives the following symmetric impulse:
1, ,
taa
∈−
⎧
ft
sym
()
⎪
=
⎨
0, ,
⎪
⎩
[]
taa
∉−
[]
. (2.3.36)
Fourier transform of (2.3.36) which is even, is
ξ=
()
sin(2 )
ˆ
f
sym
πξ
πξ
a
. (2.3.37)
Chapter 2. Fourier series, wavelets, and integral transforms
s
f
s
f
s
f
86
_____________________________________________________________________________
The plot for function (2.3.37) is given below:
2
1
0.5
In view of Scholium 2.3.1, the plot reveals that the highest amplitude value
the zero image frequency.
Example 2.3.4
Now, we modify function (2.3.36) by forming a skew symmetric impulse:
The Fourier transform of this (odd) function is
-4 -2 2 4
Figure 2.3.1. Integral Fourier transform
11
() sgn() () , ,0
ft tft ta
skew sym
==−∈−
22
0
ˆ
for a “symmetric” step-function
ym
1
,0,
+∈
⎧
⎪
⎨
⎪
⎩
ta
2
[
[
0, , 0 0,
∉− ∪
ta a
[
)
)
)
ym
ˆ
is attained at
ym
. (2.3.38)
)
[
ξ=
cos(2 )
ˆ
()
fi
skew
2
πξ
a
πξ
. (2.3.39)
The corresponding plot is given in Fig. 2.3.2
Chapter 2. Fourier series, wavelets, and integral transforms
s
s
f
s
f
87
_____________________________________________________________________________
4
2
Again, in view of Scholium 2.3.1, the plot shows that the highest (infinite) amplitude value
ˆ
f is attained at the zero image frequency.
kew
Example 2.3.5
Let real
-4 -2 2 4
-2
-4
Figure 2.3.2. Integral Fourier transform
and
0α>
2
−πα
() ,
fe
=∈
xx . (2.3.40)
x
0
ˆ
for a skew-symmetric function
kew
n
kew
Function (2.3.40) admits the following integral transform:
ˆ
() ,
fc
yy
=∈
n
()
α+
α
2
2(1)/2
n
+
y
n
, (2.3.41)
where
1
n
+
⎛⎞
Γ
⎜⎟
2
=
n
π
⎝⎠
(1)/2
n
. (2.3.42)
+
c
Function (2.3.41) is known as the Poisson kernel associated with the upper half-space
1
n
+
:( ,..., , ), 0
xxx x
111
+++
nn n
. Properties of Poisson kernel are discussed by Stein and
>
Weiss (1971, Ch. II).
Chapter 2. Fourier series, wavelets, and integral transforms
88
_____________________________________________________________________________
Example 2.3.6
Let real 0α> and
2
2
−πα
() ,
fe
=∈
xx
x
n
. (2.3.43)
Function (2.3.43) has the following Fourier image:
ˆ
fe
yy . (2.3.44)
() ,
nn
=α ∈
2
/
−π α
y
/2
−
Function (2.3.44) is known as the Weierstrass kernel.
2.3.5. Discrete Fourier Transform
The Discrete Fourier Transform (DFT) is a transformation that is applied to functions
defined at some discrete points.
Useful references can be found in Bracewell (1999), Ramirez (1986), and Walker
(1996).
Definition 2.3.3 Discrete Fourier transform; DFT)
The Discrete Fourier Transform (DFT) is defined by the following expression:
1
N
where
, 0,..., 1
nNω= −
n
ˆ
() ()
ffte
ω=
nk
∑
k
−
0
=
are the normalized frequencies:
ω= . (2.3.46)
n
N
n
2
ik
−πω
n
, (2.3.45)
Proposition 2.3.10
The inverse of the DFT (IDFT) is given by the following expression:
1
N
−
ft f e
() ( )
1
=ω∑, (2.3.47)
kn
N
ˆ
0
=
n
2
πω
ik
n
Proof
The proof flows out from the orthogonality condition for exponents:
()
nm k
N
1
−
δ=
mk
1
N
∑
n
2
e
0
=
−
π
i
N
, (2.3.48)
Chapter 2. Fourier series, wavelets, and integral transforms
89
_____________________________________________________________________________
where
Remarks 2.3.6
is Kroneker’s delta.
δ
mk
A. Orthogonality relation (2.3.48) is a discrete analogue of the corresponding relation for
continuous functions.
Sometimes, the DFT is considered as linear transformation in
B.
arbitrary vector in
N
, then its image
N
under DFT can be defined according to
∈x
N
: let
;...;
xx
=x
()
01
(2.3.45) as:
=⋅
xD x
, (2.3.49)
F
be
N
−
where
D is a square
C.
Similarly, the IDFT can be considered as linear transformation in
=⋅
where
D
:
Proposition 2.3.11
Matrices
D
F
where
is the unit diagonal matrix in
I
F
is a square
F
and
D
F
NN×
NN×
-matrix with components:
2nki
−π
N
e
D
()
=
F
nk
xD x
F
, (2.3.51)
-matrix with components:
nk
2
π
i
nk
1
=
N
e
N
D
()
F
(2.3.50)
N
:
(2.3.52)
defined by (2.3.50) and (2.3.52) respectively, satisfy identity:
DD I
⋅=
FF
.
, (2.3.53)
N
Proof
The proof relies on identity (2.3.48) and definitions (2.3.50) and (2.3.52) for matrices
.
D
F
Proposition 2.3.12 (Discrete variant of the Plancherel theorem)
Let ,
N
∈xy then the corresponding DFT-vectors ,N∈xy
satisfy a relation
D
and
F
Chapter 2. Fourier series, wavelets, and integral transforms
y
y
90
_____________________________________________________________________________
1
N
x
. (2.3.54)
Proof
⋅= ⋅x
To prove (2.3.54) we should note that according to (2.3.50) matrix
⋅≡ ⋅⋅ ⋅=⋅⋅⋅
But according to (2.3.48)
Relations (2.3.55) and (2.3.56) complete the proof.
Corollary (Discrete variant of the Parceval theorem)
xy Dx Dy xDDy
()
FF FF
()
1
⋅=
DD I
FF
N
1
22
=
xx
N
. (2.3.56)
. (2.3.57)
2.3.6. Fast Fourier Transform
The Fast Fourier Transform (FFT) is any numerically effective algorithm to compute
the DFT by not more than
logNN multiplications.
is symmetric; so
D
F
. (2.3.55)
Useful references can be found in Bracewell (1999), Ramirez (1986), and Walker
(1996).
Remark 2.3.7
Analysis of expression (2.3.49) reveals that determining N components of vector x needs in
2
multiplications (N multiplications for determining each component of vector
N
Definition 2.3.4 (Fast Fourier transform; FFT)
The Fast Fourier Transform (FFT) is any algorithm allowing to determine
vector
by no more than
x
multiplications.
logNN
).
x
N components of
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