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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
71
_____________________________________________________________________________
0.8
0.6
0.4
0.2
-4 -2 2 4
is called a (unit) step-function; Fig. 2.2.2 .
Remark 2.2.2
A. Shift of the argument of the unit step-function by some
the unit step along the horizontal axis to left if
Multiplication of the argument of the unit step-function by some
B.
stretch of the unit step along the horizontal axis, if
Definition 2.2.3 (Haar's generic wavelet)
A function
( ) (2 1) (2 )tt tψ=θ−−θ
0
Figure 2.2.2 Unit step-function
(2.2.3)
, and to right if
0a >
, and shrink, if
1c <
,0aa∈≠
results in shift of
.
0a <
,0cc∈≠
results in
.
1c >
is called Haar’s (generic) wavelet; Fig. 2.2.3 .
Chapter 2. Fourier series, wavelets, and integral transforms
j
j
j
72
_____________________________________________________________________________
1
0.8
0.6
0.4
0.2
-4 -2 2 4
Figure 2.2.3 Haar’s generic wavelet
Definition 2.2.4 (Elementary Haar's wavelet)
0
-0.2
-0.4
-0.6
-0.8
-1
Haar’s generic wavelet is a source of some other functions
, called elementary (or basic)
tψ
,()jk
Haar’s wavelets:
/2
where
,()jk
() 2 (2 )
,
jk
and k are integers, and 0j≥ . Parameter j is called the j-level of the wavelet
.
tψ
jj
ttkψ=ψ−, (2.2.4)
Definition 2.2.5 (Scaling function)
Along with wavelets, we will also need a constant function
() 1tϕ=
0,0
. (2.2.5)
In wavelet analysis such a function is called the scaling function of the zero order.
Remarks 2.2.3
A. According to (2.2.4), Haar’s elementary wavelets differ by their position along abscissa
that is defined by parameter k, and their width and height defined by parameter
.
B.
At 0
Different Haar’s wavelets are schematically drawn in Fig. 2.2.4, from where it becomes
C.
= and 0k = we arrive at the generic Haar’s wavelet, thus
() ()ttψ=ψ
0,0
.
clear that an arbitrary wavelet resembles one period of the sine or cosine functions,
possibly shifted along the horizontal axis and stretched or shrank according to a
multiplier at the argument (such trigonometric functions serve as the basis for the so
called windowed Fourier transform). Thus, Haar’s wavelets have finite support (the
region, where they do not vanish), which makes them better suited for analyzing
Chapter 2. Fourier series, wavelets, and integral transforms
73
_____________________________________________________________________________
stochastic, or aperiodic processes. For example, a function presented in Fig. 2.1.9 admits
“good” approximation by Haar’s wavelets in any closed subinterval in
word “good” will be precisely defined in the next subsection.
D.
However, Haar’s wavelets exhibit some obvious deficiencies comparing to
trigonometric or exponential functions used in ordinary Fourier transform. Possibly, the
main and the most obvious deficiency, is lack of continuity and differentiability of the
basic wavelets. This limits or prevents their use in situations where some of the
differential properties of the approximated function should be retained. The other
deficiency also follows from the wavelet definition, as the introduced wavelets have
finite support, and cannot effectively be applied to analyzing some non-trivial functions
with the infinite domain, for example polynomials defined in
polynomials do not belong to
2
(,)L −∞ ∞ -space.
0, / 2π
[
(,)−∞ ∞ , though
. The
)
Proposition 2.2.1
All Haar’s wavelets along with scaling function
Figure 2.2.4 Different Haar’s wavelets
2.2.2. Basic properties of Haar’s wavelets
are mutually orthogonal:
ϕ
0,0
∞
() () 0
ttdt
ψψ =
,,
jk mn
∫
−∞
∞
() () 0
ttdt
ψϕ =
,0,0
jk
∫
−∞
, (2.2.6)
Chapter 2. Fourier series, wavelets, and integral transforms
j
L
j
L
L
L
f
74
_____________________________________________________________________________
provided either
Proof
Both relations in (2.2.6) follow from definitions (2.2.4) and (2.2.5).
Remark 2.2.4
m≠ , or kn≠ .
Orthogonality of wavelets
wavelets is alternating. This condition is sometimes called as the oscillation condition, and this
clarifies, why name “wavelet” is used for these functions.
Proposition 2.2.2
All the Haar’s wavelets are normal with respect to
Proof
Again, performing integration and recalling the multiplier
Proposition 2.2.3
Haar’s wavelets along with the zero-order scaling function represent the complete basis in
space (Any function belonging to
wavelets and the zero-order scaling function with any desired accuracy).
to the scaling function
tψ
,()jk
2
-norm:
∞
ψ≡ψ =
,,
jk jk
L
()
2
∫
−∞
2
-space can be approximated by the suitably chosen Haar’s
2
() 1
tdt
. (2.2.7)
means that each of the basic
ϕ
0,0
/2
, we arrive at (2.2.7).
2
2
-
Proof
The proof relies on a theorem of the everywhere density in
functions.
Proposition 2.2.4
Any function
to the basic Haar’s wavelets and the zero-order scaling function:
where
2
() [ , ]ft L∈−∞∞
() ()
tc c t
=ϕ + ψ
can be expanded into absolutely convergent series with respect
∞∞
00,0 , ,
∑∑
0
jk
==−∞
jk jk
2
of the space of linear (stepwise)
, (2.2.8)
Chapter 2. Fourier series, wavelets, and integral transforms
75
_____________________________________________________________________________
∞∞
and
Proof
Actually this proposition is a consequence from Propositions 2.2.1 – 2.2.3 .
Remarks 2.2.5
A. Expressions (2.2.8) - (2.2.10) reveal that the wavelet series has much in common with
=ϕ =
00,0
the Fourier series, and indeed the wavelet series can be regarded as Fourier series, but
with some other basis functions.
() () ()c f t t dt f t dt
∫∫
−∞ −∞
∞
cfttdt
=ψ
,,
jk jk
() ()
∫
−∞
, (2.2.9)
(2.2.10)
If the interval, where function
B.
(2.2.8) (with respect to k ) becomes finite.
2.2.3. Some other wavelets
For definitions of the wavelets considered in this subsection see deOliveira and Araújo,
Delprat et al. (2005).
Definition 2.2.5 (Mexican hat function)
The generic (or mother) wavelet can be taken as the second derivative of Gaussian function
(such a derivative is known as the “Mexican hat”):
() 1 exp
t
ψ= − −
σ
2
1/4 1/2 2 2
(3 ) 2
πσ σ σ
is defined, is bounded then the second sum in
()ft
22
⎛⎞⎛⎞
tt
⎜⎟⎜⎟
, (2.2.11)
⎝⎠⎝⎠
Chapter 2. Fourier series, wavelets, and integral transforms
76
_____________________________________________________________________________
where
corresponding to different
σ is an auxiliary parameter specifying width of the Mexican hat. The Mexican hats
are presented in Fig. 2.2.5.
σ
0.4
Remark 2.2.6
The set of wavelets based on the Mexican hat generic wavelet, leads to the wavelets with the
incompact support, as their domain is the whole interval
from the generic wavelets are obtained by the following formula
-6 -4 -2 2 4 6
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
Figure 2.2.5. The generic wavelet “Mexican hat” at different σ
σ=2
σ=1
σ=1/2
t
. The wavelets constructed
,−∞ ∞
()
ψ=ψ
,
ab
where as it was with Haar’s wavelets, parameter
horizontal axis.
Definition 2.2.6 (Hermitian wavelet)
The continuous generic functions and the corresponding continuous wavelets, as in the case of
the Mexican hat, are called Hermitian wavelets.
Definition 2.2.7 (Hermitian hat)
Another Hermitian generic wavelet is known as the Hermitian hat; the corresponding generic
wavelet is:
()
t
1
⎛⎞
⎜⎟
a
⎝⎠
tb
−
, (2.2.12)
a
specifies scaling, and b shift along
a
Chapter 2. Fourier series, wavelets, and integral transforms
77
_____________________________________________________________________________
2
1/4 2 /2
() (1 )
ttit
ψ= π −+ . (2.2.13)
−−
5
Both real and imaginary parts of this function are used as the generic wavelets; see Fig. 2.2.6.
0.6
2
t
-4 -2 2 4
Definition 2.2.8 (DOG wavelet)
Another type of wavelet is known as DOG (Derivative Of Gaussian); the corresponding generic
wavelet is (Torrence and Compo, 1998):
() exp /2
t
ψ= −η
0.4
0.2
0
-0.2
-0.4
Figure 2.2.6. Hermitian hat generic wavelets
1
ь m
+
(1)
−
1/2
m
Γ+
()
d
m
d
η
2
()
. (2.2.14)
Re(ψ)
t
Im(ψ)
Remark 2.2.7
It can be shown (Torrence and Compo, 1998) that wavelet (2.2.14) generalizes Mexican hat
wavelet.
Chapter 2. Fourier series, wavelets, and integral transforms
f
f
f
f
78
_____________________________________________________________________________
2.3. Equation Chapter 2 Section 3 Fourier
integral transforms and discrete Fourier
transforms
Herein, we present the outline of one of the most useful integral transforms that has a
lot of applications in different areas of mathematics. Our analyses are mainly based on
Stein and Weiss (1971) approach; see also classical manuscripts by Bochner (1955),
Titchmarsh (1962), and a newer textbook by Vretdblad (2005).
Methods and principles of Discrete Fourier Transforms (DFT) are discussed by
Bracewell (1999), Ramirez (1986), and Walker (1996).
2.3.1. Basic definitions
Remark 2.3.1
The integral Fourier transform arises as the natural limit of a Fourier series, when a given
function
Definition 2.3.1 (Fourier integral transform)
Let
defined by the following formula:
where the new function
representation (1.3.34) for the exponential function, expression (2.3.1) gives the following real
and imaginary parts for the Fourier image:
()ft
has the unbounded domain
()ft
be integrable function belonging to
∞
ˆ
ξ=
() ()
ˆ
ξ
()
ˆ
ξ= πξ∫, (2.3.2)
Re ( ) ( )cos(2 )
ˆ
Im ( ) ( )sin(2 )
ξ=− πξ∫. (2.3.3)
fte dt
∫
−∞
is called the Fourier image of f. Taking into account Euler’s
∞
ft tdt
−∞
∞
−∞
(,)t ∈−∞∞
1
(,)L −∞ ∞
−πξ
2
it
, (2.3.1)
ft tdt
.
. The Fourier integral transform is
Formulas (2.3.2), (2.3.3) are known as cosine and sine integral transforms respectively.
Chapter 2. Fourier series, wavelets, and integral transforms
f
f
f
f
f
L
f
79
_____________________________________________________________________________
Remark 2.3.2
Thus, a real integrable function f will generally have the complex Fourier image.
Scholium 2.3.1
Considering Fourier image
ξ≡ ξ is the amplitude and
where
() ()Af
is the phase of the image.
Proposition 2.3.1
An even integrable function has necessary real Fourier image, which is also even.
B. An odd integrable function has necessary imaginary Fourier image, which is also odd.
Proof
The proofs immediately follow from formulas (2.3.2), (2.3.3).
ˆ
ξ in Eulerian form
()
()
i
ˆ
fAe
ξ= ξ
() ()
ϕξ
, (2.3.4)
ˆ
ˆ
ϕξ ≡
( ) arccos
⎛⎞
⎜⎟
⎝⎠
ξ
Re ( )
f
ξ
()
A
(2.3.5)
Proposition 2.3.2
Suppose that both f and
inverse Fourier transform:
where sign
every point of its continuity.
Proof
Firstly, we prove (2.3.6) for the case, when both
into (2.3.6) and changing sequence of integration, yields:
ˆ
belong to 2(,)L −∞ ∞ , then the following expression defines the
∞
∫
−∞
means almost everywhere. Actually, the integral in (2.3.6) converges to f at
≅
()
t
δτ−
∞∞ ∞
∫∫ ∫
−∞ −∞ −∞
⎛⎞
πξ τ−
2()
() () ( ) ( )
te ddt ft tdtf
⎜⎟
⎜⎟
⎝⎠
it
πξτ
2
ˆ
i
edf
ξξ≅τ
() ()
, (2.3.6)
and
ξ= δτ− =τ
ˆ
belong to
1
. Substituting (2.3.1)
. (2.3.7)
Chapter 2. Fourier series, wavelets, and integral transforms
L
L
L
f
f
f
f
L
f
f
80
_____________________________________________________________________________
Generalization to the case
Corollary
2
is obvious, since
1
is dense in
2
.
If both f and
Proof
The proof immediately follows from (2.3.6).
Remark 2.3.3
Definition 2.3.1 admits an obvious generalization to the n-dimensional case:
for any
1
()
fL∈
ˆ
belong to
ˆ
() () ,
ffedx
ξ= ξ∈
n
.
1
(,)L −∞ ∞
(0) ( )
x
∫
n
and
∞
=ξξ
∫
−∞
2
in
−πξ⋅
is continuous at 0t = , then
ˆ
fd
x
(2.3.8)
(2.3.9)
Proposition 2.3.3
A. Mapping
If
B.
Proof
Proof A immediately follows from Hölder’s inequality (1.1.15) and boundedness of the
exponent in (2.3.9). Proof of condition B flows out from considering the integral Fourier
transform of the characteristic function
ˆ
() () ...
χ=χ = =
ξ (2.3.10)
QQ
2.3.2. The main properties
1
ˆ
from
f→
1
n
()
fL∈ , then
−πξ⋅ −πξ⋅
22
edx edx
x
∫∫∫
n
ˆ
ii
xx
n
into
()
L
is uniformly continuous, and
χ of the unit cube Q :
Q
11
00
()
L∞
n
is bounded, and
11
(2 )
−π
ˆ
.
f∞≥
1
L
ˆ
at
→
0f
ξ→∞
i
−πξ
2
n
∏
n
i
k
=
1
k
e
−
ξ
k
.
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