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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
61
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Remark 2.1.5
A. While Proposition 2.1.1 says that the Fourier series of function ()
Dirichlet conditions converges almost everywhere (except points of discontinuity) to
that function, Proposition 2.1.3 asserts that the Fourier series of any integrable with its
square, function (i.e. belonging to class
There are some peculiar examples of functions belonging to
Fourier series do not converge in the sense of topology of simple convergence, but they
converge on average in the sense of Proposition 2.1.3.
Quite often the considered function, defined on the interval
B.
Assume that such a function satisfies the Dirichlet conditions (i) and (iii) in
its Fourier series produces a new function, which is periodic, and at the ends of the
interval, Fourier series converges to
afb++ −
Suppose that the initial function
C.
points of discontinuity. Then at these points the convergence become uneven, moreover,
while the resulting function tends to the median value at these points, in the very closed
vicinity of these points there appear anomalous jumps, known as Gibbs effect.
D.
We should also mention the non-stability of the Fourier series summation procedure.
This is known as the “ill-posed” problem of Fourier synthesis. Such instability usually
takes place, when sufficiently large number of terms of the series is retained, which
results in appearing parasitic oscillations. To avoid this, special regularization
algorithms should be used.
() ()
2
2
. (2.1.15)
satisfies the Dirichlet conditions, but has several
()ft
) converges on average to that function.
,Lab
[]
2
,ab∈ , is not periodic.
[]
t satisfying the
, for which their
,Lab
[]
. Then,
,ab
[]
Example 2.1.1
Let function
The corresponding Euler-Fourier coefficients can be obtained analytically:
2.1.3. Examples of Fourier series
in the interval
()ft
ft
11exp()
== ≠
cci k
,,0
0
22
[]
⎧
⎪
=
()
⎨
⎪
⎩
k
be
0,2π
∈π
t
1, 0,
[
t
∈π π
0, , 2
[]
−π
ik
π
k
)
. (2.1.16)
. (2.1.17)
Chapter 2. Fourier series, wavelets, and integral transforms
f
f
62
_____________________________________________________________________________
After substituting these coefficients into series (2.1.3), we get
∞
1 1 exp( ' )
() exp( ')
+
ti ikt
∼
∑
22'
'
k
=−∞
where 'k takes any integer value, except the zero. Truncating series in the right-hand side of
(2.1.18) to 10 and 50 terms respectively and performing summation yields partial sums plotted
in Fig. 2.1.1. Some defects of convergence at points of discontinuity related to Gibbs effect are
clearly visible in this figure.
ik
−π
π
k
0.8
0.6
0.4
0.2
. (2.1.18)
Example 2.1.2
The same function (2.1.16), but now expanded into Fourier series (2.1.7) containing only sine
functions (some times that is called sine Fourier transform). Performing computations of the
coefficients
Combining (2.1.7) with (2.1.19) gives Fourier series in the form:
-4 -2 2 4
Figure 2.1.1. Partial sums for the truncated Fourier series (N=10
and N=50) for function (2.1.16)
by applying (2.1.8), yields:
b
k
1cos( )
−π
b
=
k
∞
1cos( )
tkt
∼
() sin( )
−π
∑
1
k
=
0
k
k
(2.1.19)
π
k
k
π
t
(2.1.20)
Chapter 2. Fourier series, wavelets, and integral transforms
63
_____________________________________________________________________________
Truncating this series to 50 terms, yields the partial sum plotted in Fig. 2.1.2.
0.6
0.4
0.2
As can be seen, the complete (Fig. 2.1.1) and sine Fourier (Fig. 2.1.2) transform differ by
shifting the sine transform by -½.
Example 2.1.3
Now, the same function (2.1.16), but expanded into Fourier series (2.1.7) containing only
cosine functions (some times that is called cosine Fourier transform). Performing computations
of coefficients
Now, the cosine Fourier series become
-4 -2 2 4
Figure 1.4.2 Partial sum for the truncated sine Fourier series (N=50)
by applying (2.1.8), yields:
a
k
1sin()
aa k
,0,0
===>
0
2
k
k
k
π
0
-0.2
-0.4
-0.6
for function (2.1.16)
π
(2.1.21)
t
ft∼ . (2.1.22)
1
()
2
Chapter 2. Fourier series, wavelets, and integral transforms
f
f
64
_____________________________________________________________________________
1.4
1.2
0.8
0.6
0.4
0.2
Thus, we arrive at the constant function plotted in Fig. 2.1.3. It is clear that combining sine and
cosine series gives the complete series (2.1.18).
Example 2.1.4
Let function ( )ft in the interval
The corresponding Euler-Fourier coefficients can be obtained analytically:
-4 -2 2 4
-0.2
-0.4
0
x
Figure 2.1.3 Partial sum for the cosine Fourier series for function (2.1.16)
0, / 2π
[]
() cos()
tt=
2(4 exp(2 ))
ik ik
c
=
k
−π
(16 1)
π−
be
. (2.1.23)
2
k
. (2.1.24)
After substituting these coefficients into series (2.1.3), we get
∞
2(4 exp(2 ))
ik ik
tikt
∼
( ) exp(4 )
∑
k
=−∞
−π
2
k
(16 1)
π−
. (2.1.25)
Chapter 2. Fourier series, wavelets, and integral transforms
f
65
_____________________________________________________________________________
Truncating series in the right-hand side of (2.1.25) to 50 terms and performing summation,
yields the partial sum plotted in Fig. 2.1.2. Again, at points of discontinuity we can observe the
Gibbs effect.
0.8
0.6
0.4
0.2
Example 2.1.5
The same function (2.1.23), but now we expand it into sine Fourier series. This gives
Truncating this series to 50 terms yields the partial sum plotted in Fig. 2.1.5
16
0
k
2
k
0.6
0.4
0.2
t
(2.1.26)
-3 -2 -1 1 2 3
Figure 2.1.4. Truncated Fourier series (N=50) for function (2.1.23)
∞
tkt
∼
() sin( )
∑
π−
(16 1)
k
1
=
-0.2
-0.4
-0.6
0
t
-3 -2 -1 1 2 3
Figure 2.1.5. Truncated sine Fourier series (N=50) for function (2.1.23)
Chapter 2. Fourier series, wavelets, and integral transforms
f
⎡
⎢⎥⎣
⎡
⎢
⎣
66
_____________________________________________________________________________
Example 2.1.6
The same function (2.1.23), but now we expand it into cosine Fourier series. This gives
∞
24
tkt
∼
() cos( )
+
∑
π
k
=
Truncating this series to 50 terms yields the partial sum plotted in Fig. 2.1.6
-3 -2 -1 0 1 2 3
Figure 2.1.6. Truncated cosine Fourier series (N=50) for function (2.1.23)
−
2
k
(16 1)
π−
1
0.7
0.68
0.66
0.64
0.62
0.6
0.58
0.56
0.54
0.52
0.5
(2.1.27)
t
Remark 2.1.6
Example 2.1.4. is interesting in the respect that it demonstrates useless of the Fourier analysis,
as well as other expansions utilizing periodic functions, for solving the extrapolation problem
for non-periodic functions or periodic ones but with the unknown periods, or with varying in
time periods (the latter can be studied by applying the so called chirplet transform, though.
Suppose for function (2.1.23) defined initially at
some other time interval, say
analytic in some vicinity of the terminal point
analytic in the considered initial time interval), and if
in the assumed vicinity of analyticity at point
series
ππ
t
,
∈+ε
22
⎤
, if the considered function is assumed to be
⎥
⎦
, then we can expand the function into Taylor’s
t
0
π
⎤
, we need to know its behavior at
0,
t
∈
2
⎦
(in our case
t
0
is sufficiently small, and it is contained
ε
π
, and our function is
=
t
0
2
Chapter 2. Fourier series, wavelets, and integral transforms
f
f
f
L
67
_____________________________________________________________________________
()
k
∞
ft t t
() ( )
=−=
∑
=
k
()
ft
0
!
k
0
N
=−+−
∑
=
0
k
k
0
k
()
()
ft
0
()( )
k
!
k
tt ott
00
. (2.1.28)
+
1
N
Reducing the infinite sum in the right-hand side of (2.1.28) to the first
N terms, results in
constructing the desired extrapolation with the known asymptotic estimate for the residual; see
the latter part in (2.1.28) containing the residual
(Not t+−
1
, that is known as Taylor’s series
0
with the residual in Peano’s form. Let our function (2.1.23) be expanded into Taylor’s series,
and we retain only first 20 terms, then thus truncated series give the following function; see Fig.
2.1.7, where the extrapolated function obtained by the truncated Taylor’s series is marked in
blue, while dotted red curve corresponds to the cosine function. Thus as we can see from Fig.
2.1.7, even the first 10 terms of Taylor’s series give quite adequate extrapolation, at least until
.
4t =
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
12345
t
Figure 2.1.7. Truncated Taylor’s series (N=10) for function (2.1.23)
However, in reality we usually do not know the function to be extrapolated in an analytical
form. Quite often we know only some finite number of values
discrete points of the time argument:
be constructed, that takes given values
tt
1
. In such situation, an interpolation polynomial can
,...,
N
1
,...,
at the given discrete points of the time interval.
f
N
1
,...,
of the function at some
f
N
There are several different formulas for the interpolation polynomials, of which the Lagrange
and Newton interpolation polynomials are the most widely used. Herein we give an example of
the Lagrange interpolation polynomial of degree
degree taken the prescribed values
() ()
tQt
,...,
1
N
=
∑
=
k
at the given argument values
f
N
, (2.1.29)
k
1
, which is a polynomial of the lowest
1N −
:
,...,
tt
1
N
Chapter 2. Fourier series, wavelets, and integral transforms
f
f
68
_____________________________________________________________________________
where
value
are polynomials of degree
Qt
()
k
at
t
k
k
( )...( )( )...( )
tt tt tt tt
()
Qt f
=
kk
−− − −
( )...( )( )...( )
tt tt tt tt
−−− −
kkkkkkN
111
111
kk N
−+
−+
having roots at
1N −
,..., , ,...,
ttt t
111
kk N
−+
. (2.1.30)
Constructing the Lagrange polynomial of degree 10 for the considered function (2.1.23) and the
uniformly spaced points
,..., 0, /2tt∈π
111
[]
, we get function plotted in Fig. 2.1.8, where values
corresponding to the Lagrange polynomial are marked in green, while cos( )t as before in
dotted red.
1
0.8
0.6
0.4
0.2
and taking
These results clearly show that for the considered function extrapolation with the Lagrange
polynomial produces almost as good result as it was achieved with Taylor’s series.
Example 2.1.7
Herein we give an example of a function with a variable period. Let function
the interval [0, / 2]π be
0
-0.2
-0.4
-0.6
-0.8
-1
12345
t
Figure 2.1.8. Extrapolation with the Lagrange polynomial of degree 10 for function
(2.1.23)
()ft
defined on
() cos(tan())
ttt= . (2.1.31)
Chapter 2. Fourier series, wavelets, and integral transforms
L
69
_____________________________________________________________________________
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0.2 0.4 0.6 0.8 1 1.2 1.4
t
Figure 2.1.9. Function (2.1.31)
The plot of this function is given in Fig. 2.1.9.
It is interesting to say that while the presented function is continuous and even analytic
everywhere in (0, / 2)π , it cannot be expanded into convergent (in
That is because function (2.1.31) is not integrable in
(0, / 2)π
2
-topology) Fourier series.
, the latter means that the
following integral:
π
2
cos(tan( ) )tt dt
∫
0
(2.1.32)
does not exist.
2.2. Equation Chapter 2 Section 2 Wavelet
analyses
Firstly, we give a native introduction to the wavelet analysis, based on Haar’s
definition for wavelets, as Haar’s wavelets are the most convenient for applications in
the theory of mechanical vibrations, and, at the same time, these are the easiest
wavelets to begin with. A brief outline of some other wavelets will be given in the
subsequent subsections.
Chapter 2. Fourier series, wavelets, and integral transforms
70
_____________________________________________________________________________
Discussion of the main properties of Haar’s wavelets can be found in manuscripts by
Chui (1992) and Burrus, Gopinath, and Guo (1998); the original Haar’s paper (1910)
delivers possibly the best explanation of the ideas of wavelets.
2.2.1. Basic definitions for Haar’s wavelets
Definition 2.2.1 (Heaviside step-function)
A function
1, 0
t
⎧
Ht
()
⎪
=
⎨
⎪
⎩
is called Heaviside function; see Fig.2.2.1.
>
0, 0
t
<
(2.2.1)
-4 -2 2 4
Remark 2.2.1
Shift of the argument of the Heaviside function by some
corresponding graph along horizontal axis to left, if
Definition 2.2.2 (Unit step function)
0.8
0.6
0.4
0.2
0
Figure 2.2.1 Heaviside function
,0aa∈≠
, and to right, if
0a >
results in shift of the
.
0a <
A function
() () (1 )tHtH tθ= −
(2.2.2)
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