Добавил:
ivanov666
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Additional Chapters of Higher Mathematics for Masters in Civill and Geotechnical Engineering. Учебное пособие по дополнительным разделам высшей матема
.pdf
Chapter 2. Fourier series, wavelets, and integral transforms
f
f
f
91
_____________________________________________________________________________
Remarks 2.3.8
A. While no algorithms more effective than demanding logNN multiplications for DFT are
known, from theoretical point of view there are no objections against more effective
algorithms.
One of the most popular FFT algorithms is going back to Gauss (1805), see also Heideman
B.
M.T., Johnson D.H., and Burrus C.S. (1884). Later on, Gauss algorithm was rediscovered
by Cooley and Tukey (1965); it is sometimes referred to as Cooley – Tukey FFT algorithm.
The main idea of this algorithm lies in factorizing
be achieved if
DFT into a sequence of successive DFT algorithms in smaller dimensional subspaces.
Actually, it is equivalent to substituting multiplication with a single matrix
matrices.
N =
p
, where p is a positive integer. In such a case it is possible to split
2
N , if it is not prime. The best results can
to the block
D
F
2.4. Equation Chapter 2 Section 4 Laplace,
Laplace-Carson, and Mellin integral transforms
2.4.1. Laplace integral transform
Laplace integral transform is one of the main types of integral transforms, that is
widely used in solving ordinary and partial differential equations (ODE and PDE).
The main properties of Laplace transform are discussed by Arendt et al. (2002), Dyke
(1999), and Schiff (1999).
Definition 2.4.1 (Laplace integral transform)
Let
well defined). The Laplace integral transform of
following formula:
where iξ=α+ β is a complex variable, and function ( )
The exact minimum value
be a locally integrable function (the integral of f over any finite interval in
()ft
∞
() ()
ξ=
inf Re( )ξ=α
fte dt
∫
0
()
is
(0, )∞
()ft is a new function ( )
t
−ξ
, (2.4.1)
ξ is called the Laplace image of f.
, at which the improper integral
0
ξ defined by the
Chapter 2. Fourier series, wavelets, and integral transforms
f
f
f
f
92
_____________________________________________________________________________
converges, is called the abscissa of the absolute convergence.
Proposition 2.4.1
The following condition is sufficient for existence of the Laplace transform:
∞
()
∫
0
t
−ξ
te dt
(2.4.2)
Proof
The proof is obvious.
Proposition 2.4.2
Let ()
t be a locally integrable function having only finite number of extremums and points of
discontinuity of the first order (limits
finite), if ( )
left half-plane
In particular, if
Re( )ξ>α
. (2.4.3)
0
(0)ft−
0
and
(0)ft+
at such a point exist and are
0
ξ is the Laplace image and all the possible singular points of ( )fξ belong to the
Re( )ξ<α
α+∞
1
1
2
i
π
α−∞
1
is a point of continuity of f, then
t
for some
1
i
∫
i
2
ξ
() ( 0) ( 0)
fed ft ft
ξβ= −++
α+∞
1
1
∫
i
π
α−∞
1
α>α
10
t
1
i
() ()
ed ft
ξβ=
i
, then
()
2
t
ξ
. (2.4.5)
. (2.4.4)
Proof
The proof is analogous to the proof of Proposition 2.3.2.
Definition 2.4.2 (Stiltjes convolution)
Let
1
,()fg L
∈
, then the
+
fgt ft g d t
∗≡−τττ ∈
()
Stiltjes convolution of these functions is defined by
t
() ( ) () ,
∫
0
. (2.4.6)
+
Chapter 2. Fourier series, wavelets, and integral transforms
f
f
f
f
f
f
f
f
f
f
f
f
93
_____________________________________________________________________________
Proposition 2.4.3
The Stiltjes convolution is commutative:
Proof
The proof flows out from changing variables in the integral in the right-hand side of (2.4.6).
Proposition 2.4.4
D. Let 1,()
If
E.
F.
If
Proof
The proof is analogous to proof of Proposition 2.3.4.
()()
fg L∈ , then
n
()
:() ( )
tfthτ→−
h
()
τ=
:() ()
xflxμ→
l
is the multiplying operator, then
() ( )
μ=μ
l
() ()
gt g f t∗=∗
~
gfg∗=⋅
(2.4.7)
. (2.4.8)
is the shifting operator, then
~
h
~1
h
−ξ
ef
−
lf
. (2.4.9)
. (2.4.10)
1
−
l
Proposition 2.4.5
Let both
absolute convergence
Proof
Extrapolating function ()
represent
multiplied by the value of
Now, performing differentiation of (2.4.12), exploiting (2.3.13) and (2.3.26) and noting that
everywhere in
f∂=∂
0
and
∂ be locally integrable in
, then at
α
0
∂=ξξ−
()
Reξ>α
~
t defined at 0t ≥ by the zero function at negative t , allows us to
as a sum of a continuous at
()ft
at
()ft
t+=
() () (0 ) ()
tftf Ht
=+
0
, we arrive at (2.4.11).
+
0
() (0)fff
0
+
:
+
functions with the finite abscissa of the
+
. (2.4.11)
function
0t =
and the Heaviside step function
()
t
0
. (2.4.12)
Chapter 2. Fourier series, wavelets, and integral transforms
f
f
gf
f
f
f
94
_____________________________________________________________________________
Corollary
~
Proposition 2.4.6
kk kk
ff f f
∂=ξξ−ξ −−
()
( ) (0 ) ... (0 )
−−
1( 1)
++
. (2.4.13)
Let ()
t be locally integrable in
t
⎛⎞
⎜⎟
()
fd
∫
⎜⎟
0
⎝⎠
, then
+
~
ττ =
()
ξ
. (2.4.14)
ξ
Proof
Let
be the integral in the left-hand side of (2.4.14), then
()gt
∂=
transform to the last identity and taking into account (2.4.11), we arrive at (2.4.14).
Remark 2.4.1
Similarly to relation (2.4.14) it can be shown that
i
α+∞
Provided function
1()
it
π
2
α−∞ ξ
and parameter
integral in the left-hand side of (2.4.15) is convergent at every
∞
1
⎛⎞
⎜⎟
fded
()
∫∫
i
1
ζζ ξ≅
⎜⎟
⎝⎠
meet conditions of Proposition 2.4.2, and the internal
α
1
ξ
t
t
(2.4.15)
ξ∈
.
+
. Applying the Laplace
Proposition 2.4.7
Suppose that (i) both f and
exists (as is easy to see, assumption (ii) ensures that
f
(0 )
+
be integrable in
∂
lim ( ) (0 )
ξ→+∞
lim ( ) 0
ξ→
ff
ξξ=
f
ξξ=
0
functions; (ii)
+
()0f +∞ =
+
. (2.4.16)
1
()fL
∈
), then
+
Proof
In view of (2.4.11), the left-hand side in the first relation in (2.4.16) can be written in a form:
; and (iii) limit
Chapter 2. Fourier series, wavelets, and integral transforms
f
f
95
_____________________________________________________________________________
lim ( ) lim ( ) (0 )
ξ→+∞ ξ→+∞
ffedf
ξξ= ∂ τ+ =
⎛⎞
⎜⎟
⎜⎟
⎝⎠
⎛⎞
⎜⎟
⎜⎟
⎝⎠
∞
∫
0
∞
∂τ+ =
() (0) (0)
fd f f
0
∫
0
−ξτ
0
++
+
, (2.4.17)
where
is the function introduced in (2.4.12). Proof of the second relation in (2.4.16) is
0
analogous.
2.4.2. Laplace–Carson integral transform
Definition 2.4.3 (Laplace-Carson integral transform)
Sometimes instead of (2.3.1) an integral transformation
is introduced. This is called the Laplace–Carson (or Carson) transformation.
Proposition 2.4.8
∞
() ()
ξ=ξ
∫
0
−ξ
fte dt
t
(2.4.18)
The inverse of Laplace–Carson transformation is defined by
i
α+∞
1
∫
i
α−∞
1
fe
1()
2
i
πξ
t
ξ
ξ
1
dftft
β= − + +
(0) (0)
()
2
. (2.4.19)
Proof
The proof is analogous to the proof of Proposition 2.3.2.
Remark 2.4.2
Comparing Definitions 2.4.1 and 2.4.3 reveals that the main properties of Laplace and LaplaceCarson transformations are identical.
Chapter 2. Fourier series, wavelets, and integral transforms
f
f
96
_____________________________________________________________________________
2.4.3. Mellin integral transform
A lot of properties of Mellin transform are discussed in manuscripts by Debnath and
Bhatta (2007), Davice (2002), and a paper by Flajolet and Golin (1994).
Definition 2.4.4 (Mellin integral trasform)
The following integral transformation is called Mellin integral transform
For an arbitrary integrable function
Re ,abξ∈
() ( )
right-hand side of (2.4.20) is absolutely convergent, is called the fundamental strip.
Proposition 2.4.9
The inverse of Mellin transformation is defined by
provided
Example 2.4.1
. The region
α+∞
1
1
2
i
π
α−∞
1
(,)abα∈
1
∞
() ()
ξ=
∫
0
ξ−
ftt dt
1
. (2.4.20)
the integral in (2.4.20) is well-defined at complex ξ, if
Re ,abξ∈
() ( )
i
∫
i
−ξ
() ( 0) ( 0)
ftd ft ft
ξβ= −++
in the complex plane, on which the integral in the
1
()
2
.
, (2.4.21)
The following example, known as the Cahen-Mellin integral, gives a relation between Gamma
function and the exponent at the inverse Mellin transformation
t
etd
=Γξβ
2
i
π
α−∞
()
∫
i
1
. (2.4.22)
i
α+∞
1
1
−−ξ
Remark 2.4.3
Comparing Definitions 2.4.1 for Laplace transform and 2.4.4 for Mellin transform reveals
existence of the following correspondence:
t
Mellin f e Laplace f t
⎡⎤
(); ();
⎣⎦
−ξ = ξ
[]
. (2.4.23)
Thus, Mellin transform may be considered as the multiplicative version of the two-sided
Laplace transform; see a paper by Flajolet and Golin for discussions.
Chapter 2. Fourier series, wavelets, and integral transforms
f
f
97
_____________________________________________________________________________
2.5. Equation Chapter 2 Section 5 Other
integral transforms
2.5.1. Hankel integral transform
The main properties of Hankel transform are discussed in manuscripts by Debnath and
Bhatta (2007) and Davies (2002).
Definition 2.5.1 (Hankel integral transform)
The following integral transformation is called Hankel integral transform of order
where
1
()fL
∈
J
m
+
Proposition 2.5.1
The inverse of Hankel transformation is defined by the following transformation
provided
Proposition 2.5.2
m
∞
() () ( )
ξ= ξ
mm
fttJ tdt
∫
0
, (2.5.1)
is the Bessel function of order m . The integral in (2.5.1) is well-defined, if
.
∞
ˆ
() ( ) ( 0) ( 0)
fJtd ft ft
ξξ ξ ξ= − + +
mm
∫
0
.
1/2m >−
1
()
2
, (2.5.2)
Herein, we summarize some of the basic properties of Hankel transformation
ξ
⎡⎤
(1) ()(1) () ()()
mf mf fttJtdt
⎣⎦
2
m
−ξ−+ξ= ξ
+−
11
mm m
ξ
⎡⎤
() () () ( )
fftJtdt
ξ+ ξ = ξ
−+
11
mm m
⎣⎦
2
m
∞
∫
0
∞
′
∫
0
, (2.5.3)
, (2.5.4)
Chapter 2. Fourier series, wavelets, and integral transforms
f
f
98
_____________________________________________________________________________
∞∞
The last expression is known as the Parseval identity for Hankel transformation.
∫∫
00
ˆ
() () () ()
gdftgttdt
ξξξξ=
mm
, (2.5.5)
2.5.2. Hartley integral transform
Discussion of the main properties of Hartley transform can be found in Bracewell
(1986).
Definition 2.5.2 (Hartley integral transform)
The following integral transformation is called Hartley integral transform:
∞
1/2
( ) 2 ( )cas
ω= π ω
−
() ()
ft tdt
∫
−∞
, (2.5.6)
where
Remark 2.5.1
In view of (2.5.7)
Hartley integral transform becomes up to a multiplier the sine or cosine Fourier integral
transform with
cas 2 sin / 4 2 cos / 4tt tω = ω+π = ω−π
() ()()
π
±
cas cos sintttω= ω+ ω
() () ()
shift of the argument.
4
ω
(2.5.7)
(2.5.8)
2.5.3. Hilbert integral transform
Discussion of the main properties of Hilbert transform can be found in manuscripts by
Stein and Weiss (1971), Davies (2202), and Pandey (1995).
Definition 2.5.3 (Hilbert integral transform)
The following integral transformation is called Hilbert integral transform:
Chapter 2. Fourier series, wavelets, and integral transforms
ftf
f
ftf
f
f
99
_____________________________________________________________________________
∞
The integral in (2.5.9) is evaluated as the Principle Value integral. For definition and properties
of the Hilbert transform see Stein and Weiss (1971).
Proposition 2.5.3
The inverse of the Hilbert transform is given by:
The integrals in (2.5.9), (2.5.10) are evaluated as the Principle Value integrals.
Remarks 2.5.2
() P.V.
ξ=
1
(0) (0)P.V.
−+ + = ξ
tft d
()
2
1()
∫
πξ−
−∞
dt
, (2.5.9)
t
∞
1()
∫
π−ξ
−∞
f
ξ
. (2.5.10)
t
A. Formulas (2.5.9), (2.5.10) are valid for functions
validity of the Hilbert transform and its inverse are discussed by Stein [Ch.II], who uses
singular integrals, and by Stein and Weiss [Ch.II and VI], by applying an elegant construction
utilizing Poisson’s integral.
Changing variables yields to another definition of the Hilbert transform
B.
∞
() P.V.
ξ=
1()
∫
π
−∞
and its inverse
1
tft d
(0) (0) P.V.
−+ + = ξ
()
2
Proposition 2.5.4
Herein, we summarize some of the basic properties of the Hilbert transform
Hilbert transform is the unitary operator in
A.
fL p∈−∞∞≤<∞. Proofs of
ξ−
, (2.5.11)
dt
t
∞
1()
ft
−ξ
∫
πξ
−∞
2
()L , i.e.
p
,,1
()
. (2.5.12)
2
L
(2.5.13)
f=
2
L
Parseval’s identity
B.
Chapter 2. Fourier series, wavelets, and integral transforms
f
fg
f
100
_____________________________________________________________________________
∞∞
C.
Hilbert transform of Stiltjes convolution
If
is Stiltjes convolution of two functions
()ht
∫∫
−∞ −∞
() () () ()
gd ftgtdt
ξξξ=
(2.5.14)
()ft
and
()gt
, then
Examples 2.5.1
Herein we give examples of Hilbert transform
Formulas (2.5.16) and (2.5.17) can be combined, resulting:
() () ()h
ξ= ξ ξ
sin( ) cos( sin( ) cos( ))tt=− ξ =− ξ
() ()
~~
cos( ) sin( )t =ξ
()
exp( ) exp( )it i i=− ξ
()
⎛⎞
⎜⎟
⎝⎠
~
1
22
11t
++ξ
()tδ=πξ (2.5.20)
()
(2.5.15)
~
(2.5.17)
(2.5.18)
~
ξ
=
~
(2.5.19)
1
(2.5.16)
2.5.4. Poisson integral transform
Basic properties of Poisson transform are discussed by Stein (1970), Stein and Weiss
(1971), and Zayed (1996).
Definition 2.5.4 (Half-plane Poisson integral transform)
The following integral transformation is called an upper half-plane Poisson integral transform:
∞
−
where
(, ) ()
() , 1
ft L p∈≤≤∞
p
()
xy ft dt
=π
1
∫
−∞
and
x ∈
y
2
xt y
−+
()
,
y
∈
, (2.5.21)
2
.
+
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]