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Gauge Institute Journal
H. Vic Dannon
Delta Function and Hilbert
Transform
H. Vic Dannon
[email protected]
May, 2011
Abstract The Hilbert Integral Theorem guarantees that the
Hilbert Transform and its Inverse are well defined operations, so
that inversion yields the original function that generated the
Transform.
It is Fundamental to the theory of the Hilbert
Transform, but has not been obtained until now. We establish it
here for any hyper-real function in Infinitesimal Calculus, from
the properties of the Delta Function.
Nor were Conditions for the Existence of the Hilbert Transform
established. In general, it may seem that the existence of the
Hilbert Transform requires an analytic function, but conditions
have been given for functions that are absolutely integrable. That
is, functions with
∫
p
g(ξ ) d ξ < ∞ . We use the Delta Function, in
Infinitesimal Calculus to establish that any hyper-real function
has a Hilbert Transform.
The Transform gives rise to two Delta Functions, known in optical
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H. Vic Dannon
coherence, δ+(x ) , and δ−(x ) . We derive new formulas for these
functions.
Keywords: Infinitesimal, Infinite-Hyper-Real, Hyper-Real,
Cardinal, Infinity. Non-Archimedean, Non-Standard Analysis,
Calculus, Limit, Continuity, Derivative, Integral, Delta Function,
Fourier Transform, Hilbert Transform, Filter,
2000 Mathematics Subject Classification 26E35; 26E30;
26E15; 26E20; 26A06; 26A12; 03E10; 03E55; 03E17; 03H15;
46S20; 97I40; 97I30; 46F12; 46F25; 46F30; 32A55; 32A45;
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H. Vic Dannon
Introduction
0.1 Transfer Function
A Filter composed of a Resistance R , and an Inductance L in
series, driven by a harmonic signal at frequency ω , has the
impedance R + i ωL , and the Transfer Function
1
R
−ωL
=
+i
.
2
2 2
2
2 2
R + i ωL
R
+
ω
L
R
+
ω
L
u ( ω,0)
v ( ω,0)
The functions u(ω, 0) , and v(ω, 0) can be extended to the analytic
functions
u(z ) =
R
2
2 2
R +z L
,
and
v(z ) =
−zL
R 2 + z 2L2
.
It follows that
ξ =∞
Cauchy Principal Value of
∫
ξ =−∞
To see that, note that on the contour γ
3
u(ξ )
d ξ = πv(ω, 0) .
ξ−ω
Gauge Institute Journal
u(z )
H. Vic Dannon
R
1
∫> z − ω dz = L2 ∫> (z − i R )(z + i R )(z − ω) dz
γ
γ
L
L
⎧⎪
⎫⎪
1
⎪
⎪
=
2πiRes ⎨
⎬
⎪⎪ (z − i R )(z + i R )(z − ω) ⎪⎪ R
L2
L
L
⎩⎪
⎭⎪z =i L
R
⎧⎪
⎫⎪
1
⎪
⎪
=
2πi lim ⎨
⎬
R
⎪⎪
z →i R ⎪
+
−
)
(
z
i
)(
z
ω
L2
L ⎪
L
⎩⎪
⎭⎪
R
=
R
L2
=π
=
2πi
1
2i RL (i RL − ω)
1
iR − ωL
−i πR − πωL
R 2 + ω 2L2
.
On the semicircle in the upper-half plane
u(z )
∫ z − ωdz → 0 , as z → ∞ .
On the semi-circle that bypasses ω
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u(z )
∫? z − ω dz
=
H. Vic Dannon
R
1
dz
∫?
L2 (z − i R )(z + i R )(z − ω)
L
=
L
⎪⎧
⎪⎫
1
⎪⎬
(−π)iRes ⎪⎨
2
R
R
⎪⎪ (z − i )(z + i )(z − ω) ⎪⎪
L
L
L
⎩⎪
⎭⎪z = ω
R
⎧
⎫
⎪
⎪
1
⎪
⎪
= − πi lim ⎨
⎬
2
R
R
z → ω ⎪ (z − i )(z + i ) ⎪
L
⎪
⎪
⎪
L
L ⎪
⎩
⎭
R
=−
=
R
L2
πi
1
(ω − i RL )(ω + i RL )
−i πR
R 2 + ω 2L2
.
Therefore,
ξ =∞
Principal Value of
u(ξ )
−i πR − πωL
−i πR
dξ =
−
ξ−ω
R 2 + ω 2L2
R 2 + ω 2L2
ξ =−∞
∫
=π
−ωL
R 2 + ω 2L2
That is,
ξ =∞
⎧
⎫
1 ⎪⎪⎪
u(ξ, 0) ⎪
⎪
v(ω, 0) = ⎨ princial value of ∫
dξ ⎪
⎬
⎪
π ⎪⎪
ξ
ω
−
⎪
ξ =−∞
⎪⎩
⎪
⎭
ξ =∞
⎧ ξ = ω−ο
⎫
1 ⎪⎪⎪
u(ξ, 0)
u(ξ, 0) ⎪
⎪
dξ + ∫
dξ ⎪
= ⎨ ∫
⎬,
⎪
π ⎪⎪ ξ =−∞ ξ − ω
ξ
ω
−
⎪
ξ = ω +ο
⎪
⎩⎪
⎭
where ο is an infinitesimal.
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Similarly, it can be shown that
ξ =∞
⎧
⎫
1 ⎪⎪⎪
v(ξ, 0) ⎪
⎪
u(ω, 0) = − ⎨ princial value of ∫
dξ ⎪
⎬
⎪
π ⎪⎪
ξ
ω
−
⎪
ξ =−∞
⎪
⎩⎪
⎭
ξ =∞
⎧ ξ = ω−ο
⎫
1 ⎪⎪⎪
v(ξ, 0)
v(ξ, 0) ⎪
⎪
dξ + ∫
dξ ⎪
=− ⎨ ∫
⎬,
⎪
π ⎪⎪ ξ =−∞ ξ − ω
ξ
ω
−
⎪
ξ = ω +ο
⎪
⎩⎪
⎭
where ο is an infinitesimal.
0.2 Hilbert Transform
v(ω, 0) is the Hilbert Transform of u(ω, 0)
v(ω, 0) = Hu ,
and u(ω, 0) is the Inverse Hilbert Transform of v(ω, 0)
u(ω, 0) = H−1v .
In general, let
f (z ) = u(x , y ) + iv(x , y )
be analytic in y < 0 , and consider the contour
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By Cauchy Theorem,
f (z )
∫? z − x
γ
dz = 0 .
0
On the semi-circle that bypasses x 0 ,
∫>
⎧⎪ f (z ) ⎫⎪⎪
f (z )
dz = i πRes ⎪⎨
⎬
⎪⎩⎪ z − x 0 ⎪⎭⎪z =x
z − x0
0
= i π lim f (z ) = i π f (x 0 )
z →x 0
If on the semi-circle in the lower half plane
f (z )
∫ z − x dz → 0 , as z → ∞ .
0
then
ξ =∞
princial value of
f (ξ )
∫ ξ − x d ξ + i π f (x 0 ) = 0 .
0
ξ =−∞
That is,
ξ =∞
u(ξ, 0) + iv(ξ, 0)
d ξ + i π ( u(x 0 , 0) + iv(x 0 , 0) ) = 0
∫
ξ
−
x
0
ξ =−∞
p.v.
Therefore,
ξ =∞
1
u(ξ, 0)
v(x 0 , 0) = p.v. ∫
d ξ = Hu
π
ξ
x
−
0
ξ =−∞
ξ =∞
1
v(ξ, 0)
u(x 0 , 0) = − p.v. ∫
d ξ = H−1v .
π
ξ − x0
ξ =−∞
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0.3 Existence of the Hilbert Transform
In general, it may seem that the existence of the Hilbert
Transform requires an analytic function, but conditions have been
given for functions that are absolutely integrable. That is,
functions with
∫
p
g(ξ ) d ξ < ∞ . We use the Delta Function, in
Infinitesimal Calculus to establish that any hyper-real function
has a Hilbert Transform.
0.4 The Hilbert Integral Theorem
The Hilbert Integral Theorem guarantees that the Hilbert
Transform and its Inverse are well defined operations, so that
inversion yields the original function that generated the
Transform. We shall establish it for any hyper-real function in
Infinitesimal Calculus, from the properties of the Delta Function.
0.5 Delta functions and the Hilbert Transform
The Hilbert transform gives rise to two Delta Functions, known in
optical coherence, δ+(x ) , and δ−(x ) .
these Delta Functions.
8
We derive new formulas for
Gauge Institute Journal
H. Vic Dannon
1.
Hyper-real Line
Each real number α can be represented by a Cauchy sequence of
rational numbers, (r1, r2 , r3 ,...) so that rn → α .
The constant sequence (α, α, α,...) is a constant hyper-real.
In [Dan2] we established that,
1. Any totally ordered set of positive, monotonically decreasing
to zero sequences (ι1, ι2 , ι3 ,...) constitutes a family of
infinitesimal hyper-reals.
2. The infinitesimals are smaller than any real number, yet
strictly greater than zero.
3. Their reciprocals
(
1 1 1
, ,
ι1 ι2 ι3
)
,... are the infinite hyper-reals.
4. The infinite hyper-reals are greater than any real number,
yet strictly smaller than infinity.
5. The infinite hyper-reals with negative signs are smaller
than any real number, yet strictly greater than −∞ .
6. The sum of a real number with an infinitesimal is a
non-constant hyper-real.
7. The Hyper-reals are the totality of constant hyper-reals, a
family of infinitesimals, a family of infinitesimals with
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negative sign, a family of infinite hyper-reals, a family of
infinite hyper-reals with negative sign, and non-constant
hyper-reals.
8. The hyper-reals are totally ordered, and aligned along a
line: the Hyper-real Line.
9. That line includes the real numbers separated by the nonconstant hyper-reals. Each real number is the center of an
interval of hyper-reals, that includes no other real number.
10.
In particular, zero is separated from any positive real
by the infinitesimals, and from any negative real by the
infinitesimals with negative signs, −dx .
11.
Zero is not an infinitesimal, because zero is not strictly
greater than zero.
12.
We do not add infinity to the hyper-real line.
13.
The infinitesimals, the infinitesimals with negative
signs, the infinite hyper-reals, and the infinite hyper-reals
with negative signs are semi-groups with
respect to addition. Neither set includes zero.
14.
The hyper-real line is embedded in \∞ , and is not
homeomorphic to the real line. There is no bi-continuous
one-one mapping from the hyper-real onto the real line.
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15.
H. Vic Dannon
In particular, there are no points on the real line that
can be assigned uniquely to the infinitesimal hyper-reals, or
to the infinite hyper-reals, or to the non-constant hyperreals.
16.
No neighbourhood of a hyper-real is homeomorphic to
an \n ball. Therefore, the hyper-real line is not a manifold.
17.
The hyper-real line is totally ordered like a line, but it
is not spanned by one element, and it is not one-dimensional.
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2.
Integral of a Hyper-real Function
In [Dan3], we defined the integral of a Hyper-real Function.
Let f (x ) be a hyper-real function on the interval [a, b ] .
The interval may not be bounded.
f (x ) may take infinite hyper-real values, and need not be
bounded.
At each
a ≤ x ≤b,
there is a rectangle with base [x − dx2 , x + dx2 ] , height f (x ) , and area
f (x )dx .
We form the Integration Sum of all the areas for the x ’s that
start at x = a , and end at x = b ,
∑
f (x )dx .
x ∈[a ,b ]
If for any infinitesimal dx , the Integration Sum has the same
hyper-real value, then f (x ) is integrable over the interval [a,b ] .
Then, we call the Integration Sum the integral of f (x ) from x = a ,
to x = b , and denote it by
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x =b
∫
f (x )dx .
x =a
If the hyper-real is infinite, then it is the integral over [a, b ] ,
If the hyper-real is finite,
x =b
∫
f (x )dx = real part of the hyper-real . ,
x =a
2.1 The countability of the Integration Sum
In [Dan1], we established the equality of all positive infinities:
We proved that the number of the Natural Numbers,
Card` , equals the number of Real Numbers, Card \ = 2Card ` , and
we have
Card `
Card ` = (Card `)2 = .... = 2Card ` = 22
= ... ≡ ∞ .
In particular, we demonstrated that the real numbers may be
well-ordered.
Consequently, there are countably many real numbers in the
interval [a, b ] , and the Integration Sum has countably many terms.
While we do not sequence the real numbers in the interval, the
summation takes place over countably many f (x )dx .
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The Lower Integral is the Integration Sum where f (x ) is replaced
by its lowest value on each interval [x − dx2 , x + dx2 ]
2.2
∑
x ∈[a ,b ]
⎛
⎞
⎜⎜
inf
f (t ) ⎟⎟⎟dx
⎜⎝ x −dx ≤t ≤x + dx
⎠⎟
2
2
The Upper Integral is the Integration Sum where f (x ) is replaced
by its largest value on each interval [x − dx2 , x + dx2 ]
2.3
⎛
⎞⎟
⎜⎜
f (t ) ⎟⎟dx
∑ ⎜⎜ x −dxsup
⎟
dx
≤t ≤x +
⎠⎟
x ∈[a ,b ] ⎝
2
2
If the integral is a finite hyper-real, we have
2.4 A hyper-real function has a finite integral if and only if its
upper integral and its lower integral are finite, and differ by an
infinitesimal.
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3.
Delta Function
In [Dan5], we have defined the Delta Function, and established its
properties
1. The Delta Function is a hyper-real function defined from the
⎧⎪ 1 ⎫⎪⎪
hyper-real line into the set of two hyper-reals ⎪
⎨ 0, ⎬ . The
⎪⎩⎪ dx ⎭⎪⎪
0, 0, 0,... . The infinite hyper-
hyper-real 0 is the sequence
real
1
depends on our choice of dx .
dx
2. We will usually choose the family of infinitesimals that is
spanned by the sequences
1
1
1
,
,
,… It is a
n
n2
n3
semigroup with respect to vector addition, and includes all
the scalar multiples of the generating sequences that are
non-zero. That is, the family includes infinitesimals with
1
will mean the sequence n .
dx
negative sign. Therefore,
Alternatively, we may choose the family spanned by the
sequences
1
2n
,
1
3n
,
1
4n
15
,… Then,
1
dx
will mean the
Gauge Institute Journal
H. Vic Dannon
sequence 2n . Once we determined the basic infinitesimal
dx , we will use it in the Infinite Riemann Sum that defines
an Integral in Infinitesimal Calculus.
3. The Delta Function is strictly smaller than ∞
1
dx
δ(x ) ≡
4. We define,
where
χ
⎡ −dx , dx ⎤ (x ) ,
⎢⎣ 2 2 ⎥⎦
χ
⎧⎪1, x ∈ ⎡ − dx , dx ⎤
⎢⎣ 2 2 ⎥⎦ .
⎪
⎡ −dx , dx ⎤ (x ) = ⎨
⎪⎪ 0, otherwise
⎣⎢ 2 2 ⎦⎥
⎩
5. Hence,
™ for x < 0 , δ(x ) = 0
™ at x = −
1
dx
, δ(x ) jumps from 0 to
,
dx
2
1
.
x ∈ ⎡⎢⎣ − dx2 , dx2 ⎤⎦⎥ , δ(x ) =
dx
™ for
™ at x = 0 ,
™ at x =
δ(0) =
1
dx
1
dx
, δ(x ) drops from
to 0 .
dx
2
™ for x > 0 , δ(x ) = 0 .
™ x δ(x ) = 0
6. If dx =
7. If dx =
1
n
2
n
, δ(x ) =
, δ(x ) =
χ
χ
(x ), 2
[− 1 , 1 ]
2 2
1
,
χ
(x ), 3
[− 1 , 1 ]
4 4
2
,
(x )...
[− 1 , 1 ]
6 6
3
2 cosh2 x 2 cosh2 2x 2 cosh2 3x
16
,...
Gauge Institute Journal
8. If dx =
1
n
H. Vic Dannon
, δ(x ) = e −x χ[0,∞), 2e−2x χ[0,∞), 3e−3x χ[0,∞),...
x =∞
9.
∫
δ(x )dx = 1 .
x =−∞
10.
1
δ(x − ξ ) =
2π
k =∞
∫
eik (x −ξ )dk
k =−∞
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4.
δ(x ) and the Hilbert Transform
4.1 H { δ(⋅)} = −
Proof:
1
πx
For any infinitesimal d ξ , the Integration Sum for the
function
1
1 1
δ(ξ ) =
ξ −x
ξ − x dξ
χ
(ξ )
[−d ξ , d ξ ]
2
2
has only the unique hyper-real term
1 1
ξ − x dξ
χ
1
ξ −x
(ξ )d ξ =
[−d ξ , d ξ ]
2 2
χ
(ξ ) .
[−d ξ , d ξ ]
2
2
Therefore, the Hilbert Transform
ξ =∞
1
1
H { δ(⋅)} = p.v. ∫
δ(ξ )d ξ
π
ξ
−
x
ξ =−∞
exists, and equals the constant hyper-real part of
1 1
π ξ −x
χ
(ξ )
[−d ξ , d ξ ]
2 2
=−
ξ =0
That is
H { δ(⋅)} = −
18
1
.,
πx
1
.
πx
1 1
π ξ −x
χ
(ξ ) ,
[−d2ξ ,d2ξ ]
Gauge Institute Journal
H. Vic Dannon
Consequently,
4.2 δ(x ) = the inverse Hilbert Transform of the function −
=
t =∞
1
π2
∫
p.v.
t =−∞
1
πx
1
dt .
t(t − x )
t =∞
1
1 1
δ(x ) = − p.v. ∫ −
dt
π
π
t
t
−
x
t =−∞
Proof:
1
=
4.3
1
π2
1
π
2
π2
t =∞
p.v.
∫
t =−∞
t =∞
1
∫ t(t − x ) dt
t =−∞
=
p.v.
t =∞
p.v.
Proof: δ(0) =
∫
t =−∞
1
dt
t(t − x )
x =0
1
dt . ,
t(t − x )
1
= an infinite hyper-real
dx
=0
x ≠0
1
.
dx
δ(x ) x ≠0 = 0 . ,
4.4
δ(ξ − x ) =
1
π2
τ =∞
1
∫ (τ − ξ )(τ − x ) d τ .
τ =−∞
p.v.
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Gauge Institute Journal
1
H. Vic Dannon
τ = ξ −ο
τ = x −ο
τ =∞
1
1
1
=
dτ + ∫
dτ + ∫
dτ
∫
2
(τ − ξ )(τ − x )
(τ − ξ )(τ − x )
π τ =−∞ (τ − ξ )(τ − x )
τ =ξ +ο
τ =x +ο
for any infinitesimal ο .
Proof:
t =∞
1
δ(ξ − x ) =
p.v.
π2
∫
t =−∞
1
dt
t(t − [ξ − x ])
The change of variable, τ = t + x , gives
=
=
4.5 δ(ξ − x ) =
1
π2
1
π2
τ =∞
∫
p.v.
τ =−∞
1
dτ
(τ − x )(τ − ξ )
τ =∞
⎫
1 ⎧
1
1 ⎪
⎪
⎪
−
⎨
∫ ξ − x ⎪⎩⎪ τ − ξ τ − x ⎪⎬⎪⎪⎭d τ . ,
τ =−∞
p.v.
1
π2
τ =∞
⎫
1 ⎧
1
1 ⎪
⎪
⎪
−
⎨
∫ ξ − x ⎪⎩⎪ τ − ξ τ − x ⎪⎬⎪⎭⎪d τ
τ =−∞
p.v.
Proof: Apply 4.4. ,
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H. Vic Dannon
5.
Hilbert Integral Theorem
The Fundamental Theorem of the Hilbert Transform Theory is the
Hilbert Integral Theorem.
It guarantees that the Hilbert Transform and its Inverse are well
defined operations, that when inverted, yield the original function.
The use of the Delta Function, and Infinitesimal Calculus
Integration, are necessary to establish the Hilbert Integral
Theorem.
Consequently, it cannot be established in the Calculus of Limits
which is inadequate for dealing with singularities, and it does not
hold in the Calculus of Limits under any conditions.
5.1
Hilbert Integral Theorem
If u(x ) is hyper-real function,
Then, the Hilbert Integral Theorem holds.
1
u(x ) = − p.v.
π
τ =∞
∫
τ =−∞
ξ =∞
⎛
⎞⎟
1 ⎜⎜ 1
u(ξ )
d ξ ⎟⎟⎟d τ
⎜⎜ p.v. ∫
τ − x ⎝⎜ π
ξ − τ ⎠⎟⎟
ξ =−∞
Proof:
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Gauge Institute Journal
H. Vic Dannon
In Infinitesimal Calculus, the Integration Sum
ξ =∞
∫
u(ξ )δ(ξ − x )d ξ
ξ =−∞
yields u(x ) . That is,
ξ =∞
u(x ) =
∫
u(ξ )δ(ξ − x )d ξ .
ξ =−∞
By 4.4, δ(ξ − x ) equals the Integration Sum
δ(ξ − x ) =
1
π2
τ =∞
p.v.
∫
τ =−∞
1
dτ ,
(τ − ξ )(τ − x )
where the principal value means that τ has to skip both ξ , and x .
Substituting in the Integration Sum for u(x ) ,
ξ =∞
τ =∞
⎛
⎞⎟
⎜⎜ 1
1
1
u(x ) =
u(ξ )⎜ p.v. ∫
d τ ⎟⎟⎟d ξ
∫
⎜
(τ − ξ )(τ − x ) ⎠⎟⎟
π ξ =−∞
⎜⎝ π
τ =−∞
The terms in this Integration Sum are zero whenever ξ ≠ x . The
only nonzero term appears when ξ = x .
Changing the Summation order, ξ needs to skip τ ,and the
principal value is necessary in the ξ integration as well.
Therefore,
τ =∞
ξ =∞
⎛
⎞⎟
1
1 ⎜⎜ 1
u(ξ )
u(x ) = p.v. ∫
d ξ ⎟⎟⎟d τ
⎜⎜ p.v. ∫
π
τ − x ⎜⎝ π
τ − ξ ⎠⎟⎟
τ =−∞
ξ =−∞
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H. Vic Dannon
τ =∞
ξ =∞
⎛
⎞⎟
1
1 ⎜⎜ 1
u(ξ )
u(x ) = − p.v. ∫
d ξ ⎟⎟⎟d τ . ,
⎜⎜ p.v. ∫
π
τ − x ⎜⎝ π
ξ − τ ⎠⎟⎟
τ =−∞
ξ =−∞
Then, the Hilbert transform of u(ξ ) ,
ξ =∞
1
u(ξ )
p.v. ∫
dξ ,
π
ξ
τ
−
ξ =−∞
converges to a hyper-real function H (τ ) , some of its values may be
infinite hyper-reals, like the Delta Function.
And the Inverse Hilbert Transform of H (τ )
τ =∞
1
H (τ )
dτ
− p.v. ∫
π
τ
x
−
τ =−∞
converges to the hyper-real function u(x ) .
5.2 If u(x ) is hyper-real function,
Then,
ξ =∞
1
u(ξ )
d ξ converges to H (τ )
™ the hyper-real integral p.v. ∫
π
ξ
τ
−
ξ =−∞
τ =∞
1
H (τ )
d τ converges to u(x )
™ the hyper-real integral − p.v. ∫
π
τ
x
−
τ =−∞
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H. Vic Dannon
6.
δ+ (t ), δ−(t ), and the Hilbert transform
6.1 Definition of δ+(x ) , and δ−(x )
ω =0
1
δ−(x ) ≡
e i ωtd ω
∫
2π ω =−∞
1
δ+ (x ) ≡
2π
ω =∞
∫
e i ωtd ω
ω =0
δ(x ) = δ−(x ) + δ+ (x )
6.2
Proof: 3.10. ,
6.3
H {e−i ωx } = −ie−i ωx
For ω > 0 ,
That is, for ω > 0 , H delays the phase of e−i ωx by − π2 .
Proof:
H {e
−i ωx
}
ξ =∞
1
e −i ωξ
dξ
= p.v. ∫
x
π
ξ
−
ξ =−∞
In y < 0 , consider the contour γ
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Gauge Institute Journal
H. Vic Dannon
By Cauchy Theorem,
∫?
γ
e−i ωz
dz = 0 .
z −x
On the semi-circle that bypasses x ,
∫>
⎧⎪ e−i ωz ⎫⎪
e −i ωz
⎪⎬
dz = i πRes ⎪⎨
⎪
z −x
⎩⎪ z − x ⎪⎭⎪z =x
= i π lim e−i ωz
z →x
= i πe−i ωx .
By Jordan’s Lemma, on the semi-circle in the lower half plane
∫
e−i ωz
dz → 0 , as z → ∞ .
z −x
Therefore,
ξ =∞
e−i ωξ
d ξ + i πe−i ωx = 0 .
princial value of ∫
ξ −x
ξ =−∞
That is,
ξ =∞
1
e −i ωξ
p.v. ∫
d ξ = −ie −i ωx . ,
π
ξ −x
ξ =−∞
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Gauge Institute Journal
H. Vic Dannon
H { δ−(x )} = −i δ−(x )
6.4
That is, H rotates δ−(x ) by − π2
Proof:
ω =0
⎧⎪
⎫⎪
⎪⎪ 1
⎪⎪
i ωt
H { δ−(x )} = H ⎨
e
d
ω
⎬
⎪⎪ 2π ∫
⎪⎪
⎩⎪ ω =−∞
⎭⎪
ω =∞
⎧
⎫⎪
⎪
1
⎪
⎪⎪
−i ωt
⎪
e
d
ω
= H⎨
⎬
∫
⎪
⎪⎪
2
π
⎪
ω =0
⎪
⎩
⎭⎪
1
=
2π
1
=
2π
ω =∞
∫
H {e −i ωt }d ω
ω =0
ω =∞
∫
−ie−i ωtd ω
ω =0
1
= −i
2π
ω =0
∫
e i ωtd ω
ω =−∞
= −i δ−(x ) . ,
6.5
For ω > 0 ,
H {ei ωx } = iei ωx
That is, for ω > 0 , H advances the phase of ei ωx by
Proof:
26
π
2
.
Gauge Institute Journal
H. Vic Dannon
H {e
i ωx
}
ξ =∞
1
e i ωξ
dξ
= p.v. ∫
x
π
ξ
−
ξ =−∞
In y > 0 , consider the contour γ
By Cauchy Theorem,
∫>
γ
ei ωz
dz = 0 .
z −x
On the semi-circle that bypasses x ,
∫?
⎧⎪ ei ωz ⎫⎪
ei ωz
⎪⎬
dz = −i πRes ⎪⎨
⎪
z −x
⎩⎪ z − x ⎪⎭⎪z =x
= −i π lim ei ωz
z →x
= −i πei ωx .
By Jordan’s Lemma, on the semi-circle in the upper half plane
∫
e i ωz
dz → 0 , as z → ∞ .
z −x
Therefore,
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Gauge Institute Journal
H. Vic Dannon
ξ =∞
e i ωξ
d ξ − i πe i ωx = 0 .
princial value of ∫
ξ −x
ξ =−∞
That is,
ξ =∞
1
ei ωξ
p.v. ∫
d ξ = ie i ωx . ,
π
ξ −x
ξ =−∞
H { δ+ (x )} = i δ+ (x )
6.6
That is, H rotates δ+ (x ) by
π
2
Proof:
⎧⎪ ω =∞
⎫
⎪
⎪⎪ 1
⎪
i ωt
⎪
H { δ+ (x )} = H ⎨
e
d
ω
⎬
∫
⎪⎪ 2π
⎪
⎪
ω =0
⎪
⎩⎪
⎭
1
=
2π
1
=
2π
ω =∞
∫
H {ei ωt }d ω
ω =0
ω =∞
∫
ie i ωtd ω
ω =0
1
=i
2π
ω =∞
∫
e i ωtd ω
ω =0
= i δ+(x ) . ,
6.7
H { δ(x )} = i ( δ+(x ) − δ−(x ) )
Proof: 6.4, and 6.6. ,
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Gauge Institute Journal
H. Vic Dannon
δ+ (x ) − δ−(x ) = i
6.8
1
πx
Proof: By 6.7,
δ+(x ) − δ−(x ) = −i H { δ(x )}
ξ =∞
1
δ(ξ )
dξ
= −i p.v ∫
π ξ =−∞ ξ − x
= −i
=i
6.9
Proof:
1 ⎛⎜ 1 ⎞⎟
⎟
⎜
π ⎜⎝ −x ⎠⎟
1
.,
πx
δ+ (x ) =
1
1
δ(x ) + i
2
πx
δ−(x ) =
1
1
δ(x ) − i
2
πx
6.2, and 6.8. ,
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H. Vic Dannon
References
[Dan1] Dannon, H. Vic, “Well-Ordering of the Reals, Equality of all Infinities,
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[Dan2] Dannon, H. Vic, “Infinitesimals” in Gauge Institute Journal Vol.6 No
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[Dan3] Dannon, H. Vic, “Infinitesimal Calculus” in Gauge Institute Journal
Vol.7 No 1, February 2011;
[Dan4] Dannon, H. Vic, “Riemann’s Zeta Function: the Riemann Hypothesis
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[Dan5] Dannon, H. Vic, “The Delta Function” in Gauge Institute Journal
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Preprint Nr. 2053, August 1999
30
Gauge Institute Journal
H. Vic Dannon
[Mikus] Mikusinski, J. and Sikorski, R., “The elementary theory of
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[Rand] Randolph, John, “Basic Real and Abstract Analysis”, Academic Press,
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(1)
In “Collected Papers, Bernhard Riemann”, translated from
the 1892 edition by Roger Baker, Charles Christenson, and
Henry Orde, Paper XII, Part 5, Conditions for the existence of a
definite integral, pages 231-232, Part 6, Special Cases, pages
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(2)
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[Schwartz] Schwartz, Laurent, Mathematics for the Physical Sciences,
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