Download Delta Function and the Poisson Summation Formula

Delta Function and the
Poisson Summation Formula
H. Vic Dannon
[email protected]
July, 2015
Abstract In the Calculus of Limits, the Poisson Summation
Formula holds under unnecessary conditions.
Using Delta function, and Periodic Delta Functions, we present
here an Infinitesimal Calculus proof, that is free of unnecessary
conditions.
Then, we apply the formula to a particular series.
Keywords:
Infinitesimal,
Infinite-Hyper-Real,
Hyper-Real,
infinite Hyper-real, Infinitesimal Calculus, Delta Function,
Fourier Transform, Periodic Delta Function, Delta Comb, Fourier
Series, Dirichlet Kernel,
2000 Mathematics Subject Classification 26E35; 26E30;
26E15; 26E20; 26A06; 26A12; 03E10; 03E55; 03E17; 03H15;
46S20; 97I40; 97I30.
1
Contents
0.
Calculus of Limits Limitations on the Poisson Summation
Formula
1.
Hyper-real Line
2.
Integral of a Hyper-real Function
3.
Delta Function
4.
Periodic Delta Function dPeriodic (x - x )
5.
The Poisson Summation Formula in Infinitesimal Calculus
6.
An Inquiry
7.
Applying the Formula to 6.
References
2
Calculus of Limits Limitations on
the Poisson Summation Formula
In the Calculus of Limits, the Poisson Summation Formula is
claimed to hold under restricting conditions: For instance, the
version presented in [Apostol, p.332]:
Theorem 11.24
If
f (x ) ³ 0 , for -¥ < x < ¥
x =¥
ò
f (x )dx exists as an improper Riemann Integral
x =-¥
f increases on -¥ < x £ 0
f decreases on
0£x <¥
m =¥
f (m +) + f (m-)
,
2
m =-¥
å
Then
and
m =¥
x =¥
å ò
f (x )e -2 pimxdx
m =-¥ x =-¥
are absolutely convergent to the same limit.
In fact, it is likely that even under these restrictions, the formula
does not hold.
However, since no scientist may ever care about these restrictions,
we will skip an examination that will render them meaningless,
and establish the Formula in Infinitesimal Calculus, free of
restricting conditions
3
For a Hyper-real Function in Infinitesimal Calculus, we obtain
under no limitations
m =¥
å
m =-¥
m =¥
å
f (m ) =
m =-¥
4
F (2pm ) .
1.
Hyper-real Line
The minimal domain and range, needed for the definition and
analysis of a hyper-real function, is the hyper-real line.
Each real number a can be represented by a Cauchy sequence of
rational numbers, (r1, r2 , r3 ,...) so that rn  a .
The constant sequence (a, a, a,...) is a constant hyper-real.
In [Dan2] we established that,
1. Any totally ordered set of positive, monotonically decreasing
to zero sequences (i1, i2 , i3 ,...) 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
, ,
i1 i2 i3
)
,... 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.
5
7. The Hyper-reals are the totality of constant hyper-reals, a
family of infinitesimals, a family of infinitesimals with
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.
6
15.
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.
7
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 ,
å
f (x )dx .
x Î[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
x =b
ò
f (x )dx .
x =a
If the hyper-real is infinite, then it is the integral over [a,b ] ,
8
If the hyper-real is finite,
x =b
ò
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
= ... º ¥ .
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 .
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
å
x Î[a ,b ]
æ
ö
çç
inf
f (t ) ÷÷÷dx
çè x -dx £t £x + dx
÷ø
2
2
9
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
æ
÷÷ö
çç
f (t ) ÷dx
å çç x -dxsup
÷÷
dx
£
t
£
x
+
ø
x Î[a ,b ] è
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.
10
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
ïì 1 ïü
hyper-real line into the set of two hyper-reals ïí 0, ïý . The
ïîï dx ïþï
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
2
3
n
n
n
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
,… Then,
1
dx
will mean the
sequence 2n . Once we determined the basic infinitesimal
11
3. The Delta Function is strictly smaller than ¥
1
dx
d(x ) º
4. We define,
where
c
é -dx , dx ù (x ) ,
ëê 2 2 úû
c
ïïì1, x Î éê - dx , dx ùú
ë 2 2 û.
é -dx , dx ù (x ) = í
êë 2 2 úû
ïï 0, otherwise
î
5. Hence,
 for x < 0 , d(x ) = 0
 at x = -
dx
1
, d(x ) jumps from 0 to
,
2
dx
1
x Î éêë - dx2 , dx2 ùúû , d(x ) =
.
dx
 for
 at x = 0 ,
 at x =
d(0) =
1
dx
dx
1
, d(x ) drops from
to 0 .
2
dx
 for x > 0 , d(x ) = 0 .
 x d(x ) = 0
6. If dx =
7. If dx =
8. If dx =
1
n
2
n
1
n
, d(x ) =
, d(x ) =
c
c
(x ), 2
[- 1 , 1 ]
2 2
1
,
c
(x ), 3
[- 1 , 1 ]
4 4
2
,
(x )...
[- 1 , 1 ]
6 6
3
2 cosh2 x 2 cosh2 2x 2 cosh2 3x
,...
, d(x ) = e -x c[0,¥), 2e-2x c[0,¥), 3e-3x c[0,¥),...
12
x =¥
9.
ò
d(x )dx = 1 .
x =-¥
10.
1
d(x - x ) =
2p
k =¥
ò
e -ik (x -x )dk
k =-¥
13
4.
Periodic Delta Function dPeriodic (x - x )
In [Dan5], we defined the Periodic Delta Function, and related it
to the Dirichlet Kernel in Infinitesimal Calculus
1)
Periodic Delta Function Definition
dPeriodic (x - x ) = ... + d(x - x + 2) + d(x - x ) + d(x - x - 2) + ...
is a periodic hyper-real Delta function, with period T = 2 .
2)
Fourier Transform of dPeriodic (x )
 { dPeriodic (x )} = ... + e -i 4 pn + 1 + e i 4 pn + ...
3) Fourier Integral Theorem for dPeriodic (x )
 -1 { dPeriodic (x )} = dPeriodic (x )
4)
Dirichlet Sequence Definition
The Fourier Series partial sums
x =1
n { f (x )} =
f (x ) { 12 e -in p(x -x ) + ... + 12 e -i p(x -x ) + 12 + 12 e i p(x -x ) + ... + 12 e in p(x -x ) }d x .

x =-1
ò
Dirichlet Sequence
give rise to the Dirichlet Sequence
Dn (x - x ) = 12 e -in p(x -x ) + ... + 12 e -i p(x -x ) + 12 + 12 e i p(x -x ) + ... + 12 e in p(x -x )
14
=
=
5)
1
2
+ cos p(x - x ) + cos 2p(x - x ) + ... + cos n p(x - x )
sin(n + 12 )p(x - x )
n = 0,1, 2,..
,
2 sin 12 p(x - x )
Dirichlet Sequence is a Periodic Delta Sequence
Each Dn (x ) =
sin(n + 12 )px
, n = 0,1, 2, 3,...
2 sin 12 px
 has the sifting property on each interval,
x =-3
..
ò
x =-1
Dn (x )dx = 1 ;
x =-5
ò
x =1
Dn (x )dx = 1 ;
ò
Dn (x )dx = 1 ..
x =-1
x =-3
 is a continuous function
 peaks on each of these intervals to
Dirichlet Sequence Represents dPeriodic (x - x )
6)
dPeriodic (x - x ) =
=
=
7)
lim Dn (x ) = n + 12 .
x 2m
1
2
sin 2n2+1 p(x - x )
2 sin 12 p(x - x )
+ cos p(x - x ) + cos 2p(x - x ) + ... + cos n p(x - x )
1 e -in p(x -x )
2
+ ... + 12 e-i p(x -x ) + 12 + 12 ei p(x -x ) + ... + 12 ein p(x -x ) .
Hyper-real Dirichlet Kernel
15
ìï
Dirichlet (x - x ) = ïí
ïï
î
8) Let N =
1
dx
Dirichlet (x - x ) =
=
1
2
1
2
+ n , x - x = 2m
x - x ¹ 2m
0,
be an infinite Hyper-real. Then
sin(N + 12 )p(x - x )
2 sin 12 p(x - x )
+ cos p(x - x ) + cos 2p(x - x ) + ... + cos N p(x - x )
= 12 e -iN p(x -x ) + .. + 12 e -i p(x -x ) + 12 + 12 e i p(x -x ) + .. + 12 e iN p(x -x )
= d(x - x + 2N ) + ... + d(x - x + 2) + d(x - x ) + d(x - x - 2) + ... + d(x - x - 2N )
= dPeriodic (x - x )
9)
... + d(x - x + 2) + d(x - x ) + d(x - x - 2) + ... =
= .. + 12 e -i 2p(x -x ) + 12 e-i p(x -x ) + 12 + 12 e i p(x -x ) + 12 e i 2p(x -x ) + ..
=
10)
1
2
+ cos p(x - x ) + cos 2p(x - x ) + cos 3p(x - x ) + ...
... + d(q - f + 2p) + d(q - f) + d(q - f - 2p) + ... =
= .. + 12 e -i 2(q -f) + 12 e -i (q -f) + 12 + 12 ei (q -f) + 12 ei 2(q -f) + ..
=
1
2
+ cos(q - f) + cos 2(q - f) + cos 3(q - f) + ...
16
5.
The Poisson Summation Formula
in Infinitesimal Calculus
By [Dan4], in Infinitesimal Calculus, a Hyper-real function has a
Fourier Transform
 {f }(w) = F (w) ,
without the Calculus of Limits conditions that in fact do not
guarantee the Fourier Transform for any function.
For such hyper-real function in Infinitesimal Calculus we need no
Calculus of Limits conditions to prove the Poisson Summation
Formula:
5.1 Poisson Summation Formula
For a Hyper-real function f (x ) in Infinitesimal Calculus
... + f (-2) + f (-1) + f (0) + f (1) + f (2) + ... =
= ... + F (-4p) + F (-2p) + F (0) + F (2p) + F (4p) + ...
Proof:
To apply the Fourier Transform to the f (n ) Summation, we define
S (x ) = ... + f (x - 2) + f (x - 1) + f (x ) + f (x + 1) + f (x + 2) + ...
u =¥
= ... +
ò
u =¥
f (u )d(x + 2 - u ])du +
u =-¥
ò
u =-¥
17
f (u )d(x + 1 - u )du +
u =¥
+
ò
f (u )d(x - u )du +
u =-¥
u =¥
+
ò
u =¥
f (u )d(x - 1 - u )du +
u =-¥
ò
f (u )d(x - 2 - u )du + ...
u =-¥
u =¥
=
ò
f (u ){.. + d(x + 2 - u ) + d(x + 1 - u ) + d(x - u ) +
u =-¥
+d(x - 1 - u ) + d(x - 2 - u ) + ..}du .
= f (x ) * {...
+ d(x + 2) + d(x + 1) + d
(x ) + d(x - 1) + d(x - 2) + ...}


Periodic Delta
Thus, S (x ) is the convolution of f with a Periodic Delta Function,
which was defined, and discussed extensively in [Dan5]
By the Convolution Theorem, S (x ) Fourier Transforms into the
product of the Transforms of f , and the Periodic Delta
 {S (x )} w =  {f (x )} w  {... + d(x + 2) + d(x + 1) +
+d(x ) + d(x - 1) + d(x - 2) + ..} w
= F (w){.. + d(x + 2) + d(x + 1) +
+d(x ) + d(x - 1) + d(x - 2) + ..} w
x =¥
Since
ò
d(x + 2 ) w =
d(x + 2)e -i wxdx = e 2i w ,
x =-¥
x =¥
d(x + 1) w =
ò
d(x + 1)e -i wxdx = e i w ,
x =-¥
x =¥
d(x ) w =
ò
d(u - x )e -i wxdu = e 0 = 1 ,
x =-¥
18
x =¥
d(x - 1) w =
ò
d(x - 1)e -i wxdx = e -i w ,
x =-¥
x =¥
d(x - 2 ) w =
ò
d(x - 2)e -i wxdu = e 2i w
x =-¥
 {S (u )} w = F (w){... + e 2i w + e i w + 1 + +e -i w + e -2i w + ...}
By [Dan5, 8.4, and 8.5], the Hyper-real Dirichlet Kernel
..
+ e 2i w + e i w + 1
+ +e -i w + e -2i w + ...

Dirichlet Kernel
equals the Periodic Delta
æw
ö
æw
ö
æwö
æw
ö
æw
ö
... + 2d çç + 4 ÷÷÷ + 2d çç + 2 ÷÷÷ + 2d çç ÷÷÷ + 2d çç - 2 ÷÷÷ + 2d çç - 4 ÷÷÷ + ...
çè p
çè p ø
çè p
çè p
èç p
ø
ø
ø
ø
= 2p{.. + d ( w + 4p ) + d ( w + 2p ) + d ( w ) + d ( w - 2p ) + d ( w - 4p ) + ..}
Substituting the Periodic Delta into  {S (u )} w ,
 {S (x )} w = F (w)2p{... + d ( w + 4p ) + d ( w + 2p ) +
+d ( w ) + d ( w - 2p ) + d ( w - 4p ) + ...}
= {.. + F (-4p)2pd ( w + 4p ) + F (-2p)2pd ( w + 2p ) +
+F (0)2pd(w) + F (2p)2pd(w - 2p) + F (4p)2pd(w - 4p) + ...} .
Since
w =¥
 {2pd ( w + 4p )} =
-1
ò d ( w + 4 p )e
i wx
d w = e i (-4 p )x
w =-¥
w =¥
 {2pd ( w + 2p )} =
-1
ò d ( w + 2p ) e
w =-¥
19
i wx
d w = e i (-2 p )x
w =¥
 {2pd ( w )} =
ò d ( w )e
-1
i wx
dw = e0 = 1
w =-¥
w =¥
 {2pd ( w - 2p )} =
-1
ò d ( w - 2p )e
i wx
d w = e i 2 px
w =-¥
w =¥
 {2pd ( w - 4p )} =
-1
ò d ( w - 4 p )e
i wx
d w = e i 4 px
w =-¥
S (x ) = .. + F (-4p)e i (-4 p )x + F (-2p)e i (-2 p )x +
+F (0) + F (2p)e i 2 px + F (4p)e i 4 px + ...
S (0) = ... + F (-4p) + F (-2p) + F (0) + F (2p) + F (4p) + ...
Therefore,
... + f (-2) + f (-1) + f (0) + f (1) + f (2) + ... = S (0) =
= ... + F (-4p) + F (-2p) + F (0) + F (2p) + F (4p) + ... . 
The Calculus of Limits proofs under their bizarre conditions, mask
the fact that
5.2 the Formula does not indicate which terms in the f (n )
summation contribute to whichever terms in the F (n )
summation
Many statements of the formula attempt to hint at 1.2 by using an
index
n for the f summation, and an index m for the F
summation.
This does not clarify the fact that the whole f summation equals
the whole F summation.
20
6.
An Inquiry
Professor Thomas Radil inquired about applying the Formula to the summation of a series:
21
22
7.
Applying Poisson Summation to 6.
F ( d1 2pm ) =
p -dv 2 pm
e
v
1
1
p -v 2 pm
p
F (0) = lim e d
=
d
d m 0 v
vd
The rest of the series transforms into what you have
2 pv
2 pv
2 pv
ö 2p
-2
-3
-4
2p -2dpv æç
1
d
d
d + ... ÷
÷
1
+
+
+
=
e
e
e
e
çç
÷
÷ø vd 2 pv
vd
è

e d -1
1
1-e
-
2 pv
d
Therefore, by the Poisson Summation Formula,
æ
2p çç 1
S =
ç +
vd ççè 2
÷÷ö
÷÷ . 
÷
- 1 ÷ø
1
e
23
2 pv
d
References
[Apostol] Tom Apostol, “Mathematical Analysis”, Second Edition, Addison
Wesley, 1974.
[Dan1] Vic Dannon, “Infinitesimals” in Gauge Institute Journal Vol.6 No 4,
November 2010;
[Dan2] Vic Dannon,
“Infinitesimal Calculus” in Gauge Institute Journal
Vol.7 No 4, November 2011;
[Dan3] Vic Dannon, “The Delta Function” in Gauge Institute Journal Vol.8
No 1, February 2012;
[Dan4] Vic Dannon, “Delta Function, the Fourier Transform, and Fourier
Integral Theorem”, in Gauge Institute Journal Vol.8 No 2, May 2012.
[Dan5] Vic Dannon, “Periodic Delta Function and Dirichlet Summation of
Fourier Series” in Gauge Institute Journal Vol.9, No 2, May 2013.
[Spiegel] Murray Spiegel, “Mathematical Handbook of Formulas and Tables”,
Schaum’s Outline Series, McGraw-Hill, 1968. p.109.
24