Document 2438674

Gauge Institute Journal
H. Vic Dannon
Periodic Delta Function,
and Fejer-Cesaro Summation
of Fourier Series
H. Vic Dannon
[email protected]
June, 2012
Abstract
The Fejer Summation Theorem supplies the
conditions under which the Fejer-Cesaro Summation of Fourier
Series, associated with a function f (x ) equals f (x ) .
It is believed to hold in the Calculus of Limits. In fact,
The Theorem cannot be proved in the Calculus of Limits
under any conditions,
because the Fejer Summation requires integration of the singular
Fejer Kernel.
In Infinitesimal Calculus, the Fejer Kernel is the Periodic Delta
Function,
δperiodic (x ) = ... + δ(x + 4) + δ(x + 2) + δ(x ) + δ(x − 2) + δ(x − 4) + ... .
This function violates the Calculus of Limits Conditions
™ The Hyper-real δ(x ) , is not defined in the Calculus of Limits,
and δ(x ) is not integrable in any bounded interval.
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™
1
2
H. Vic Dannon
( δ(x + 0) + δ(x − 0) ) = 0
does not replace
δ(x )
at its
discontinuity point, x = 0 .
But δPeriodic (x ) equals its Fejer Summation, and the Fejer
Summation associated with any periodic hyper-real f (x ) , equals
f (x ) .
Keywords:
Infinitesimal,
Infinite-Hyper-Real,
Hyper-Real,
infinite Hyper-real, Infinitesimal Calculus, Delta Function,
Periodic Delta Function, Delta Comb, Fourier Series, Dirichlet
Kernel, Fejer Kernel, Fejer-Cesaro Summation, Fejer Summation
Theorem,
2000 Mathematics Subject Classification 26E35; 26E30;
26E15; 26E20; 26A06; 26A12; 03E10; 03E55; 03E17; 03H15;
46S20; 97I40; 97I30
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Contents
0. The Origin of the Fejer Summation Theorem
1. The Divergence of the Fejer Kernel in the Calculus of Limits
2. Hyper-real line.
3. Integral of a Hyper-real Function
4. Delta Function
5. Periodic Delta Function, δperiodic (ξ − x )
6. Convergent Series
7. Fejer Sequence and δperiodic (ξ − x )
8. Fejer Kernel and δperiodic (ξ − x )
9. Fejer Summation and δperiodic (ξ − x )
10. Fejer Summation Theorem
References
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The Origin of the Fejer-Cesaro
Summation Theorem
Let f (x ) be a function defined on [−1,1] , so that f (1) = f (−1) .
The Fourier Coefficients of f (x ) are
ξ =1
1
f (ξ )e−in πξd ξ ≡ cn , n = ..., −2, −1, 0,1, 2,... ,
∫
2 ξ =−1
The Fourier Series partial sums
ξ =1
Sn { f (x )} =
f (ξ ) { 12 e −in π(ξ −x ) + ... + 12 e −i π(ξ −x ) + 12 + 12 ei π(ξ −x ) + ... + 12 e in π(ξ −x ) }d ξ ,
ξ =−1
∫
Dirichlet Sequence
give rise to the Dirichlet Sequence
Dn (x ) = 12 e −in πx + ... + 12 e −i πx + 12 + 12 e i πx + ... + 12 e in πx
=
=
1
2
+ cos πx + cos 2πx + ... + cos n πx
sin(n + 12 )πx
2 sin 12 πx
,
n = 0,1, 2,..
0.1 Cesaro
To assign a numerical value to the divergent series
1 − 1 + 1 − 1 + 1 − 1 + ... ,
Cesaro suggested to consider the convergence of the Arithmetic
Means of its Partial Sums
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σ0 = s 0 = 1 ,
σ1 =
σ2 =
σ3 =
s0 + s1
1 + (1 − 1)
= 12 ,
=
2
2
s0 + s1 + s2
1 + (1 − 1) + (1 − 1 + 1)
=
=
3
3
2
3
,
s 0 + s1 + s2 + s 3
1 + (1 − 1) + (1 − 1 + 1) + (1 − 1 + 1 − 1)
=
=
4
4
1
2
,
……………………………………………………………………………..
Thus,
σ2k +1 = 12 ,
σ2k =
k +1
2k +1
→
1
2
and the series converges to 12 .
we conclude that
the infinite series 1 − 1 + 1 − 1 + 1 − 1 + ... has Cesaro Sum of
For any series
a0 + a1 + a2 + a3 + ... , with partial sums s 0 , s1, s2 ,...
If
s 0 + s1 + ... + sm
→σ
m +1
Then σ is the Cesaro Sum of a0 + a1 + a2 + a3 + ...
0.2
Fejer
applied Cesaro summation to Fourier Series.
The Fejer Summation partial sums are the Arithmetic Means
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1
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Fej Sn { f (x )} =
ξ =1
=
∫
ξ =−1
f (ξ )
H. Vic Dannon
S0 { f (x )} + S1 { f (x )} + ... + Sn { f (x )}
n +1
1
{(n + 1) 12 + n cos[π(ξ − x )] + ... + cos[πn(ξ − x )]}d ξ .
1
n
+
Fejer Sequence
The Fejer Summation associated with the function f (x ) is clearly
different from the Fourier Series associated with f (x ) , but it may
nevertheless converge to f (x ) .
The equality of the Fejer Summation associated with f (x ) , to f (x )
is the Fejer Summation Theorem.
The question is under which conditions does the Theorem hold.
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1.
The
Divergence
of
the
Fejer
Kernel in the Calculus of Limits
The Fejer Summation is believed to converge to f (x ) provided that
1. f (x ) is integrable on [−1,1]
2. f (x ) is periodic with period T = 2
3.
1
( f (x + 0) + f (x − 0) ) replaces f (x ) at a discontinuity point.
2
These Conditions reflect the belief that the equality depends only
on the function, regardless of the singularity of the Fejer Kernel.
The
Fejer
Summation
Sn +1 = Sn + an+1 .
is
not
an
infinite
series,
where
It has a singular Kernel, and it raises the
question whether it equals f (x ) .
In the Calculus of Limits, no smoothness of the function
guarantees the convergence of the Fejer Summation.
1.1 The Fejer Kernel is either singular or zero
In the Calculus of Limits, the Fejer Summation is the limit of the
Fej Sn { f (x )} = n1 c−ne −in πx + ... + nn−1 c−1e−i πx + c0 + n n−1 c1ei πx + ... + n1 cnein πx
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⎛ ξ =1
⎞⎟
⎛ ξ =1
⎞⎟
⎛ ξ =1
⎞⎟
⎜⎜ 1
⎜⎜ 1
⎜⎜ 1
⎟
⎟
in
x
in
−in πξ
π
πξ
= ⎜ 2n ∫ f (ξ )e
+ ... + ⎜ 2 ∫ f (ξ )d ξ ⎟⎟ + ... + ⎜ 2n ∫ f (ξ )e d ξ ⎟⎟⎟e −in πx
d ξ ⎟⎟e
⎜⎜
⎜
⎜
⎟⎟
⎟
⎟
⎝ ξ =−1
⎠
⎝⎜ ξ =−1
⎠⎟
⎝⎜ ξ =−1
⎠⎟
ξ =1
=
f (ξ ) { 21n e −in π(ξ −x ) + ... + n2−n1 e −i π(ξ −x ) + 12 + n2−n1 e i π(ξ −x ) + ... + 21n ein π(ξ −x ) } d ξ .
ξ =−1
∫
Fejer Sequence
As n → ∞ , the Fejer Sequence becomes the Fejer Kernel, which is
singular, and diverges at any ξ − x = 2k .
Thus, the Fejer Summation does not converge in the Calculus of
Limits.
Avoiding the singularity at ξ = x , by using the Cauchy Principal
Value of the integral does not recover the Theorem, because at any
ξ − x ≠ 2k , the Fejer Kernel vanishes, and the integral is
identically zero, for any function f (x ) .
Plots of the Fejer sequence confirm that
In the Calculus of Limits,
the Fejer Kernel is either singular or zero
1.2
Plots of Fejer Sequence
plots the spikes at x = 0 , x = −2 , x = 2
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gives 9 spikes
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Thus, the Fejer Summation Theorem does not hold in the Calculus
of Limits.
1.3 Infinitesimal Calculus Solution
By resolving the problem of the infinitesimals [Dan2], we obtained
the Infinite Hyper-reals that are strictly smaller than ∞ , and
constitute the value of the Delta Function at the singularity.
The controversy surrounding the Leibnitz Infinitesimals derailed
the development of the Infinitesimal Calculus, and the Delta
Function could not be defined and investigated properly.
In Infinitesimal Calculus, [Dan3], we can differentiate over jump
discontinuities, and integrate over singularities.
The Delta Function, the idealization of an impulse in Radar
circuits, is a Discontinuous Hyper-Real function which definition
requires Infinite Hyper-reals, and which analysis requires
Infinitesimal Calculus.
In [Dan5], we show that in infinitesimal Calculus, the hyper-real
ω =∞
1
δ(x ) =
ei ωxd ω
∫
2π ω =−∞
is zero for any x ≠ 0 ,
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it spikes at x = 0 , so that its Infinitesimal Calculus
x =∞
integral is
∫
δ(x )dx = 1 ,
x =−∞
and δ(0) =
1
< ∞.
dx
Here, we show that in Infinitesimal calculus, the Fejer Kernel is
the periodic hyper-real Delta Function: A periodic train of Delta
Functions.
And the Fejer Summation Fej S { f (x )} associated with a Hyperreal periodic function f (x ) , equals f (x ) .
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2.
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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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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3.
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
3.1 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 .
The Lower Integral is the Integration Sum where f (x ) is replaced
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by its lowest value on each interval [x − dx2 , x + dx2 ]
3.2
∑
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 ]
3.3
⎛
⎞⎟
⎜⎜
f (t ) ⎟⎟dx
∑ ⎜⎜ x −dxsup
⎟
dx
≤t ≤x +
⎠⎟
x ∈[a ,b ] ⎝
2
2
If the integral is a finite hyper-real, we have
3.4 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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4.
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 ⎪
⎬ . The
⎨ 0, ⎪
⎪
⎪ 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
18
,… Then,
1
dx
will mean the
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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 ) = ⎨
⎪
⎣⎢ 2 2 ⎦⎥
0, otherwise
⎪
⎩
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
19
,...
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 =−∞
In [Dan6], we obtained
k =∞
10.
1
δ(ξ − x ) =
e −ik (ξ −x )dk
∫
2π k =−∞
In [Dan8], we defined the Periodic Delta Function, and obtained
11.
δPeriodic (x ) = ... + δ(x + 4) + δ(x + 2) + δ(x ) + δ(x − 2) + δ(x − 4) + ...
= ... + 12 e −in πx + ... + 12 e −i πx +
20
1
2
+ 12 e i πx + ... + 12 e in πx + ...
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H. Vic Dannon
5.
Periodic Delta Function δperiodic (ξ − x )
5.1 Periodic Delta Function
δPeriodic (ξ − x ) = ... + δ(ξ − x + 2) + δ(ξ − x ) + δ(ξ − x − 2) + ...
is a periodic hyper-real Delta function, with period T = 2 .
In [Dan8], we obtained
δPeriodic (ξ − x ) = .. + 12 e −in π(ξ −x ) + .. + 12 e −i π(ξ −x ) + 12 + 12 e i π(ξ −x ) + .. + 12 e in π(ξ −x ) + ..
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6.
Convergent Series
In [Dan10], we defined convergence of infinite series in
Infinitesimal Calculus
6.1
Sequence Convergence to a finite hyper-real a
an → a iff
6.2
Sequence Convergence to an infinite hyper-real A
an → A iff
6.3
an
represents the infinite hyper-real A .
Series Convergence to a finite hyper-real s
a1 + a2 + ... → s iff
6.4
an − a = infinitesimal .
a1 + ... + an − s = infinitesimal .
Series Convergence to an Infinite Hyper-real S
a1 + a2 + ... → S iff
a1 + ... + an
represents the infinite hyper-real S .
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7.
Fejer Sequence and δperiodic (ξ − x )
7.1 Fejer Sequence Definition
Let f (x ) be an integrable function on [−1,1] .
Then, for each n = ..., −3, −2, −1, 0,1, 2, 3,... , the integrals
ξ =1
1
f (ξ )e−in πξd ξ ≡ cn
∫
2 ξ =−1
are the Fourier Coefficients of f (x ) .
The Fourier Series partial sums
ξ =1
Sn { f (x )} =
f (ξ ) { 12 e −in π(ξ −x ) + ... + 12 e −i π(ξ −x ) + 12 + 12 e i π(ξ −x ) + ... + 12 e in π(ξ −x ) }d ξ ,
ξ =−1
∫
Dirichlet Sequence
give rise to the Dirichlet Sequence
Dn (x ) = 12 e −in πx + ... + 12 e −i πx + 12 + 12 ei πx + ... + 12 e in πx
=
=
1
2
+ cos πx + cos 2πx + ... + cos n πx
sin(n + 12 )πx
2 sin 12 πx
,
n = 0,1, 2,..
The Fejer Summation partial sums are the Arithmetic Means
Fej Sn { f (x )} =
S0 { f (x )} + S1 { f (x )} + ... + Sn { f (x )}
n +1
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ξ =1
=
∫
f (ξ )
ξ =−1
H. Vic Dannon
1
{(n + 1) 12 + n cos[π(ξ − x )] + ... + cos[πn(ξ − x )]}d ξ .
n
+1
Fejer Sequence
They give rise to the Fejer Sequence
Fn (x ) =
7.2
Fm −1(x ) =
=
1
2
1
2
+ n n+1 cos[π(ξ − x )] + ... +
+ mm−1 cos πx + ... +
1
cos[πn(ξ
n +1
m −(m −1)
cos(m
m
− x )]
− 1)πx ,
D0(x ) + D1(x ) + ...Dm −1(x )
,
m
2 1
1 sin ( 2 m πx )
=
,
2m sin2 (1 πx )
2
m = 1, 2,..
Proof:
Fm −1(x ) =
D0(x ) + D1(x ) + ...Dm −1(x )
m
1
sin 23 πx
sin(m − 12 )πx ⎫⎪⎪
1 ⎧⎪⎪ sin 2 πx
= ⎨
+
+ ... +
⎬
1 πx
⎪⎪
m ⎪⎪ 2 sin 1 πx 2 sin 1 πx
2
sin
2
2
2
⎩
⎭
=
1
sin 12 πx + sin 23 πx + ... + sin(m − 12 )πx }
{
1
2m sin 2 πx
=
1
2m sin 12 πx
=
{
1
4m sin2 12 πx
cos 0−cos πx
2 sin 1 πx
2
+
cos πx −cos 2 πx
2 sin 1 πx
2
{1 − cos m πx }
24
+ ... +
cos(m −1)πx −cos m πx
2 sin 1 πx
2
}
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H. Vic Dannon
2 1
1 sin ( 2 m πx )
=
.,
2m sin2 ( 1 πx )
2
7.3 Fejer Sequence is a Periodic Delta Sequence, and
represents a Periodic Delta Function
2 1
1 sin ( 2 m πx )
, m = 1, 2, 3,...
Each Fm −1(x ) =
2m sin2 (1 πx )
2
1. has the sifting property on each interval,
x =−3
...
∫
x =−1
Fm −1(x )dx = 1 ;
x =−5
∫
x =1
Fm −1(x )dx = 1 ;
x =−3
∫
Fm −1(x )dx = 1 …
x =−1
2. is a continuous function
3. peaks on each of these intervals to lim Fm (x ) = 12 m .
x →2k
Proof of (1)
x =1
∫
x =1
Fm −1(x )dx =
x =−1
∫
x =−1
⎡ 1 + m −1 cos πx + ... + 1 cos(m − 1)πx ⎤ dx
⎢⎣ 2
⎥⎦
m
m
x =1
= ⎡⎢ 12 x + mm−π1 sin πx + ... + m(m1−1)π sin(m − 1)πx ⎤⎥
⎣
⎦ x =−1
= 1 .,
Proof of (3)
As x → 0 ,
2 1
1 sin ( 2 m πx )
0
→
2m sin2 (1 πx )
0
2
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Applying Bernoulli’s rule,
1
2m
⎡ sin2 1 m πx ⎤ '
1
1
1
1 (2 sin 2 m πx )cos 2 m πx ( 2 m π)
⎢⎣
⎥⎦
2
→
⎡ sin2 1 πx ⎤ ' x →0 2m (2 sin 1 πx )cos 1 πx ( 1 π)
2
2
2
2
x =0
⎣⎢
⎦⎥
=
1 sin m πx
2 sin πx
Applying Bernoulli’s rule to
=
x =0
0
0
1 sin m πx
,
2 sin πx
1 [sin m πx ]'
1 πm cos m πx
→
2 [sin πx ]' x →0 2 π cos πx
x =0
= 12 m . ,
7.4 Fejer Sequence Represents a Periodic Delta Function
δperiodic (ξ − x ) =
2
m
1 ⎜⎛ sin 2 π(ξ − x ) ⎞⎟⎟
⎜
⎟
2m ⎜⎜⎝ sin 1 π(ξ − x ) ⎠⎟⎟
2
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H. Vic Dannon
8.
Fejer Kernel and δperiodic (ξ − x )
8.1 Fejer Kernel in the Calculus of Limits
2 1
1 sin 2 m π(ξ − x )
Fejer (ξ − x ) = lim
m →∞ 2m sin2 1 π(ξ − x )
2
= lim
m →∞
{
1
2
+ mm−1 cos π(ξ − x ) + ... +
m −(m −1)
cos(m
m
− 1)π(ξ − x )
}
8.2 In the Calculus of Limits, the Fejer Kernel does not have
the sifting property
Proof:
By 4.3, as ξ − x → 2k ,
2 1
1 sin 2 m π(ξ − x )
→ 12 m .
2
2m sin 1 π(ξ − x )
2
Hence,
2 1
1 sin 2 m π(ξ − x )
→ lim 12 m = ∞ . ,
lim lim
m →∞ ξ −x →2k 2m sin2 1 π(ξ − x )
m →∞
2
8.3
Hyper-real Fejer Kernel in Infinitesimal Calculus
⎧⎪
Fejer (ξ − x ) = ⎪⎨
⎪⎪
⎩
1
n
2
, ξ − x = 2k
.
0, ξ − x ≠ 2k
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H. Vic Dannon
Proof: at any ξ − x = 2k ,
2 1
1 sin 2 m π(ξ − x )
2m sin2 1 π(ξ − x )
2
=
1m
2
.,
ξ −x =2k
For ξ − x ≠ 2k , and for any m ,
sin2 12 m π(ξ − x )
sin2 12 π(ξ − x )
is bounded by
M (x ) . Therefore,
21
1 sin 2 m π(ξ − x )
≤
0≤
2m sin2 1 π(ξ − x )
1
2m
M (ξ − x )
2
Hence, for ξ − x ≠ 2k ,
2 1
1 sin 2 m π(ξ − x )
= infinitesimal . ,
2m sin2 1 π(ξ − x )
2
8.4
Let
1
N
2
=
1
dx
be an infinite Hyper-real. Then,
2 1
1 sin 2 N π(ξ − x )
Fejer (ξ − x ) =
2N sin2 1 π(ξ − x )
2
=
1
2
+
N −1 cos π(ξ
N
− x ) + ... +
N −(N −1)
cos(N
N
− 1)π(ξ − x )
= ... + δ(ξ − x + 2) + δ(ξ − x ) + δ(ξ − x − 2) + ...
= δperiodic (ξ − x )
Proof:
Fejer (ξ − x ) =
1
2
+
N −1
cos π(ξ
N
− x ) + ... +
28
N −(N −1)
cos(N
N
− 1)π(ξ − x )
Gauge Institute Journal
H. Vic Dannon
By 8.3,
⎧⎪ N , ξ − x = −2 ⎧⎪ N , ξ = x ⎧⎪ N , ξ − x = 2
= ... + ⎨⎪ 2
+ ⎨⎪ 2
+ ⎨⎪ 2
+ ...
⎪⎪ 0, ξ − x ≠ −2
⎪⎪ 0, ξ ≠ x
⎪⎪ 0, ξ − x ≠ 2
⎩
⎩
⎩
⎪⎧⎪ 1 , ξ − x = −2 ⎪⎧⎪ 1 , ξ = x ⎪⎧⎪ 1 , ξ − x = 2
= ... + ⎨ dx
+ ⎨ dx
+ ⎨ dx
+ ...
⎪⎪ 0, ξ − x ≠ −2
⎪⎪ 0, ξ ≠ x
⎪⎪ 0, ξ − x ≠ 2
⎩
⎩
⎩
= ... + δ(ξ − x + 2) + δ(ξ − x ) + δ(ξ − x − 2) + ...
= δPeriodic (ξ − x ) . ,
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H. Vic Dannon
9.
Fejer Summation and δperiodic (ξ − x )
9.1 Fejer Summation of a Hyper-real Function
Let f (x ) be a hyper-real function integrable on [−1,1] .
Then, for each n = ..., −3, −2, −1, 0,1, 2, 3,... , the integrals
ξ =1
1
f (ξ )e−in πξd ξ ≡ cn
∫
2 ξ =−1
exist, with finite, or infinite hyper-real values. The cn are the
Fourier Coefficients of f (x ) .
The Fourier Series partial sums
ξ =1
Sn { f (x )} =
f (ξ ) { 12 e −in π(ξ −x ) + ... + 12 e −i π(ξ −x ) + 12 + 12 ei π(ξ −x ) + ... + 12 e in π(ξ −x ) } d ξ ,
ξ =−1
∫
Dirichlet Sequence
give rise to the Dirichlet Sequence
Dn (x ) = 12 e −in πx + ... + 12 e −i πx + 12 + 12 ei πx + ... + 12 e in πx
=
=
1
2
+ cos πx + cos 2πx + ... + cos n πx
sin(n + 12 )πx
2 sin 12 πx
,
n = 0,1, 2,..
The Fejer Summation partial sums are the Arithmetic Means
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Gauge Institute Journal
H. Vic Dannon
Fej Sn { f (x )} =
ξ =1
=
∫
f (ξ )
ξ =−1
S0 { f (x )} + S1 { f (x )} + ... + Sn { f (x )}
n +1
1
{(n + 1) 12 + n cos[π(ξ − x )] + ... + cos[πn(ξ − x )]}d ξ .
1
n
+
Fejer Sequence
They give rise to the Fejer Sequence
Fn (x ) =
1
2
1 cos[ πn(ξ − x )]
+ n n+1 cos[π(ξ − x )] + ... + n +
1
By 4.2, for m = 1, 2,.. ,
Fm −1(x ) =
=
1
2
+
m −1
cos πx
m
+ ... +
1
m
cos(m − 1)πx ,
D0(x ) + D1(x ) + ...Dm −1(x )
,
m
2 1
1 sin ( 2 m πx )
=
.
2m sin2 (1 πx )
2
Let
1
N
2
=
1
dx
be an infinite Hyper-real.
The Hyper-real Fejer Kernel is
Fejer (x ) =
1
2
+
N −1
cos πx
N
+ ... +
1
N
cos(N − 1)πx
= ... + δ(x + 4) + δ(x + 2) + δ(x ) + δ(x − 2) + δ(x + 4)...
The Fejer Summation associated with f (x ) is
Fejer S { f (x )} =
1 c e −Ni πx
N −N
+ ... +
N −1 c e −i πx
−1
N
+ c0 +
N −1 c e i πx
1
N
+ ... +
1
N
cN e Ni πx
For each x , it may assume finite or infinite hyper-real values.
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H. Vic Dannon
Fejer S { δPeriodic (ξ − x )} = δPeriodic (ξ − x )
9.2
Proof: Let N be an infinite hyper-real.
Fejer S { δPeriodic (ξ − x )} =
1c
e −Ni π(ξ −x )
N −N
+c0 +
+ .. +
N −1 c e −i π(ξ −x )
−1
N
N −1
c1e i π(ξ −x )
N
+ .. +
1
N
cN e Ni π(ξ −x ) ,
where
u =1
ƒ
N −n c
n
N
=
1
2
∫
δPeriodic (u )e−in πudu ,
u =−1
ƒ T = 2 is the period,
ƒ and c = 0 .
For δPeriodic (u ) with T = 2 ,
u =1
N −n
cn
N
=
1
2
∫
⎡ ... + δ(u + 2) + δ(u ) + δ(u − 2) + ... ⎤e −in πudu
⎣
⎦
u =−1
u =1
=
1
2
∫
u =−1
δ(u )e −in πudu = 12 .
Therefore,
Fejer S { δPeriodic (ξ − x )} = 12 e −iN π(ξ −x ) + ... + 12 e−i π(ξ −x )
+ 12 + 12 e i π(ξ −x ) + ... + 12 e iN π(ξ −x ) ,
= δPeriodic (ξ − x ) ,
by 5. ,
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H. Vic Dannon
10.
Fejer Summation Theorem
The Fejer Summation Theorem for a hyper-real function, f (x ) , is
the Fundamental Theorem of Fejer Summation.
It supplies the conditions under which the Fejer Summation
associated with f (x ) equals f (x ) .
It is believed to hold in the Calculus of Limits under some
Conditions. In fact,
The Theorem cannot be proved in the Calculus of Limits
under any conditions,
because the Fejer Summation requires integration of the singular
Fejer Kernel.
10.1 Fejer Summation Theorem cannot be proved in
the Calculus of Limits
Proof:
Take L = 1 , and c = 0 .
In the Calculus of Limits, the Fejer Summation is the limit of
Fej Sn { f (x )} = n1 c−ne −in πx + ... + nn−1 c−1e −i πx
+c0 + nn−1 c1e i πx + ... + n1 cnein πx ,
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Gauge Institute Journal
H. Vic Dannon
ξ =1
⎛ ξ =1
⎞⎟
⎛
⎞⎟
⎜⎜ 1
⎜⎜ n −1
⎟
in πξ
in
x
−
π
= ⎜ 2n ∫ f (ξ )e d ξ ⎟⎟e
+ ... + ⎜ 2n ∫ f (ξ )ei πξd ξ ⎟⎟⎟e −i πx
⎜⎜
⎜
⎟
⎟
⎠⎟
⎝ ξ =−1
⎠⎟
⎝⎜
ξ =−1
ξ =1
⎛ ξ =1
⎞⎟ ⎛
⎞⎟
⎛ ξ =1
⎞⎟
⎜⎜ 1
⎜⎜ n −1
⎜⎜ 1
⎟
⎟
i
i
x
in
−
πξ
π
−
πξ
d ξ ⎟⎟⎟ein πx
+ ⎜ 2 ∫ f (ξ )d ξ ⎟⎟ + ⎜ 2n ∫ f (ξ )e d ξ ⎟⎟e + ... + ⎜ 2n ∫ f (ξ )e
⎜⎜
⎜⎜
⎟⎟ ⎜⎜
⎟⎟
⎟
⎝ ξ =−1
⎠ ⎝
⎠
⎝ ξ =−1
⎠⎟
ξ =−1
ξ =1
=
∫
ξ =−1
f (ξ ) { 21n ein π(ξ −x ) + ... + n2−n1 ei π(ξ −x ) + 12 +
+ n2−n1 e−i π(ξ −x ) + ... +
1 e −in π(ξ −x )
2n
}d ξ .
As n → ∞ , the Fejer Sequence
Fn (ξ − x ) =
1 in π(ξ −x )
e
2n
+ ... + n2−n1 e i π(ξ −x )
+ 12 + n2−n1 e −i π(ξ −x ) + ... +
1 e −in π(ξ −x )
2n
becomes the Fejer Kernel, the infinite series
... +
1 in π(ξ −x )
e
2n
+ ... + n2−n1 e i π(ξ −x )
+ 12 +
By 8.2,
1 e −i π(ξ −x )
2n
+ ... +
n −1 e −in π(ξ −x )
2n
+ ... ,
The Fejer Kernel is singular whenever ξ − x = 2k , and
the Fejer Summation diverges in the Calculus of Limits.
Avoiding the singularity at ξ − x = 2k , by using the Cauchy
Principal Value of the integral does not recover the Theorem,
because at any ξ − x ≠ 2k , the Fejer Kernel is zero, and the
integral is identically zero, for any function f (x ) .
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Gauge Institute Journal
H. Vic Dannon
Thus, the Fejer Summation Theorem cannot be proved the
Calculus of Limits. ,
10.2 Calculus of Limits Conditions are irrelevant to Fejer
Summation Theorem
Proof: The Fejer Conditions are
1. f (x ) is integrable on [c − L, c + L ]
2. f (x ) is periodic with period T = 2L
3.
1
( f (x + 0) + f (x − 0) ) replaces f (x ) at a discontinuity point.
2
It is clear from 10.1 that the Fejer conditions on f (x ) do not
resolve the singularity of the Fejer kernel, and are not sufficient
for the Fejer Summation Theorem. ,
In Infinitesimal Calculus, by 8.4, the Fejer Kernel is the Periodic
Delta Function, and by 9.2, it equals its Fejer Summation.
Then, the Fejer Summation Theorem holds for any periodic
integrable Hyper-Real Function:
10.3 Fejer Summation Theorem for Hyper-real f (x )
If f (x ) is hyper-real function integrable on [c − L, c + L ] , so that
f (c − L) = f (c + L)
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Gauge Institute Journal
H. Vic Dannon
f (x ) = Fejer S { f (x )}
Then,
Proof: Take L = 1 , c = 0 , and 12 N =
1
dx
an infinite Hyper- real.
ξ =1
f (x ) =
∫
ξ =−1
f (ξ ) {... + δ(ξ − x + 2) + δ(ξ − x ) + δ(ξ − x − 2) + ...}dξ
δPeriodic (ξ −x ), where the period of Delta is T=2
By 8.4, δPeriodic (ξ − x ) = Fej (ξ − x )
ξ =1
=
∫
f (ξ ) { 21N eiN π(ξ −x ) + ... +
N −1 e i π(ξ −x )
2N
+ 12 +
ξ =−1
+ N2N−1 e −i π(ξ −x ) + ... +
1
2N
e−iN π(ξ −x ) } d ξ .
In [Dan6], we established that the Fourier Transform,
ξ =∞
∫
f (ξ )e −i 2πνξd ξ ,
ξ =−∞
exists for any Hyper-real function f (x ) . That is, the summation
ξ =∞
∑
f (ξ )e −i 2πνξd ξ
ξ =−∞
exists for any Hyper-real function f (x ) .
Consequently, the
summations over intervals exist, and we may write the integral as
the sum of integrals over intervals
=
1
N
⎛ ξ =1
⎞⎟
⎜
iN πξ
d ξ ⎟⎟⎟e−iN πx + ... +
⎜⎜⎜ 12 ∫ f (ξ )e
⎟⎟
⎜⎝ ξ =−1
⎠
⎛
⎜
⎞⎟
⎛ ξ =1
⎞⎟
⎜⎜ 1
⎟
⎟⎟ +
i
x
−
π
⎟
f
(
ξ
)
e
d
ξ
e
f
(
ξ
)
d
ξ
+
⎜
⎟⎟
⎟⎟
∫
2 ∫
⎜
⎟
⎝⎜
⎠⎟
⎝⎜
⎠
ξ =−1
ξ =−1
N −1 ⎜ 1
⎜
2N ⎜ 2
c−N
ξ =1
i πξ
c−1
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c0
Gauge Institute Journal
H. Vic Dannon
⎛
⎜
⎞⎟
f (ξ )e d ξ ⎟⎟⎟e i πx + ... +
∫
⎟
⎝⎜ ξ =−1
⎠⎟
+ N2N−1 ⎜⎜ 12
⎜
ξ =1
−i πξ
1
2N
⎛ ξ =1
⎞⎟
⎜⎜ 1
−iN πξ
d ξ ⎟⎟⎟e iN πx
⎜⎜ 2 ∫ f (ξ )e
⎟
⎝⎜ ξ =−1
⎠⎟
c1
=
−iN πx
1
c
e
−
N
N
+ ... +
cN
−i πx
N −1
c
e
1
−
N
+ c0 +
i πx
N −1
c
e
1
N
+ ... +
iN πx
1
c
e
N
N
= Fej S { f (x )} . ,
In particular, the Periodic Delta Function violates the Fejer
Conditions
™ The Hyper-real δ(x ) , is not defined in the Calculus of Limits,
and δ(x ) is not integrable in any bounded interval.
™
1
2
( δ(x + 0) + δ(x − 0) ) = 0
does not replace
δ(x )
at its
discontinuity point, x = 0 .
But by 9.2, δPeriodic (x ) satisfies the Fejer Summation Theorem.
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H. Vic Dannon
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[Achieser] Achieser, N. I., Theory of Approximation, Ungar, 1956.
[Carslaw] Carslaw, H. S., “Introduction to the Theory of Fourier Series and
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[Dan1] Dannon, H. Vic, “Well-Ordering of the Reals, Equality of all Infinities,
and the Continuum Hypothesis” in Gauge Institute Journal Vol. 6 No. 2, May
2010;
[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
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[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 Vol.
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Gauge Institute Journal
H. Vic Dannon
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