Periodic Delta Function and Fejer-Cesaro - Gauge-institute.org

Gauge Institute Journal 

H. Vic Dannon 

Periodic Delta Function, 

and Fejer-Cesaro Summation 

of Fourier Series 

H. Vic Dannon 

vic0@comcast.net 

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, 

δ periodic( 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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H. Vic Dannon 

 

1 

( 

2 

δ( x + 0) + δ( x − 0) ) = 0 does not replace δ( x) 

at its 

discontinuity point, x = 0 . 

But δ Periodic( 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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H. Vic Dannon 

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, δ ( ξ − x) 

6. Convergent Series 

periodic 

7. Fejer Sequence and δ ( ξ − x) 

periodic 

8. Fejer Kernel and δ ( ξ − x) 

periodic 

9. Fejer Summation and δ ( ξ − x) 

periodic 

10. Fejer Summation Theorem 

References 

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H. Vic Dannon 

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 

1 

2 

ξ= 

1 

−inπξ 

∫ f () ξ e dξ 

≡ c n 

, n = ..., −2, −1, 0,1,2,... , 

ξ=−1 

The Fourier Series partial sums 

ξ = 1 

{ } { 1 −inπξ ( −x) 1 −i πξ ( −x) 1 1 iπξ ( −x) 1 inπξ 

( −x) 

( ) = ∫ ( ξ) + ... + + + + ... + 

2 2 2 2 2 

} 

S n 

fx f e e e e dξ 

, 

 

ξ =−1 

give rise to the Dirichlet Sequence 

0.1 Cesaro 

Dirichlet Sequence 

1 −inπx 1 −i πx 1 1 iπx inπ 

x 

2 2 2 2 

Dn 

( x) = e + ... + e + + e + ... + 1 e 

2 

= + cos πx 

+ cos 2 πx 

+ ... + cosnπ 

x 

1 

2 

sin( n + ) πx 

= , n = 0,1, 2,.. 

2sin x 

1 

2 

1 

π 

2 

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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H. Vic Dannon 

σ 

σ 

σ 

σ 

0 

= s0 = 1, 

s + s 1 + (1−1) 

= = = , 

2 2 

0 1 1 

1 2 

s + s + s 1 + (1− 1) + (1− 1+ 

1) 

= = = , 

3 3 

0 1 2 2 

2 3 

s + s + s + s 1 + (1− 1) + (1− 1+ 1) + (1− 1+ 1−1) 

= = = , 

4 4 

0 1 2 3 1 

3 2 

…………………………………………………………………………….. 

Thus, 

σ + 

= , 

1 

2k 

1 2 

and the series converges to 1 2 . 

we conclude that 

σ 

k+ 

1 1 

2k 

= → 

2k 

+ 1 2 

the infinite series 1− 1 + 1− 1+ 1− 1 + ... has Cesaro Sum of 

1 

2 

For any series 

a0 + a1 + a2 + a3 + ..., with partial sums s 0 

, s 1 

, s 2 

,... 

s0 + s1 + ... + s 

If m 

m + 1 

→ 

σ 

0.2 Fejer 

Then σ is the Cesaro Sum of a0 + a1 + a2 + a3 + ... 

applied Cesaro summation to Fourier Series. 

The Fejer Summation partial sums are the Arithmetic Means 

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H. Vic Dannon 

FS 

ej 

n 

{ fx ( )} 

= 

{ f ( x ) } + { f ( x ) } + ... + { f ( x ) } 

S S S 

0 1 

n + 1 

n 

ξ= 

1 

1 

= ( ) {( 1) 

1 

∫ f ξ n + + ncos[ π( ξ − x)] + ... + cos[ πn( ξ −x)]} 

dξ. 

n + 1 

2 

 

ξ=−1 

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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H. Vic Dannon 

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 

1 

3. ( 

2 

fx ( + 0) + fx ( − 0) ) replaces f ( x ) at a discontinuity point. 

These Conditions reflect the belief that the equality depends only 

on the function, regardless of the singularity of the Fejer Kernel. 

The Fejer Summation is not an infinite series, where 

S n+ 1 

= S n 

+ a n+ 1. 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 

1 − 

n−1 − 

n−1 

{ } 

in x i x i x in x 

ej n 

fx ( ) = π π π 

c 

ne ... c 

1e c0 ce 

1 

... ce 

n − 

+ + n − 

+ + + + π 

n 

n n 

FS 

1 

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H. Vic Dannon 

⎛ ⎞ ⎛ ⎞ ⎛ 

= + + + + 

⎜ ⎜ ⎝ ⎠ ⎝ ⎠ ⎝⎜ 

ξ= 1 ξ= 1 ξ= 

1 

1 ∫ −inπξ inπx () ... 

1 

() ... 

1 

inπξ inπx 

f ξe dξ e f ξdξ − 

f() 

ξe dξ 

e 

2n 

∫ 2 ∫ 2n 

ξ=− 1 

⎟ 

ξ=− 1 

⎟ 

ξ=−1 

ξ = 1 

∫ 

ξ =−1 

{ 1 −inπξ ( −x) n− 

1 −iπξ ( −x) 1 n− 

1 iπξ ( −x) 1 inπξ 

( −x) 

} 

= f() ξ e + ... + e + + e + ... + e dξ 

 

2n 2n 2 2 

n 2n 


⎞⎟⎠ 

. 

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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H. Vic Dannon 

gives 9 spikes 

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H. Vic Dannon 

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 

( x) 

e i ω 

δ = x ω 

2π 

∫ d 

ω=−∞ 

is zero for any x ≠ 0 , 

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H. Vic Dannon 

it spikes at 

x = 0 , so that its Infinitesimal Calculus 

x =∞ 

∫ 

integral is δ( xdx ) = 1, 


x =−∞ 

1 

δ (0) = < ∞. 

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 FS { f ( x) 

associated with a Hyperreal 

periodic function f ( x ), equals f ( x ). 

ej 

} 

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H. Vic Dannon 

2. 

Hyper-real Line 

Each real number α can be represented by a Cauchy sequence of 

rational numbers, ( r , r , r ,...) so that r → α . 

1 2 3 

The constant sequence ( ααα , , ,...) is a constant hyper-real. 

In [Dan2] we established that, 

1. Any totally ordered set of positive, monotonically decreasing 

n 

to zero sequences 

infinitesimal hyper-reals. 

( ι1, ι2, ι3,...) 

constitutes a family of 

2. The infinitesimals are smaller than any real number, yet 

strictly greater than zero. 

1 1 1 

3. Their reciprocals ( , , ,... 

ι 1 

ι 2 

ι 3 

) 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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H. Vic Dannon 

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 

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. 

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H. Vic Dannon 

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 [ ab] , . 

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 

dx dx 

[ x − , x + 2 

], height f () x , and area 

2 

f ( xdx. ) 

We form the Integration Sum of all the areas for the x ’s that 

start at x = a, and end at x = b, 

∑ f ( xdx ) . 

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 [ ab] , . 

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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H. Vic Dannon 

x= 

b 

∫ f ( xdx ) . 

x= 

a 

If the hyper-real is infinite, then it is the integral over [, ab] , 

If the hyper-real is finite, 

x= 

b 

∫ fxdx ( ) = 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, 

2 Card 

Card = , and 

we have 

2 Card 

2 

Card 

Card = ( Card) = .... = 2 = 2 = ... ≡ ∞. 

In particular, we demonstrated that the real numbers may be 

well-ordered. 

Consequently, there are countably many real numbers in the 

interval [ ab] , , 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 ( xdx. ) 

The Lower Integral is the Integration Sum where f ( x ) is replaced 

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H. Vic Dannon 

by its lowest value on each interval 

3.2 

∑ 

x∈[ a, b] 

⎛ 

⎜⎝ 

dx dx 

2 2 

[ x − , x + ] 

⎞ 

inf f ( t) 

dx 

⎠⎟ 

x− ≤t≤ x+ 

dx dx 

2 2 

The Upper Integral is the Integration Sum where f ( x ) is replaced 

by its largest value on each interval 

3.3 

∑ 

x∈[ a, b] 

⎛ 

⎜⎝ 

dx dx 

2 2 

[ x − , x + ] 

⎞ sup f ( t) 

dx 

⎠⎟ 

x− ≤t≤ x+ 

dx dx 

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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H. Vic Dannon 

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 

hyper-real line into the set of two hyper-reals 

⎧ 

⎪ 1 ⎫ 

⎨0, ⎪ 

⎬ 

⎪⎩ 

dx ⎪ . The 

⎭ 

hyper-real 

0 is the sequence 0, 0, 0,... . The infinite hyperreal 

1 

dx 

depends on our choice of dx . 

2. We will usually choose the family of infinitesimals that is 

spanned by the sequences 

1 

n , 1 

2 

n 

, 

1 

n 

3 

,… It is a 

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 

negative sign. Therefore, 

1 

dx 

will mean the sequence n . 

Alternatively, we may choose the family spanned by the 

sequences 

1 

2 n , 

1 

3 n , 

1 

4 n ,… Then, 1 

dx 

will mean the 

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H. Vic Dannon 

sequence 2 n 

. 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 ∞ 

4. We define, 

1 

χ δ ( x) ≡ dx ( ) 

, 

dx x 

dx 

⎡ ⎤ , 

⎢− 

⎣ 2 2 ⎥⎦ 

where 

χ ⎡ 

⎢− 

⎣ 

dx, 

dx 

2 2 

⎧ dx dx 

1, x ∈ ⎡− 

, ⎤ 

( x) 

= ⎪ ⎢ 2 2 ⎥ 

⎨ ⎣ ⎦ . 

⎪⎪ 0, otherwise 

⎩ 

⎤ 

⎥⎦ 

5. Hence, 

for x < 0 , δ ( x) = 0 

at 

for 

dx 

x =− , δ( x) 

jumps from 0 to 

2 

dx dx 

⎢ ⎣ 

, 

2 2 ⎥ ⎦ , 1 

( x) 

x ∈ ⎡− 

⎤ 

δ = . 

dx 

1 

dx , 

at x = 0 , 

δ (0) = 

1 

dx 

at 

dx 

x = , δ( x) 

drops from 

2 

for x > 0 , δ ( x) = 0. 

xδ ( x) = 0 

1 

dx to 0. 

6. If dx = 

1 

, ( x) = 1 1( x),2 1 1( x),3 1 1( x )... 

n 

[ − , ] [ − , ] [ − , ] 

δ χ χ χ 

2 2 4 4 6 6 

7. If dx = 

2 

, 

n 

1 2 3 

δ ( x) = , , ,... 

2 2 2 

2 cosh x 2 cosh 2x 2 cosh 3x 

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H. Vic Dannon 

8. If dx = 

1 

, 

n 

− x − 2x − 3x 

[0, ∞) [0, ∞) [0, ∞) 

δ( x) = e χ ,2 e χ , 3 e χ ,... 

x =∞ 

∫ 

9. δ( xdx ) = 1. 

x =−∞ 

In [Dan6], we obtained 

10. 

k =∞ 

1 −ik( ξ−x 

) 

δξ ( − x) 

= e 

2π 

∫ dk 

k =−∞ 

In [Dan8], we defined the Periodic Delta Function, and obtained 

11. 

δ Periodic( x ) = ... + δ ( x + 4) + δ ( x + 2) + δ ( x ) + δ ( x − 2) + δ ( x − 4) + ... 

1 −inπx 1 −i πx 1 1 iπx 1 inπx 

e e e e 

2 2 2 2 2 

= ... + + ... + + + + ... + + ... 

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H. Vic Dannon 

5. 

Periodic Delta Function δ ( ξ − x) 

5.1 Periodic Delta Function 

periodic 

δPeriodic( ξ− x) = ... + δξ ( − x + 2) + δξ ( − x) + δξ ( −x 

− 2) +... 

is a periodic hyper-real Delta function, with period T = 2 . 

In [Dan8], we obtained 

1 inπξ ( x ) 1 iπξ ( x) 1 1 iπξ ( x) 1 inπξ 

( x ) 

2 2 2 2 2 

− − − − − − 

δPeriodic( ξ − x) = .. + e + .. + e + + e + .. + e + .. 

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H. Vic Dannon 

6. 

Convergent Series 

In [Dan10], we defined convergence of infinite series in 

Infinitesimal Calculus 

6.1 Sequence Convergence to a finite hyper-real a 

a → a iff a − a = infinitesimal . 

n 

n 

6.2 Sequence Convergence to an infinite hyper-real A 

a → A iff a 

n 

represents the infinite hyper-real A. 

n 

6.3 Series Convergence to a finite hyper-real s 

a1 + a2 + ... → s iff a1 + ... + an 

− s = infinitesimal . 

6.4 Series Convergence to an Infinite Hyper-real S 

a1 + a2 + ... → S 

iff 

a 

1 

+ ... +a 

n 

represents the infinite hyper-real S . 

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H. Vic Dannon 

7. 

Fejer Sequence and δ ( ξ − x) 

periodic 

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 

1 

2 

ξ= 

1 

∫ 

ξ=−1 

−inπξ 

f () ξ e dξ 

≡ c n 

are the Fourier Coefficients of f ( x ). 


ξ = 1 

{ } { 1 −inπξ ( −x) 1 −iπξ ( −x) 1 1 iπξ ( −x) 1 inπξ 

( −x) 

( ) = ∫ ( ξ) + ... + + + + ... + 

2 2 2 2 2 

} 

S n 


, 

 

ξ =−1 




x 

2 2 2 2 

Dn 

( x) = e + ... + e + + e + ... + 1 e 

2 

= + cos πx 

+ cos 2 πx 

+ ... + cosnπ 

x 

1 

2 


= , n = 0,1, 2,.. 

2sin x 

1 

2 

1 

π 

2 


FS 

ej 

n 

{ fx ( )} 

= 

{ f ( x) } + { f( x) } + ... + { f( x) 

} 

S S S 

0 1 

n + 1 

n 

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H. Vic Dannon 

ξ= 

1 

1 

= ( ) {( 1) 

1 


dξ. 

n + 1 

2 

 

ξ=−1 

They give rise to the Fejer Sequence 


F ( x ) = + cos[ πξ ( − x )] + ... + cos[ π n ( ξ− 

x )] 

1 n 

1 

n 2 n+ 1 n+ 

1 

7.2 

1 m−1 

m−( m−1) 

m 1 

π 

2 m m 

F 

− 

( x) = + cos x + ... + cos( m − 1) πx 

, 

D ( x ) + D ( x ) + ... D ( x ) 

m 

0 1 m−1 

= , 

2 1 

2 

2 1 

2 

1 sin ( mπx) 

= , m = 1, 2, .. 

2m 

sin ( πx) 

Proof: 

F 

m−1 

D0( x) + D1( x) + ... Dm 

−1( x) 

( x) = 

m 

1 3 1 

1 

⎧ 

sin πx sin πx sin( m ) ⎫ 

⎪ 

− πx 

2 2 2 

= ... 

⎪ 

⎨ + + + 

⎬ 

m 2sin1πx 2sin1πx 

2sin1 

⎪ 

π 

⎩ 

x 

2 2 2 ⎪⎭ 

1 

= + + + x 

2 2 2 

2msin 

πx 

1 

2 

{ sin 1 πx sin 3 πx ... sin( m − 

1 ) π } 

cos 0−cos πx cos πx−cos2πx 

cos( m−1) πx−cosmπx 

{ ... 

1πx 1πx 1πx 

} 

1 

= + + + 

2msin 

πx 

1 2sin 2sin 2sin 

2 

1 

= − 

4msin 

πx 

2 1 

2 

2 2 2 

{ 1 cosmπx} 

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H. Vic Dannon 

2 1 

2 

2 1 

2 

1 sin ( mπx) 

= . 

2m 

sin ( πx) 

7.3 Fejer Sequence is a Periodic Delta Sequence, and 

represents a Periodic Delta Function 

Each 

F 

m 

2 1 

1 2 

1( x) 

m 2 1 

2 

sin ( mπx) 

− 

= , m = 1, 2, 3,... 

2 sin ( πx) 

1. has the sifting property on each interval, 

x =−3 

∫ 

... F ( x) dx = 1; F ( x) dx = 1; F ( x) dx = 1… 

x =−5 

m−1 

2. is a continuous function 

x =−1 

∫ 

x =−3 

m−1 

x = 1 

∫ 

x =−1 

m−1 

1 

3. peaks on each of these intervals to lim F ( x) 

= m. 

Proof of (1) 

x= 1 x= 

1 

⎡1 m−1 1 

m−1 ⎢⎣ 

π 

2 m m 

x=− 1 x=−1 

x→2k 

∫ ∫ x 

F ( x) dx = + cos x + ... + cos( m −1) 

πx⎤ 

⎥⎦ 

d 

= 1. 

Proof of (3) 

1 

1 x x 

m 1 1 

sin x ... sin( m 1) x 

= 

− 

π 

π 

⎤ 

2 mπ 

m( m− 1) π 

⎥⎦x 

=−1 

= 

⎡ 

⎢ + + + − 

⎣ 

m 

2 

As x → 0 , 

2 1 

2 

2 1 

2 

1 sin ( mπx) 

0 

→ 

2m 

sin ( πx) 

0 

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H. Vic Dannon 

Applying Bernoulli’s rule, 

⎡sin ⎤ 

1 ⎢ ' 

⎣ ⎦ 1 (2 sin )cos ( ) 

→ 

2m ⎡ 

⎢sin 

2 m (2 sin x)cos x( ) 

⎣ ⎦ 

2 1 

mπx 1 1 1 

2 ⎥ 

mπx mπx mπ 

2 2 2 

2 1 0 

1 1 

πx 

⎤' x→ 

π π 

1π 

2 ⎥ 

2 2 2 x = 0 

1sinmπx 

0 

= = 

2 sin 0 

π x 

x = 0 

Applying Bernoulli’s rule to 1sin mπx 

2 sinπx 

, 

1[sin mπx]' 1πmcosmπx 

1 

→ = m . 

πx 

π πx 

2 

2 [sin ]' x→0 

2 cos 

x = 0 

7.4 Fejer Sequence Represents a Periodic Delta Function 

δ 

periodic 

1 

⎛sin πξ ( x) 

⎞ 

− 

( ξ − x) 

= 2m 

⎜ sin πξ ( x) 

⎜⎝ − ⎠⎟ 

m 

2 

1 

2 

2 

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H. Vic Dannon 

8. 

Fejer Kernel and δ ( ξ − x) 

periodic 

8.1 Fejer Kernel in the Calculus of Limits 

2 1 

2 

2 1 

2 

1 sin ( ) 

ejer ( ) lim mπξ− 

x 

F ξ − x = 

m→∞ 

2 m sin πξ ( − x) 

1 m−1 

m−( m−1) 

{ πξ x 

m πξ } 

= lim + cos ( − ) + ... + cos( −1) ( −x ) 

m→∞ 

2 

m 

m 

8.2 In the Calculus of Limits, the Fejer Kernel does not have 

Proof: 

the sifting property 

By 4.3, as ξ −x 

→ 2k, 

2 1 

1 sin mπξ 

( − x) 

2m 

2 

sin 

1 

πξ ( − x) 

Hence, 

2 → 

1 

2 

2 

m . 

2 1 

1 sin mπξ 

( x) 

2 1 

lim lim lim 

m→∞ ξ−x→2k 2m 

2 1 

2 

sin πξ ( − x) 

m→∞ 

2 

− 

→ m = 

∞. 

8.3 Hyper-real Fejer Kernel in Infinitesimal Calculus 

F 

ejer 

⎧⎪ 

1 

n , ξ x 2 

( ξ x) 

⎪ − = k 

− = 2 

⎨ . 

⎪ 0, ξ − x ≠ 2k 

⎪⎩ 

27


H. Vic Dannon 

Proof: at any ξ − x = 2k, 

2 1 

mπξ− 

x 

2 

= 

1 

2 1 

2 

πξ− 

x 

2 ξ− x= 

2k 

1 sin ( ) 

2m 

sin ( ) 

m 

. 

For ξ −x 

≠ 2k, and for any m , 

Mx ( ). Therefore, 

2 1 

2 

2 1 

2 

sin mπξ 

( − x) 


is bounded by 

2 1 

mπξ− 

x 

2 

≤ 

1 

2 1 

2m 

πξ− 

x 

2 

1 sin ( ) 

0 ≤ 

M( 

ξ 

2m 

sin ( ) 

− x) 

Hence, for ξ −x 

≠ 2k, 

2 1 

2 

2 1 

2 

1 sin mπξ 

( − x) 

= 

2m 


infinitesimal . 

8.4 Let 1 N = 1 be an infinite Hyper-real. Then, 

2 dx 

F 

ejer 

( ξ − x) 

= 

2 1 

2 

2 1 

2 

1 sin Nπξ 

( − x) 

2N 


1 N −1 

2 N 

N−( N−1) 

N 

= + cos πξ ( − x) + ... + cos( N −1) πξ ( −x ) 

= ... + δξ ( − x + 2) + δξ ( − x) + δξ ( −x 

− 2) + ... 

= δ ( ξ −x) 

periodic 

Proof: 

1 N −1 

N−( N−1) 

ejer 

ξ 

π ξ 

2 N N 

F ( − x) = + cos ( − x) + ... + cos( N −1) π( ξ −x) 

28


H. Vic Dannon 

By 8.3, 

⎧⎪ , ξ − x = − 2 ⎧⎪ , ξ = x ⎧⎪ 

, ξ − x = 2 

= + ⎨ + ⎨ + ⎨ 

+ 

⎪ 0, ξ −x ≠ −2 ⎪ 0, ξ ≠ x ⎪ 0, ξ −x 

≠ 2 

⎩ ⎩ ⎩ 

N N N 

... ⎪ 2 ⎪ 2 ⎪ 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 

⎩ ⎩ ⎩ 

= ... + δξ ( − x + 2) + δξ ( − x) + δξ ( −x 

− 2) + ... 

= δ ( ξ −x). 

Periodic 

29


H. Vic Dannon 

9. 

Fejer Summation and δ ( ξ − x) 

periodic 

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 

1 

2 

ξ= 

1 

∫ 

ξ=−1 

−inπξ 

f () ξ e dξ 

≡ c n 

exist, with finite, or infinite hyper-real values. The 

c n 

are the 

Fourier Coefficients of 

f ( x ). 


ξ = 1 

{ } { 1 −inπξ ( −x) 1 −i πξ ( −x) 1 1 iπξ ( −x) 1 inπξ 

( −x) 

( ) = ∫ ( ξ) + ... + + + + ... + 

2 2 2 2 2 

} 

S n 


, 

 

ξ =−1 




x 

2 2 2 2 

Dn 

( x) = e + ... + e + + e + ... + 1 e 

2 

= + cos πx 

+ cos 2 πx 

+ ... + cosnπ 

x 

1 

2 


= , n = 0,1, 2,.. 

2sin x 

1 

2 

1 

π 

2 


30


H. Vic Dannon 

FS 

ej 

n 

{ fx ( )} 

= 

{ f ( x ) } + { f ( x ) } + ... + { f ( x ) } 

S S S 

0 1 

n + 1 

n 

ξ= 

1 

1 

= ( ) {( 1) 

1 


dξ. 

n + 1 

2 

 

ξ=−1 

They give rise to the Fejer Sequence 


F ( x ) = + cos[ πξ ( − x )] + ... + cos[ π n ( ξ− 

x )] 

1 n 

1 

n 2 n+ 1 n+ 

1 

By 4.2, for m = 1, 2, .., 

F ( x) cos πx ... cos( m 1) π 

1 m−1 1 

m 1 2 m m 

− 

= + + + − x, 

D ( x ) + D ( x ) + ... D ( x ) 

m 

0 1 m−1 

= , 

2 1 

2 

2 1 

2 

1 sin ( mπx) 

= . 

2m 

sin ( πx) 

Let 1 N = 1 be an infinite Hyper-real. 

2 dx 

The Hyper-real Fejer Kernel is 

F ( x) = + cos πx + ... + cos( N −1) 

πx 

1 N −1 1 

ejer 2 N N 

= ... + δ( x + 4) + δ( x + 2) + δ( x) + δ( x − 2) + δ( x + 4)... 

The Fejer Summation associated with f ( x ) is 

1 − 

N−1 − 

N−1 

{ } 

Ni x i x i x Ni x 

ejer 

fx ( ) = π π π 

c 

Ne ... c 

1e c0 ce 

1 

... ce 

N − 

+ + N − 

+ + + + π 

N 

N N 

F S 

1 

For each x , it may assume finite or infinite hyper-real values. 

31


H. Vic Dannon 

9.2 F S{ x } 

δ ( ξ − ) = δ ( ξ −x) 

ejer Periodic Periodic 

Proof: Let N be an infinite hyper-real. 

( ) ( 

{ ( x )} 1 c e πξ 

− = + .. + 

N − 1c 

e 

ejer Periodic N −N 

Ni x iπξ 

x 

N −1 

− − − − ) 

F S δ ξ 

N −1 iπξ 

( −x) 1 Niπξ 

( −x) 

0 1 

.. 

N 

N N 

+ c + ce + + c e , 

where 

u= 

1 

N−n 1 

π 

• c n 

= ∫ δ Periodic 

( u) 

e du, 

N 

2 

u=−1 

−in u 

• T = 2 is the period, 

• and c = 0. 

For δ ( ) with T = 2 , 

Periodic u 

N−n N n 

u= 

1 

1 

2 

u=−1 

−inπu 

c = ⎡... + δ( u + 2) + δ( u) + δ( u − 2) + ... ⎤ 

∫ ⎣ 

⎦ 

e du 

Therefore, 

u= 

1 

1 −inπu 

1 

δ( ue ) du 

2 2 

u=−1 

= ∫ = . 

( ) ( ) 

{ } 1 −iN πξ−x ( ) ... 

1 −iπξ−x 

δ ξ − = + + 

FejerS 


x e e 

by 5. 

2 2 

= δ ( ξ −x), 


1 1 i πξ ( − x ) 1 ( ) 

... 

iN πξ− 

e 

e 

x 

2 2 2 

+ + + + , 

32


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 

FS 

1 − 

1 

{ ( )} in π 

= x + ... + 

n− 

fx c e c e 

ej n n −n 

n 

−1 

−i πx 

n−1 iπx 

1 

n 

n 

inπx 

+ c 0 

+ ce 1 

+ ... + c e , 

n 

33


H. Vic Dannon 

⎛ ⎞ ⎛ 

= + + 

⎜⎝ ⎠ ⎝ 

ξ= 1 ξ= 

1 

1 inπξ inπx ( ) ... 

n 1 

iπξ iπx 

f ξe dξ − 

e − 

f( ξ) 

e dξ 

− 

e 

2n 

∫ 

2n 

∫ 

ξ=− 1 ⎟ 

⎜ 

ξ=−1 

⎟ 

⎛ ⎞ ⎛ ⎞ ⎛ 

+ + + + 

⎜ ⎜ ⎝ ⎠ ⎝ ⎠ ⎝⎜ 

ξ= 

1 

∫ 

ξ=−1 

ξ= 1 ξ= 1 ξ= 

1 

1 ∫ f () d n 1 f i i x 

() e πξ d e π 

... 

1 

f in in 

() e πξ d e π 

ξ ξ − 

− 

ξ ξ − 

ξ ξ 

2 ∫ 2n 

∫ 

x 

2n 

ξ=− 1 ⎟ ξ=− 1 ⎟ 

ξ=−1 

{ 

1 inπξ 

( −x ) n−1 

iπξ 

( −x 

) 

2n 

2n 

= f ( ξ) e + ... + e 

As n 

→∞, the Fejer Sequence 

1 inπξ 

( −x ) n−1 

iπξ 

( −x 

n( ξ ) ... 

2n 2n 

) 

F − x = e + + e 

⎞ 

⎠ 

1 

+ + 

2 

n−1 −iπξ 

( −x) 1 −inπξ 

( −x) 

e ... e 

2n 

2n 

+ + + d ξ . 

1 n−1 −iπξ 

( −x) 1 −inπξ 

( −x) 

e ... e 

2 2n 

2n 

+ + + + 

becomes the Fejer Kernel, the infinite series 

1 inπξ 

( −x ) n−1 

iπξ 

( −x 

) 

e 

e 

2n 

2n 

... + + ... + 

1 1 −iπξ 

( −x) n 1 in ( x) 

e 

− − πξ− 

e 

2 2n 

2n 

+ + + ... + + ..., 

By 8.2, 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 ). 

34


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 

1 

3. ( 

2 

fx ( + 0) + fx ( − 0) ) replaces f ( x ) at a discontinuity point. 

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) 

35


H. Vic Dannon 

Then, f ( x) = F S{ f( x )} 

ejer 

Proof: Take L = 1, c = 0, and 

1N = 1 an infinite Hyper- real. 

ξ= 

1 

∫ 

ξ=−1 

2 dx 

{ } 

f ( x) = f( ξ) ... + δ( ξ − x + 2) + δ( ξ − x) + δ( ξ −x 

− 2) + ... 

dξ 

δ 


By 8.4, δ ( ξ − x) = F ( ξ −x) 

ξ = 1 

∫ 

ξ=−1 


{ 

1 iN πξ ( −x) N −1 

iπξ 

( −x) 

2N 

2N 

= f ( ξ) e + ... + e 

ej 

( ξ−x 

), where the period of Delta is T=2 

+ 

1 

+ 

2 

N − 1 −iπξ 

( −x) 1 −iNπξ 

( −x 

e ... e ) 

2N 

2N 

} 

+ + + d ξ . 

In [Dan6], we established that the Fourier Transform, 

ξ=∞ 

−i 

2πνξ 

∫ f () ξ e dξ, 

ξ=−∞ 

exists for any Hyper-real function 

fx ( ). That is, the summation 

ξ=∞ 

∑ 

ξ=−∞ 

−i 

2πνξ 

f () ξ e dξ 

exists for any Hyper-real function 

fx ( ) 

. Consequently, the 

summations over intervals exist, and we may write the integral as 

the sum of integrals over intervals 

⎛ ξ= 1 ⎞ ⎛ ξ= 

1 ⎞ ⎛ ξ= 1 

⎞ 1 1 iN πξ iN πx () ... 

N 1 1 iπξ iπx 

f ξe dξ − 

e 

− 

− 

f() ξe dξ = e 

1 

+ + + 

f() 

ξdξ 

N 

2 ∫ 

2N 

2 ∫ 

⎟+ 

2 

⎜⎝ ξ=− 1 ⎠⎟ 

⎝⎜ 

ξ=− 1 ⎠⎟ 

∫ 

⎝⎜ 

ξ=−1 

⎟ 

⎠ 

c 

−N 

c 

−1 0 

c 

36


H. Vic Dannon 

⎛ ⎞ ⎛ 

⎞ 

+ + + 

⎝ ⎠ ⎝ 

⎠ 

 

 

ξ= 1 ξ= 

1 

N −1 1 −iπξ iπx () ... 

1 1 

iNπξ iNπ 

f ξe dξ e − 

f() 

ξe dξ 

x 

e 

2N 

2 ∫ 

2N 

2 ∫ 

⎜ 

ξ=− 1 

⎟ 

⎜ 

ξ=−1 

⎟ 

c 

1 

1 c iN x N 1 i x N 1 i x iN x 

N 

e − π − 

... c 1 e − π c − 

0 ce π 

1 

... c N − 

N − 

N 

N 

e π 

= + + + + + + 1 

N 

ej 

{ f ( x) 

} 

= FS . 

c 

N 

In particular, the Periodic Delta Function violates the Fejer 

Conditions 

The Hyper-real 

δ( x) 

, is not defined in the Calculus of Limits, 


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, 

Periodic( x ) 

satisfies the Fejer Summation Theorem. 

δ 

37


H. Vic Dannon 

References 

[Achieser] Achieser, N. I., Theory of Approximation, Ungar, 1956. 

[Carslaw] Carslaw, H. S., “Introduction to the Theory of Fourier Series and 

integrals” Third Edition, Macmillan, 1930. 

[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. 

4, November 2010; 

[Dan3] Dannon, H. Vic, “Infinitesimal Calculus” in Gauge Institute Journal 

Vol. 7 No. 4, November 2011; 

[Dan4] Dannon, H. Vic, “Riemann’s Zeta Function: the Riemann Hypothesis 

Origin, the Factorization Error, and the Count of the Primes”, in Gauge 

Institute Journal of Math and Physics, Vol. 5, No. 4, November 2009. 

[Dan5] Dannon, H. Vic, “The Delta Function” in Gauge Institute Journal Vol. 

8, No. 1, February, 2012; 

[Dan6] Dannon, H. Vic, “Delta Function, the Fourier Transform, and the 

Fourier Integral Theorem” in Gauge Institute Journal Vol. 8, No. 2, May, 

2012; 

[Dan7] Dannon, H. Vic, “Riemannian Trigonometric Series”, Gauge Institute 

Journal, Volume 7, No. 3, August 2011. 

[Dan8] Dannon, H. Vic, “Periodic Delta Function and Dirichlet Summation of 

Fourier Series” posted to www.gauge-institue.org; 

[Dan9] Dannon, H. Vic, “Lebesgue Integration” Gauge Institute Journal Vol.7 

No. 1, February 2011; 

38


H. Vic Dannon 

[Dan10] Dannon, H. Vic, “Infinite Series with Infinite Hyper-real Sum ” in 

Gauge Institute Journal Vol. 8, No. 3, August, 2012; 

[Davis] Davis, Philip, “Interpolation and Approximation” Blaisdell, 1963. 

[Hardy] Hardy, G. H., Divergent Series, Chelsea 1991. 

[Kreider] Kreider, Kuller, Ostberg, Perkins, “an introduction to real analysis” 

Addison Wesley, 1966 

[Natanson] Natanson, I. P., “Constructive Function Theory” Ungar, 1964. 

[Rogosinski] Rogosinski, Werner, “Fourier Series” Chelsea, 1950. 

[Tolstov] Tolstov, Georgi, “Fourier Series” Prentice-Hall,1962 

[Zygmund] Zygmund, A., “Trigonometric Series”, Second Edition, Cambridge 

University Press, 1968. 

39

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