Hermite Delta

Gauge Institute Journal H. Vic Dannon

Delta Function, and Expansion in Hermite

Functions H. Vic Dannon

[email protected] June, 2012

Abstract Let ( )f x be defined on the real numbers, and let

be the Hermite Polynomials on the real numbers, ( )nH x

0( ) 1H x = , , , ,… 1( ) 2H x x= 22( ) 4 2H x x= − 3

3( ) 8 12H x x x= −

The Hermite Series associated with ( )f x is

0 0 1 1 2 2( ) ( ) ( ) ....a H x a H x a H x+ + + where

21( ) ( )

2 !n nna e f

n

ξξ

ξ

ξ ξπ

=∞−

=−∞

= ∫ H dξ

are the Hermite coefficients.

The Hermite Series Theorem supplies the conditions under which

the Hermite Series associated with ( )f x equals ( )f x .

It is believed to hold in the Calculus of Limits for smooth enough

function. In fact,

1


The Theorem cannot be proved in the Calculus of Limits

under any conditions,

because the summation of the Hermite Series requires integration

of the singular Hermite Kernel.

Plots of partial sums of the Hermite Series speak volumes about

the sensibility of the claims to have infinity bound by epsilon.

In Infinitesimal Calculus, the Hermite Kernel

{ }21 10 0 2 !( ) ( ) ... ( ) ( ) ...

n n nne H H x H H xξ

πξ ξ− + + +

is the Delta Function, . ( )xδ ξ −

( xδ ξ − ) equals its Hermite Series, and the Hermite Series

associated with any hyper-real integrable ( )f x , equals ( )f x

Keywords: Infinitesimal, Infinite-Hyper-Real, Hyper-Real,

infinite Hyper-real, Infinitesimal Calculus, Delta Function,

Hermite Polynomials, Hermite Coefficients, Delta Function,

Hermite Series, Hermite Kernel, Expansion in Hermite Functions,

2000 Mathematics Subject Classification 26E35; 26E30;

26E15; 26E20; 26A06; 26A12; 03E10; 03E55; 03E17; 03H15;

46S20; 97I40; 97I30.

2


Contents

0. The Origin of the Hermite Series Theorem

1. Divergence of the Hermit Kernel in the Calculus of Limits

2. Hyper-real line.

3. Integral of a Hyper-real Function

4. Delta Function

5. Convergent Series

6. Hermite Sequence and ( )xδ ξ −

7. Hermite Kernel and . ( )xδ ξ −

8. Hermite Series of ( )xδ ξ −

9. Hermite Series Theorem

References

3


The Origin of the Hermite Series

Theorem The Hermite Polynomials on ( ,−∞ ∞)

0( ) 1H x = , , , ,…, 1( ) 2H x x= 22( ) 4 2H x x= − 3

3( ) 8 12H x x x= −

are orthogonal so that

2( ) ( ) 2 !

xx n

m n mx

e H x H x dx n πδ=∞

−

=−∞

=∫ n .

The Hermite Polynomials can be generated by expanding 22 2 2 21 1

2! 3!1 [2 ] [2 ] [2 ] ...xe x x xα α α α α α α α− = + − + − + − +2 3

2 2 2 3 412!

1 2 [4 4 ]x x xα α α α α= + − + − + +

3 3 2 4 5 613!

[8 12 6 ] ...x x xα α α α+ − + − +

the coefficient of 010!α is

0( ) 1H x = ,


1( ) 2H x x= ,


22( ) 4 2H x x= − ,


33( ) 8 12H x x x= − ,

…………………………

4


0.1 Schrodinger Equation for atomic size particle in

linear harmonic motion

An atomic size particle with mass m , oscillates along a segment

of wire [ , at frequency ν , under the force . ,A A− ]

ν

t

x

kx−

The particle’s position is

( ) cosx t A tω= , . 2ω π=Thus,

sinx Aω ω= −

2 2cosx A tω ω ω= − = −

The force equation is 2( )kx mx m xω− = = − .

Hence, the force constant is 2k mω= ,

and the potential energy of the particle is

2 21 12 2

V kx mω= = 2x .

De Broglie associated with the moving particle a wave of length

hmv

λ = ,

where v is the velocity of the particle, and h is Planck’s constant.

The wave’s frequency is

2

hmv

v v mvh

νλ

= = = .

The wave’s angular frequency is

5


2

2 2mvh

ω πν π= =

In terms of the De Broglie wave, the particle’s energy is a multiple

of Planck’s radiation energy,

E hε ν ε ω= = , 2hπ

= , is the multiplier. ε

The kinetic energy of the particle is

212mv E V= − .

Hence, 2 ( )mv m E V= − ,

2 ( )

h

m E Vλ =

−,

12

2 ( )v

m E Vπω

ωλ ν= =

−.

2 2 2

1 2 (m E V

v ω

−=

)

Schrodinger postulated a complex valued potential

( , ) ( ) i tx t x e ωψΨ =

that satisfies the wave equation

2 22

1( , ) ( , )x tx t x t

v∂ Ψ = ∂ Ψ .

Then,

2 22

10 ( , ) (x tx t x t

v= ∂ Ψ − ∂ Ψ , )

6


22 2

2 ( )"( ) ( )( )i t i tm E Vx e x eω ωψ ψ

ω

−= − −ω .

The Schrodinger equation for the linear harmonic oscillator is

2

2"( ) ( ) ( ) 0

mx E V xψ ψ+ − = .

Substituting E , and V ,

2 2122

2" ( )m

m xψ ε ω ω ψ+ − 0=

Multiplying by mω

,

2

2"( ) (2 ) ( ) 0m

x x xm

ξ

ωψ ε ψ

ω+ − = .

The change of variable m xωξ = , gives

'( ) md d ddx d dx

ωψ ψ ξψ ξ

ξ= = ,

{ }2

2'( ) ''( )m md d d

d dxdxω ωψ ξ

ψ ξ ψ ξξ

= = ,

and the equation becomes

2"( ) (2 ) ( ) 0xψ ξ ε ξ ψ+ − = .

0.2 Hermite Differential Equation

The Schrodinger equation

7


2"( ) ( ) 2 ( )x xψ ξ ξ ψ εψ− = − can be factored

( )( ) ( ) 2D D xξ ξξ ξ ψ εψ− + = − ( )x

0

D xξ ξ ψ− =

.

To solve the homogeneous equation

( )( ) ( )D D xξ ξξ ξ ψ− + = ,

we solve

( ) ( ) 0 ⇒ 'ψξ

ψ= ⇒ 21

2log cψ ξ= + ⇒

212

1 Ce ξψ = .

As , , and is discarded. ξ → ∞ 1ψ → ∞

( ) ( ) 0D xξ ξ ψ+ = ⇒ 'ψξ

ψ= − ⇒ 21

2log cψ ξ= − + ⇒

212

2 Ceξψ −= .

Now, substituting 21

2( ) ( )H eξψ ξ ξ −=

in , we have 2"( ) (2 ) ( ) 0ψ ξ ε ξ ψ ξ+ − =

( )2 21 12 22 20 ( ) (2 ) ( )D H e H eξ ξ

ξ ξ ε ξ ξ− −= + −

( )2 21 12 2 2'( ) ( ) (2 ) ( )D H e H e H eξ ξ

ξ ξ ξ ξ ε ξ ξ− −= − + −21

2ξ−

2 2 21 1 1

2 2 2 2''( ) 2 '( ) ( ) ( )H e H e H e H eξ ξ ξξ ξ ξ ξ ξ ξ− − −= − − +21

2ξ− +

2122(2 ) ( )H e ξε ξ ξ −+ −

21

2''( ) 2 '( ) (2 1) ( )H H H e ξξ ξ ξ ε ξ −⎡ ⎤= − + −⎣ ⎦ .

The Schrodinger equation becomes Hermit Differential Equation

''( ) 2 '( ) (2 1) ( ) 0H H Hξ ξ ξ ε ξ− + − = ,

8


Substituting in it

2 10 1 2 1 2( ) ... ...l l l

l l lH c c c c c cξ ξ ξ ξ ξ ξ+ ++ += + + + + + + +2

0l =

0

,

we have

2 1

2 1

2

0 0 0

( 1)

2 (2 1)

l ll l

l ll l

l l ll l

l l ll l l

l lc lc

D c D c cξ ξ

ξ ξ

ξ ξ ξ ε ξ

=∞ =∞− −

= =

=∞ =∞ =∞

= = =

−

− + −

∑ ∑

∑ ∑ ∑ ,

20

{( 1)( 2) 2 (2 1) } 0l

ll l l

l

l l c lc cε ξ=∞

+=

+ + − + − =∑ ,

2( 1)( 2) [2 1 2 ]l ll l c l cε++ + − + − =

22 1 2

( 1)( 2)l ll

c cl l

ε+

+ −=

+ +

The solution is

2 31 2 3 20 1 0 11 2 2 3

( )H c c c cε εξ ξ ξ− −⋅ ⋅

= + + + +ξ

(1 2 )(5 2 ) (3 2 )(7 2 )4 50 11 2 3 4 2 3 4 5

...c cε ε ε εξ ξ− − − −⋅ ⋅ ⋅ ⋅ ⋅ ⋅

+ + +

(1 2 )(5 2 )2 41 20 1 2 1 2 3 4{1 ...}c ε εε ξ ξ− −−

⋅ ⋅ ⋅ ⋅= + + + +

(3 2 )(7 2 )2 43 21 2 3 2 3 4 5

{1 ...}c ε εεξ ξ ξ− −−⋅ ⋅ ⋅ ⋅

+ + + + .

To keep the solution from diverging at ξ , → ∞

for , the series terms vanish for 2n = k 0c

2 1,5,9,13,...4 1,...kε = + ,

and we obtain the Hermite Polynomials. 2 ( )kH ξ

9


for , the series terms vanish for 2n k= + 1 1c

2 3,7,11,..., 4 3,...kε = +

and we obtain the Hermite Polynomials. 2 1( )kH ξ+

A solution for is the infinite linear combination ( )ψ ξ

2 21 1 12 2 2

0 0 1 1 2 2( ) ( ) ( ) ....H e H e H eξ ξ ξα ξ α ξ α ξ− − −+ +2

+

0.3 The Hermite Series Associated with ( )f x

Let ( )f x be defined on ( , , and let be the Hermite

Polynomials )

)

−∞ ∞ ( )nH x

0( ) 1H x = , , , ,… 1( ) 2H x x= 22( ) 4 2H x x= − 3

3( ) 8 12H x x x= −

The Polynomials are orthogonal on ( . That is, ,−∞ ∞

2( ) ( ) 2 !

xx n

m n mx

e H x H x dx n πδ=∞

−

=−∞

=∫ n

We define the Orthonormalized Hermite Functions

212

12

1( ) ( )

(2 ! )

xn n

nx e

nϕ

π

−= H x

If ( )f x can be expanded in the , ( )n xϕ

0 0 1 1 2 2( ) ( ) ( ) ( ) ...f x x x xα ϕ α ϕ α ϕ= + + + ,

Then,

10


0 0 1 1 2 2( ) ( ) { ( ) ( ) ( ) ...} ( )x x

n nx x

f x x dx x x x x dxϕ α ϕ α ϕ α ϕ=∞ =∞

=−∞ =−∞

= + + +∫ ∫ ϕ

n +

0 1 2

0 0 1 1 2 2( ) ( ) ( ) ( ) ( ) ( ) ..

n n n

x x x

n nx x x

x x dx x x dx x x dx

δ δ δ

α ϕ ϕ α ϕ ϕ α ϕ ϕ=∞ =∞ =∞

=−∞ =−∞ =−∞

= + +∫ ∫ ∫

. nα=

Thus, the Hermite coefficients with respect to the are ( )n xϕ

( ) ( )n nf dξ

ξ

α ξ ϕ=∞

=−∞

= ∫ ξ ξ .

The Orthonormal Hermite Series associated with ( )f x is

0 0 1 1 2 2( ) ( ) ( ) ...x x xα ϕ α ϕ α ϕ+ + + .

11


1.

Divergence of the Hermit Kernel

in the Calculus of Limits

Calculus of Limits Conditions for the Hermite Series to equal its

function reflect the belief that a smooth enough function equals its

Hermite Series.

In fact, in the Calculus of Limits, no smoothness of the function

guarantees even the convergence of the Hermite Series.

1.1 The Hermite Kernel is either singular or zero

In the Calculus of Limits, the Hermite Series is the limit of the

sequence of Partial Sums

{ } 0 0( ) ( ) ... ( )ermite n n nf x xα ϕ α ϕ= + +H S x

0 0( ) ( ) ( ) .. ( ) ( ) ( )n nf d x f dξ ξ

ξ ξ

ξ ϕ ξ ξ ϕ ξ ϕ ξ ξ ϕ=∞ =∞

=−∞ =−∞

⎛ ⎞ ⎛⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟= + +⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎝ ⎠ ⎝∫ ∫ x

⎞

⎠

{ }0 0( ) ( ) ( ) ... ( ) ( )n nf x xξ

ξ

ξ ϕ ξ ϕ ϕ ξ ϕ ξ=∞

=−∞

= + +∫ d .

As n , the orthonormal Hermite Sequence → ∞

0 0( ) ( ) ... ( ) ( )n nx xϕ ξ ϕ ϕ ξ ϕ+ +

12


becomes the orthonormal Hermite Kernel,

0 0( ) ( ) ... ( ) ( ) ...n nx xϕ ξ ϕ ϕ ξ ϕ+ + + ,

To see that it is singular at , we apply the Christoffel

Summation Formula, [Sansone, p.371],

xξ =

1 10 0

( ) ( ) ( ) ( )1( ) ( ) ... ( ) ( )

2n n n n

n n

x xnx x

x

ϕ ξ ϕ ϕ ξ ϕϕ ξ ϕ ϕ ξ ϕ

ξ+ +−+

+ + =−

.

For , xξ →

2 20 0 0( ) ( ) ... ( ) ( ) ( ) ... ( )n n nx x xϕ ξ ϕ ϕ ξ ϕ ϕ ϕ+ + → + + x ,

and

1 1( ) ( ) ( ) ( )1 12 2

n n n nx xn nx

ϕ ξ ϕ ϕ ξ ϕξ

+ +−+ +→

−00

.

Applying Bernoulli’s rule to the indeterminate limit,

1 11 1 ( ) ( ) ( ) ( )( ) ( ) ( ) ( )lim lim

( )n n n nn n n n

x x

D x D xx x

x Dξ ξ

ξ ξ ξ


ξ ξ+ ++ +

→ →

−−=

− − x

1 1lim[ '( ) ( ) '( ) ( )]n n n nxx x

ξϕ ξ ϕ ϕ ξ ϕ+ +→

= −

1 1'( ) ( ) '( ) ( )n n n nx x x xϕ ϕ ϕ ϕ+ += −

Therefore,

2 20 1

1( ) ... ( ) [ '( ) ( ) '( ) ( )]

2n n n nn

x x x x xϕ ϕ ϕ ϕ ϕ ϕ+ ++

+ + = − 1n x .

Since , and solve the differential equation, [Szego,

p.105, #5.5.2],

( )n xϕ 1( )n xϕ +

13


2

( ) ( )

1 ''( ) 0 '( ) (2 1 ) ( ) 0a x b x

z x z x n x z x⋅ + ⋅ + + − = ,

we have, ( )

( )1 1'( ) ( ) '( ) ( ) ( )

b xa xdx

n n n nx x x x const eϕ ϕ ϕ ϕ−

+ +∫− =

0( )

dxconst e

⋅∫=

, const=

for any . x−∞ < < ∞

Hence,

2 20

1( ) ... ( )

2nn

x xϕ ϕ+

+ + = const

and the Hermite Kernel diverges to ∞ at any . xξ =

Therefore, while the partial sums of the Hermite Series exist,

their limit does not. That is, due to the singularity at , the

Hermite Series does not converge in the Calculus of Limits.

xξ =

Avoiding the singularity at , by using the Cauchy Principal

Value of the integral does not recover the Theorem, because at any

, the Hermite Kernel vanishes, and the integral will be

identically zero, for any function

xξ =

xξ ≠

( )f x .

To see that the kernel vanishes for any , we apply the

Christoffel Summation Formula, with .

xξ ≠

xξ ≠

1 110 0 2

( ) ( ) ( ) ( )( ) ( ) ... ( ) ( ) n n n nn

n n

x xx x

x


ξ+ ++ −

+ + =−

.

14


We have

21

212

1( ) ( )

(2 ! )

xn n

nx e

nϕ

π

−= H x

21 22

12

1( )

(2 ! )

x xn

ne e H x

n π

−=

By [Szego, p. 105, #5.5.3], 2 2

( ) ( 1)x nn xe H x D e− −= − n x .

Thus, 21 2

212

( 1)( ) { }

(2 ! )

nx n x

n xn

x e Dn

ϕπ

−−= e ,

21 2

212

11

11

( 1)( ) { }

(2 ( 1)! )

nx n x

n xn

x en

ϕπ

++ −

++

−=

+D e

and

112

( ) ( ) ( ) ( )n n n nn x x

x

ϕ ξ ϕ ϕ ξ ϕξ

++ −=

−1+

{ }2 2 22 1

1 112

0, 0,

1 ( 1){ } { } { } { }

2 2 !( 1)!

nn n x n nn

x xn

nn

D e D e D e D ex n n

ξ ξξ ξξ π

++ − − − + −+

→ →∞→ →∞

−= −

− +

2x

. 0, as n→ → ∞

That is, the Hermite Kernel vanishes for any . xξ ≠

Plots of the Hermite Sequence confirm that

In the Calculus of Limits,

the Hermite Kernel is either singular or zero

15


1.2 Plots of 21 1

0 0 2 !{ ( ) ( ) ... ( ) ( )}


πξ ξ− + +

In Maple,

2

2231 1

2 !0

( * ( ,.5) * ( , ), 23..23)ix

ii

plot e HermiteH i HermiteH i x xπ

−

=

= −∑

In Maple, 2223

1 12 !

0

( * ( , 1) * ( , ), 2i

xi

i


−

=

− =∑ 3..23)−

16


2xe− that suppresses oscillations away from the origin, enhances

them at the origin. Thus, a singularity away from the origin

needs more terms

In Maple, 2223

1 12 !

0

( * ( ,2) * ( , ), 23..23)i

xi

i


−

=

= −∑

The plots confirm that the Hermite Series Theorem cannot be

proved 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

17


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

1( )

2i xx e

ωω

ω

δ ωπ

=∞

=−∞

= ∫ d

is zero for any , 0x ≠

it spikes at , so that its Infinitesimal Calculus

integral is ,

0x =

( ) 1x

x

x dxδ=∞

=−∞

=∫

and 1(0)

dxδ = < ∞

}

.

Here, we show that in Infinitesimal calculus, the Hermite Kernel

is a hyper-real Delta Function.

And the Hermite Series { ( )egendre f xL S associated with a Hyper-

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

18


2.

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

rational numbers, so that . 1 2 3( , , ,...)r r r nr α→

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

In [Dan2] we established that,

1. Any totally ordered set of positive, monotonically decreasing

to zero sequences constitutes a family of

infinitesimal hyper-reals.

1 2 3( , , ,...)ι ι ι

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

strictly greater than zero.

3. Their reciprocals (1 2 3

1 1 1, , ,...ι ι ι ) 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

19


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 non-

constant 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.

∞

20


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 hyper-

reals.

16. No neighbourhood of a hyper-real is homeomorphic to

an ball. Therefore, the hyper-real line is not a manifold. n

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.

21


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 2

[ ,dx dxx x− +2], 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 , and end at x b , a= =

[ , ]

( )x a b

f x dx∈∑ .

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 ,

to x , and denote it by

x a=

b=

22


( )x b

x a

f x dx=

=∫ .

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

If the hyper-real is finite,

( ) real part of the hyper-realx b

x a

f x dx=

=

=∫ .

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, , and

we have

2CardCard =

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

In particular, we demonstrated that the real numbers may be

well-ordered.

Consequently, there are countably many real numbers in the

interval [ , , and the Integration Sum has countably many terms. ]a b

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

23


by its lowest value on each interval 2 2

[ ,dx dxx x− + ]

3.2 2 2[ , ]

inf ( )dx dxx t xx a b

f t dx− ≤ ≤ +∈

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜⎝ ⎠∑

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

by its largest value on each interval 2 2

[ ,dx dxx x− + ]

3.3 2 2[ , ]

sup ( )dx dxx t xx a b

f t dx− ≤ ≤ +∈

⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∑

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.

24


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 is the sequence 0 0,0, 0,... . The infinite hyper-

real 1dx

depends on our choice of dx .

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

spanned by the sequences 1n

,2

1

n,

3

1

n,… 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, 1dx

will mean the sequence n .

Alternatively, we may choose the family spanned by the

sequences 1

2n,

1

3n,

1

4n,… Then, 1

dx will mean the

25


sequence 2n . Once we determined the basic infinitesimal

, we will use it in the Infinite Riemann Sum that defines

an Integral in Infinitesimal Calculus.

dx

3. The Delta Function is strictly smaller than ∞

4. We define, 2 2,

1( ) ( )dx dxx x

dxδ χ⎡ ⎤−⎢ ⎥⎣ ⎦

≡ ,

where 2 2

2 2,

1, ,( )

0, otherwisedx dx

dx dxxxχ⎡ ⎤−⎢ ⎥⎣ ⎦

⎧ ⎡ ⎤⎪ ∈ −⎢ ⎥⎪ ⎣ ⎦= ⎨⎪⎪⎩.

5. Hence,

for , 0x < ( ) 0xδ =

at 2dx

x = − , jumps from to ( )xδ 01dx

,

for 2 2

,dx dxx ⎡ ⎤∈ −⎢ ⎥⎣ ⎦ , 1

( )xdx

δ = .

at , 0x =1

(0)dx

δ =

at 2dx

x = , drops from ( )xδ1dx

to . 0

for , . 0x > ( ) 0xδ =

( ) 0x xδ =

6. If 1n

dx = , 1 1 1 1 1 12 2 4 4 6 6

[ , ] [ , ] [ , ]( ) ( ),2 ( ), 3 ( )...x x xδ χ χ χ− − −= x

7. If 2n

dx = , 2 2 2

1 2 3( ) , , ,...

2 cosh 2cosh 2 2cosh 3x

x x xδ =

26


8. If 1n

dx = , 2 3[0, ) [0, ) [0, )( ) ,2 , 3 ,...x x xx e e eδ χ χ χ− − −

∞ ∞ ∞=

9. . ( ) 1x

x

x dxδ=∞

=−∞

=∫

10. ( )1( )

2

kik x

k

x e ξδ ξπ

=∞− −

=−∞

− = ∫ dk

27


5.

Convergent Series In [Dan8], we defined convergence of infinite series in

Infinitesimal Calculus

5.1 Sequence Convergence to a finite hyper-real a

na → a iff infinitesimalna a− = .

5.2 Sequence Convergence to an infinite hyper-real A

iff na → A na represents the infinite hyper-real A .

5.3 Series Convergence to a finite hyper-real s

1 2 ...a a+ + → s iff 1 ... infinitesimalna a s+ + − = .

5.4 Series Convergence to an Infinite Hyper-real S

iff 1 2 ...a a+ + → S

1 ... na + + a represents the infinite hyper-real S .

28


6.

Hermite Sequence and ( )xδ ξ −

6.1 Hermite Sequence Definition

If ( )f x can be expanded in the , ( )nH x

0 0 1 1 2 2( ) ( ) ( ) ( ) ...f x a H x a H x a H x= + + + ,

Then,

2( ) ( )

xx

nx

e f x H x dx=∞

−

=−∞

=∫

2

0 0 1 1 2 2( ) { ( ) ( ) ( ) ...} ( )x

xn

x

f x e a H x a H x a H x H x dx=∞

−

=∞

= + + +∫

2 2

0 10 1

0 0 1 1

2 0! 2 1!

( ) ( ) ( ) ( ) ..

n n

x xx x

n nx x

a e H x H x dx a e H x H x dx

πδ πδ

=∞ =∞− −

=−∞ =−∞

= +∫ ∫ +

2 !nnn aπ= .

The Hermite Series partial sums

{ } 0 0( ) ( ) ... ( )ermite n n nf x a H x a H= + +H S x

{ }21 10 0 2 !

( ) ( ) ( ) ... ( ) ( )n n nn

f e H H x H H x dξ

ξπ

ξ

ξ ξ ξ=∞

−

=−∞

= + +∫ ξ .

give rise to the Hermite Sequence

29


{ }21 10 0 2 !

( , ) ( ) ( ) ... ( ) ( )nn nn

H x e H H x H H xξπ

ξ ξ−= + + nξ .

6.2 Hermite Sequence is a Delta Sequence

For each 0,1,2,3,...n =

{ }21 10 0 2 !

( , ) ( ) ( ) ... ( ) ( )nn nn

H x e H H x H H xξπ

ξ ξ−= + + nξ ,

1. has the sifting property

{ }21 10 0 2 !( ) ( ) ... ( ) ( ) 1

n n nne H H x H H x d

ξξ

πξ

ξ ξ=∞

−

=−∞

+ + =∫ ξ

2. is a continuous function

3. peaks for each to xξ → 1const n⋅ +

Proof of (1)

{ }21 10 0 2 !( ) ( ) ... ( ) ( )


ξξ

πξ

ξ ξ=∞

−

=−∞

+ + =∫ ξ

2 21 1 10 0 2 !1 1

1

( ) ( ) ... ( ) ( )nn nn

H x e H d H x e H dξ ξ

ξ ξπ πξ ξ

ξ ξ ξ ξ=∞ =∞

− −

=−∞ =−∞

= + +∫ ∫

By [Spanier, p.222, #24:10:5], for , 1,2,...,k n=

2

122

0, 1, 3,5,...( )

!( 1) , 2, 4,6,... nk

ke H b d

n b k

ξξ

ξ

ξ ξπ

=∞−

=−∞

⎧ =⎪⎪⎪= ⎨⎪ − =⎪⎪⎩∫

Therefore, for , 1,2,...,k n=

30


2( ) 0ke H d

ξξ

ξ

ξ ξ=∞

−

=−∞

=∫ .

Hence,

{ }21 10 0 2 !( ) ( ) ... ( ) ( ) 1


ξξ

πξ

ξ ξ=∞

−

=−∞

+ + =∫ ξ .

Proof of (3)

By the Christoffel Summation Formula, [Sansone, p.371],

1 10 0

( ) ( ) ( ) ( )1( ) ( ) ... ( ) ( )

2n n n n

n n

x xnx x

x


ξ+ +−+

+ + =−

,

where 21

212

1( ) ( )

(2 ! )

xn n

nx e

nϕ

π

−= H x .

For , xξ →

1 1( ) ( ) ( ) ( )1 12 2

n n n nx xn nx

ϕ ξ ϕ ϕ ξ ϕξ

+ +−+ +→

−00

.

Applying Bernoulli’s rule to the indeterminate limit,

1 11 1 ( ) ( ) ( ) ( )( ) ( ) ( ) ( )lim lim

( )n n n nn n n n

x x

D x D xx x

x Dξ ξ

ξ ξ ξ

ϕ ξ ϕ ϕ ξ ϕϕ ξ ϕ ϕ ξ ϕξ ξ

+ ++ +

→ →

−−=

− − x

1 1lim[ '( ) ( ) '( ) ( )]n n n nxx x

ξϕ ξ ϕ ϕ ξ ϕ+ +→

= −

1 1'( ) ( ) '( ) ( )n n n nx x x xϕ ϕ ϕ ϕ+ += −

Therefore,

31


2 20 1

1( ) ... ( ) [ '( ) ( ) '( ) ( )]

2n n n nn

x x x x xϕ ϕ ϕ ϕ ϕ ϕ+ ++

+ + = − 1n x .

Since , and solve the differential equation, [Szego,

p.105, #5.5.2],

( )n xϕ 1( )n xϕ +

2

( ) ( )

1 ''( ) 0 '( ) (2 1 ) ( ) 0a x b x

z x z x n x z x⋅ + ⋅ + + − = ,

we have, ( )

( )1 1'( ) ( ) '( ) ( ) ( )

b xa xdx

n n n nx x x x const eϕ ϕ ϕ ϕ−

+ +∫− =

0( )

dxconst e

⋅∫=

, const=

for any . x−∞ < < ∞

Hence,

2 20( ) ... ( ) 1nx x nϕ ϕ+ + = + const

Therefore, substituting 21 1

2 2( ) (2 ! ) ( )xn

n nH x n e xπ ϕ= ,

21 12 2( ) (2 ! ) ( )n

n nH n e ξξ π= ϕ ξ

21 10 0 2 !

{ ( ) ( ) ... ( ) ( )}n n nn

e H H x H H xξπ

ξ ξ− + + =

{ }2 21

2( )

0 0( ) ( ) ... ( ) ( )xn ne xξ ϕ ξ ϕ ϕ ξ ϕ− −= + + x

{ }2 20( ) ... ( )nxx x

ξϕ ϕ

→→ + + 1n const= + .

32


7.

Hermite Kernel and ( )xδ ξ −

7.1 Hermite Kernel in the Calculus of Limits The Hermite Series partial sums

{ } { }21 10 0 2 !

Hermite Sequence

( ) ( ) ( ) ( ) ... ( ) ( )nermite n n nn

f x f e H H x H H xξ

ξπ

ξ

ξ ξ ξ=∞

−

=−∞

= + +∫H S dξ .

give rise to the Hermite Sequence.

The limit of the Hermite Sequence is an infinite series called the

Hermite Kernel

{ }21 10 0 2 !

( ) ( ) ( ) ... ( ) ( ) ...nermite n nn

x e H H x H H xξπ

ξ ξ ξ−− = + + +H

7.2 In the Calculus of Limits, the Hermite Kernel does not have

the sifting property

Proof: for , xξ →

{ }21 10 0 2 !( ) ( ) ... ( ) ( ) .. lim 1

n n nn ne H H x H H x n conξ

πξ ξ−

→∞+ + + = + st

n→∞→ ∞

That is, for , xξ →

{ }21 10 0 2 !( ) ( ) ... ( ) ( ) ...


πξ ξ− + + + is singular.

33


7.3 Hyper-real Hermite Kernel in Infinitesimal Calculus

{ }21 10 0 2 !

( ) ( ) ( ) ... ( ) ( ) ...nermite n nn

x e H H x H H xξπ

ξ ξ ξ−− = + + +H

,

0 ,

n x

x

ξ

ξ

⎧⎪ =⎪= ⎨⎪ ≠⎪⎩

. ( xδ ξ= − )

Proof:

{ }21 10 0 2 !

( ) ( ) ( ) ... ( ) ( ) ...nermite n nn

x e H H x H H xξπ

ξ ξ ξ−− = + + +H

,

0 ,

n x

x

ξ

ξ

⎧⎪ =⎪= ⎨⎪ ≠⎪⎩.

Denoting by 1dx

the infinite hyper-real n ,

1

0,

,dx

x

x

ξ

ξ

⎧ ≠⎪⎪= ⎨⎪ =⎪⎩

. ( )xδ ξ= −

34


8.

Hermite Series and ( )xδ ξ −

8.1 Hermite Series of a Hyper-real Function

Let ( )f x be a hyper-real function integrable on ( . ,−∞ ∞)

Then, for each , the integrals 0,1,2, 3,...n =

212 !

( ) ( )n

xx

n nnx

a e f xπ

=∞−

=−∞

= ∫ H x dx .

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

Hermite Coefficients of

na

( )f x .

The Hermite Series associated with ( )f x is

{ } 0 0 1 1 2 2( ) ( ) ( ) ( ) ...ermite f x a H x a H x a H x= + +H S +

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

8.2 { }( ) ( )ermite x xδ ξ δ ξ− = −H S

Proof:

{ } 0 0 1 1 2 2( ) ( ) ( ) ( ) ...ermite x a H x a H x a H xδ ξ − = + + +H S

where

212 !

( ) ( )n

xx

n nnx

a e xπ

δ ξ=∞

−

=−∞

= −∫ H x dx

35


21

2 !( )

n nne Hξ

πξ−= .

Therefore,

{ } { }21 10 0 2 !

( ) ( ) ( ) ... ( ) ( ) ...nermite n nn

x e H H x H H xξπ

δ ξ ξ ξ−− = + + +H S

, by 7.3, (ermite xξ= H )−

) , by 7.3. ( xδ ξ= −

36


9.

Hermite Series Theorem

The Hermite Series Theorem for a hyper-real function, ( )f x , is the

Fundamental Theorem of Hermite Series.

It supplies the conditions under which the Hermite Series

associated with ( )f x equals ( )f x .

It is believed to hold in the Calculus of Limits under the Picone

Conditions, or under the Hobson Conditions [Sansone]. In fact,

The Theorem cannot be proved in the Calculus of Limits

under any conditions,

because the summation of the Hermite Series requires integration

of the singular Hermite Kernel.

9.1 Hermite Series Theorem cannot be proved in the

Calculus of Limits

Proof: Let ( )f x be integrable on ( , . )−∞ ∞

In the Calculus of Limits, the Hermite Series is the limit of the

sequence of Partial Sums

{ } 0 0( ) ( ) ... ( )ermite n n nf x a H x a H= + +H S x

37


21

0 0( ) ( ) ( ) ...f e H d H xξ

ξπξ

ξ ξ ξ=∞

−

=−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟= +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫

21 1

2 !... ( ) ( ) ( )

n n nnf e H d H x

ξξ

πξ

ξ ξ ξ=∞

−

=−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟+ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫

{ }21 10 0 2 !

( ) ( ) ( ) ... ( ) ( )n n nn

f e H H x H H x dξ

ξπ

ξ

ξ ξ ξ=∞

−

=−∞

= + +∫ ξ .

As n , the Hermite Sequence → ∞

{ }21 10 0 2 !( ) ( ) ... ( ) ( )


πξ ξ− + +

becomes the Hermite Kernel,

{ }21 10 0 2 !( ) ( ) ... ( ) ( ) ...


πξ ξ− + + + ,

By 7.2, the Hermite Kernel diverges to infinity at any . xξ =

Therefore, while the partial sums of the Hermite Series exist,

their limit does not. Conditions by Uspensky [Sansone] failed to

comprehend the sifting through the values of ( )f ξ by the Hermite

Kernel, and the picking of ( )f ξ at . xξ =

Avoiding the singularity at , by using the Cauchy Principal

Value of the integral does not recover the Theorem, because for

any , the Hermite Kernel vanishes, and the integral is

identically zero, for any function

xξ =

xξ ≠

( )f x .

38


Thus, the Hermite Series Theorem cannot be proved in the

Calculus of Limits.

9.2 Calculus of Limits Conditions are irrelevant to Hermite

Series Theorem

Proof: The Uspensky Conditions [Sansone, p.371] are

1. ( )f x integrable in any bounded interval

2. ( )f x integrable in ( ,−∞ ∞)

It is clear from 9.1 that these conditions on ( )f x do not resolve the

singularity of the Hermite kernel, and are not sufficient for the

Hermite Series Theorem.

In Infinitesimal Calculus, by 7.3, the Hermite Kernel is the Delta

Function, and by 8.2, it equals its Hermite Series.

Then, the Hermite Series Theorem holds for any Hyper-Real

Function:

8.3 Hermite Series Theorem for Hyper-real ( )f x

If ( )f x is hyper-real function integrable on( ,−∞ ∞)

Then, { }( ) ( )ermitef x f= H S x

Proof:

39


( ) ( ) ( )f x f x dξ

ξ

ξ δ ξ ξ=∞

=−∞

= −∫

Substituting from 7.3,

{ }21 10 0 2 !

( ) ( ) ( ) ... ( ) ( ) ...n n nn

x e H H x H H xξπ

δ ξ ξ ξ−− = + + + ,

{ }21 10 0 2 !

( ) ( ) ( ) ( ) ... ( ) ( ) ...n n nn

f x f e H H x H H xξ

ξπ

ξ

ξ ξ ξ=∞

−

=−∞

= + +∫ dξ+

This Hyper-real Integral is the summation,

{ }21 10 0 2 !

( ) ( ) ( ) ... ( ) ( ) ...n n nn

f e H H x H H x dξ

ξπ

ξ

ξ ξ ξ=∞

−

=−∞

+ + +∑ ξ

which amounts to the hyper-real function ( )f x ,and is well-defined.

Hence, the summation of each term in the integrand exists, and

we may write the integral as the sum

2

0

0 01

( ) ( ) ( ) ...

a

f e H d H xξ

ξ

ξ

ξ ξ ξπ

=∞−

=−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟= +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫

21

... ( ) ( ) ( ) ...2 !

n

n nn

a

f e H d H xn

ξξ

ξ

ξ ξ ξπ

=∞−

=−∞

⎛ ⎞⎟⎜ ⎟⎜ ⎟+ +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠∫

0 0 1 1 2 2( ) ( ) ( ) ...a H x a H x a H x= + + +

{ }( )ermite f x= H S .

40


In particular, the Delta Function violates Uspensky’s Conditions

The Hyper-real , is not defined in the Calculus of Limits,

and is not integrable in any interval.

( )xδ

But by 8.2, satisfies the Hermite Series Theorem. ( xδ ξ − )

41


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43

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Hermite Delta