Delta Function and Expansion in Hermite Functions

Gauge Institute Journal
H. Vic Dannon
Delta Function, and
Expansion in Hermite
Functions
H. Vic Dannon
vic0@comcast.net
June, 2012
Abstract Let f ( x ) be defined on the real numbers, and let
Hn ( x )
be the Hermite Polynomials on the real numbers,
H 0 ( x ) = 1,
H 1 ( x) = 2x,
H 2
2 ( x) = 4x
−2,
H 3
3 ( x) = 8x −12x
,…
The Hermite Series associated with f ( x ) is
where
aH( x) + aH( x) + aH( x) + ....
0 0 1 1 2 2
ξ=∞
a 1
2
n
() ()
n
n
2 !
e −ξ
= ∫
n
f ξ H ξ dξ
π
ξ=−∞
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,
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Gauge Institute Journal
H. Vic Dannon
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
2
{ ( ξ ) ( ) ... ( ) ( ) ...
n n
ξ
n }
1 ξ
1
π 0 0 2 n !
e − H H x + + H H x +
is the Delta Function, δξ− ( x)
.
δξ− ( x) 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.
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Gauge Institute Journal
H. Vic Dannon
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 δξ− ( x)
7. Hermite Kernel and δξ− ( x)
.
8. Hermite Series of δξ− ( x)
9. Hermite Series Theorem
References
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H. Vic Dannon
The Origin of the Hermite Series
Theorem
The Hermite Polynomials on ( −∞,
∞)
H 0 ( x ) = 1,
H 1 ( x) = 2x,
H 2
2 ( x) = 4x
−2,
H 3
3 ( x) = 8x −12x
,…,
are orthogonal so that
x =∞
2
−x
n
∫ e Hm( x) Hn( x) dx = 2 n!
πδm
n .
x =−∞
The Hermite Polynomials can be generated by expanding
2
2xα−α
2 1 2 2 1
2! 3!
e = 1 + [2 xα− α ] + [2 xα− α ] + [2 xα− α ] + ...
2 1 2 2 3 4
2!
= 1+ 2 xα− α + [4x α − 4 xα + α ] +
1 0
the coefficient of α is 0! H 0 ( x ) = 1,
1 1
the coefficient of α is 1! H 1 ( x) = 2x,
1 2
the coefficient of α is 2! H 2
2 ( x) = 4x
− 2,
3
the coefficient of
1
α is 3! H 3
3 ( x) = 8x − 12x
,
1
3!
2 3
3 3 2 4 5 6
+ [8 x α − 12 x α + 6 xα
− α ] + ...
…………………………
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H. Vic Dannon
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 [ − AA , ], at frequency ν , under the force −kx .
The particle’s position is
Thus,
xt () = Acosωt, ω = 2πν.
x
=−ωAsin
ωt
The force equation is
x
=− ω Acos
ωt
=−ω
2 2
− kx = mx = m( −ω x)
.
2
x
Hence, the force constant is
k = mω 2 ,
and the potential energy of the particle is
1 2 1 2
2 2
2
V = kx = mω x .
De Broglie associated with the moving particle a wave of length
h
λ = ,
mv
where v is the velocity of the particle, and h is Planck’s constant.
The wave’s frequency is
ν
= v v mv
λ
= = h
.
h
mv
2
The wave’s angular frequency is
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H. Vic Dannon
ω = 2πν = 2π
mv h
2
In terms of the De Broglie wave, the particle’s energy is a multiple
of Planck’s radiation energy,
E
= εν h = εω ,
= , ε is the multiplier.
h
2π
The kinetic energy of the particle is
Hence,
2
1
mv = E − V .
2
mv = 2 m( E − V ),
λ =
h
2 mE ( − V)
,
v
= λν
=
1
ω 2π ω
.
2 mE ( −V)
1 2 mE ( −V)
=
2 2 2
v ω
Schrodinger postulated a complex valued potential
Ψ (,) xt = ψ() xe iωt
that satisfies the wave equation
Then,
2 1 2
∂Ψ
x
(,) xt = ∂Ψ (,)
2 t
xt .
v
2 1 2
0 =∂xΨ( xt , ) − ∂Ψ (
2 t
xt , )
v
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H. Vic Dannon
iωt
2 mE ( −V)
2 iωt
= ψ"( xe ) − ψ( x)( −ω ) e .
2 2
ω
The Schrodinger equation for the linear harmonic oscillator is
2m
ψ"( x) + ( E − V) ψ( x) = 0 .
2
Substituting E , and V ,
2m
1 2 2
ψ" + ( εω − mω x ) ψ
2
2
=
0
Multiplying by mω
,
2
mω
2
ψ"( x) + (2 ε− x ) ψ( x) = 0 .
mω
ξ
The change of variable
ξ = mω
x , gives
dψ dψdξ
= =
dx dξ
dx
ψ'( ξ) m
ω
,
and the equation becomes
2
mω
{ }
d ψ d dξ
= ψ'( ξ) = ψ''( ξ)
2
dx dξ
dx
mω
,
ψ"( ξ) + (2 ε− ξ ) ψ( x) = 0 .
2
0.2 Hermite Differential Equation
The Schrodinger equation
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H. Vic Dannon
can be factored
2
ψ"( ξ) − ξ ψ( x) = − 2 εψ( x)
( D − ξ)( D + ξ) ψ( x) = − 2εψ( x)
.
ξ
ξ
To solve the homogeneous equation
( D − ξ)( D + ξ) ψ( x)
= 0,
ξ
we solve
ψ '
( Dξ
− ξψ ) ( x) = 0 ⇒ ξ
ψ = ⇒ 1 2
log ψ = ξ + c ⇒
2
As ξ →∞, ψ 1
→∞, and is discarded.
ξ
ψ
2
1
= Ce ξ
.
1
2
( D + ξψ ) ( x) = 0 ⇒
ξ
Now, substituting
ψ '
ξ
ψ =− ⇒ 1 2
log ψ =− ξ + c ⇒
2
ψξ () = H()
ξe
−
1
2
ξ
2
−
1
2
ξ
ψ2 = Ce .
2
in
2
ψ"( ξ) + (2 ε− ξ ) ψ( ξ) = 0 , we have
− ξ
( )
1 2 1 2
− ξ
2 2
2 2
0 = Dξ H( ξ) e + (2 ε−
ξ ) H( ξ)
e
2 2
−1 1
( ξ
− ξ
)
= Dξ H '( ξ) e − ξH( ξ) e + (2 ε−ξ ) H( ξ)
e
2 2 2
1 ξ 2 1 ξ 2 1
ξ 2
2 2 2 2
− − −
1
− 2 ξ
= H ''() ξ e −2 ξH '() ξ e − H() ξ e + ξ H()
ξ e
−
= ⎡H ''( ξ) 2 ξH '( ξ) (2ε 1) H( ξ)
⎤
⎣
− + −
⎦
e ξ
1
2
2
.
2
1
ξ 2
2
− +
+ (2 − ) ( )
1
2
2
ξ
ε ξ H ξ e
−
The Schrodinger equation becomes Hermit Differential Equation
H ''( ξ) − 2 ξH '( ξ) + (2ε− 1) H( ξ) = 0,
2
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H. Vic Dannon
Substituting in it
we have
2 l l+ 1 l+
2
0 1 2 l l+ 1 l+
2
H ( ξ ) = c + c ξ + c ξ + ... + c ξ + c ξ + c ξ + ... ,
l=∞ l=∞ l=∞
2 l
l
ξ∑ l
− 2
ξ∑ l
+ (2 −1)
∑ l
l= 0 l= 0 l=
0
D cξ ξD cξ ε cξ
l=∞ l=∞
l−2 l−1
∑ ( l−1)
lclξ
∑ lclξ
l= 2 l=
1
l
=
0,
l =∞
l
∑ {( l + 1)( l + 2) cl+
2
− 2 lcl + (2ε− 1) cl} ξ = 0 ,
l = 0
( l + 1)( l + 2) c − [2 l + 1− 2 ε]
c =
l+
2
l
0
c
l+
2
= 2l
+ 1−2ε
c
( l + 1)( l + 2)
l
The solution is
1−2 2 3 2 3
0 1 0 ε
− ε
12 1
ξ
⋅
23 ⋅
H()
ξ = c + c ξ + c ξ + c +
1−2
2 (1 2 )(5 2 ) 4
0 ε − ε − ε
ξ
ξ
12 ⋅
1234 ⋅ ⋅ ⋅
= c {1 + + + ...} +
(1−2 ε)(5−2 ε) 4 (3−2 ε)(7−2 ε)
5
0
ξ c
1234 1
ξ
⋅⋅⋅
2345 ⋅⋅⋅
+ c
+ + ...
3−2ε
2 (3−2 ε)(7−2 ε)
4
1ξ ξ ξ
23 ⋅
2345 ⋅ ⋅ ⋅
To keep the solution from diverging at ξ →∞,
for n = 2k, the c0
series terms vanish for
+ c {1 + + + ...} .
2ε = 1,5, 9,13,...4k
+ 1,... ,
and we obtain the
H2 k
() ξ
Hermite Polynomials.
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H. Vic Dannon
for n = 2k
+1, the c1
series terms vanish for
2ε = 3,7,11,..., 4k
+ 3,...
and we obtain the
H
2k
+ 1 () ξ
Hermite Polynomials.
A solution for
ψξ ()
is the infinite linear combination
1 2 1 2
1 2
2 2 2 +
− ξ − ξ − ξ
0H0 e +
1H1 e +
2H2
e
α () ξ α () ξ α () ξ ....
0.3 The Hermite Series Associated with f ( x )
Let f ( x ) be defined on ( −∞, ∞ ) , and let H ( x)
be the Hermite
Polynomials
H 0 ( x ) = 1,
H 1 ( x) = 2x,
H 2
2 ( x) = 4x
−2,
H 3
3 ( x) = 8x −12x
,…
The Polynomials are orthogonal on ( −∞,
∞)
. That is,
x =∞
∫
x =−∞
−x
2
n
m n mn
e H ( x) H ( x) dx = 2 n!
πδ
We define the Orthonormalized Hermite Functions
1
1 2
− x
ϕ ( )
2
n
x = e H ( )
1 n
x
n
(2 n ! π)
2
If f ( x ) can be expanded in the ϕ n( x ),
Then,
f ( x) = αϕ( x) + αϕ( x) + αϕ( x) + ...,
0 0 1 1 2 2
n
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H. Vic Dannon
x=∞ x=∞
∫ ϕn
= ∫ α0ϕ0 + αϕ
1 1
+ α2ϕ2
+ ϕn
x=−∞
x=−∞
f ( x ) ( x ) dx { ( x ) ( x ) ( x ) ...} ( x ) dx
x=∞ x=∞ x=∞
∫ ∫ ∫
= α ϕ ( x ) ϕ ( x ) dx + α ϕ ( x ) ϕ ( x ) dx + α ϕ ( x ) ϕ ( x ) dx ..
= α n
.
0 0 n
1 1 n
2 2
x=−∞ x=−∞ x=−∞
δ δ δ
0n 1n 2n
n
+
Thus, the Hermite coefficients with respect to the
ϕ n( x )
are
α
n
ξ=∞
= ∫ f () ξ ϕ () ξ dξ.
ξ=−∞
n
The Orthonormal Hermite Series associated with f ( x ) is
αϕ( x) + αϕ( x) + αϕ( x) + ....
0 0 1 1 2 2
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H. Vic Dannon
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
{ f x } αϕ
0 0
H S ( ) = ( x) + ... + αϕ( x )
ermite n n n
As n
⎛ ξ=∞ ⎞ ξ=∞
() 0 () ⎛
⎞
f d
0
() x .. ξϕ ξ ξ ϕ f () ξϕn
() = ξdξ + +
ϕ
∫ ∫ n
⎜
()
⎝ξ=−∞
⎠⎟
⎝⎜ξ=−∞
⎟
x
⎠
ξ=∞
∫ f () ξ { ϕ0 () ξ ϕ0
() x ... ϕn
() ξ ϕn
() x } d ξ.
ξ=−∞
= + +
→∞, the orthonormal Hermite Sequence
ϕ () ξ ϕ () x + ... + ϕ () ξ ϕ () x
0 0
n
n
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H. Vic Dannon
becomes the orthonormal Hermite Kernel,
ϕ ( ξ ) ϕ ( x ) + ... + ϕ ( ξ ) ϕ ( x ) + ... ,
0 0
n
n
To see that it is singular at
ξ = x , we apply the Christoffel
Summation Formula, [Sansone, p.371],
ϕ () ξ ϕ () x + ... + ϕ () ξ ϕ () x =
0 0
n
n
n + 1 ϕn+ 1() ξ ϕn() x − ϕn() ξ ϕn+
1()
x
.
2
ξ − x
For ξ → x ,
2 2
0 0
x
n n
x
0
x
n
ϕ ( ξ) ϕ ( ) + ... + ϕ ( ξ) ϕ ( ) → ϕ ( ) + ... + ϕ ( x ),
and
n
+ 1ϕn+ 1() ξ ϕn() x − ϕn() ξ ϕn+
1()
x n + 10
→
2 ξ − x
2 0 .
Applying Bernoulli’s rule to the indeterminate limit,
ϕn+ 1() ξ ϕn() x − ϕn() ξ ϕn+
1()
x
lim
ξ→x
ξ −x
=
Dξϕn+ 1() ξ ϕn() x − Dξϕn() ξ ϕn+
1()
x
lim
ξ→x
D ( ξ −x)
ξ
= lim[ ϕ '( ξ) ϕ ( x) −ϕ '( ξ) ϕ ( x)]
ξ→x
n+ 1 n n n+
1
= ϕ '( x ) ϕ ( x ) −ϕ '( x ) ϕ ( x )
n+ 1 n n n+
1
Therefore,
2 2 n + 1
ϕ0( x) + ... + ϕn( x) = [ ϕn+ 1'( x) ϕn( x) −ϕn
'( x) ϕn+
1( x )].
2
Since ϕ n( x ), and ϕ n 1 ( x ) solve the differential equation, [Szego,
+
13
p.105, #5.5.2],

Gauge Institute Journal
H. Vic Dannon
we have,
1 ⋅ z''( x) + 0 ⋅ z'( x) + (2n + 1 − x ) z( x) = 0,
ax ( ) bx ( )
ϕ '( x) ϕ ( x) − ϕ '( x) ϕ ( x) = ( const)
e
n+ 1 n n n+
1
2
−
∫
bx ( )
dx a( x)
=
( const)
e∫
0⋅dx
= const ,
for any −∞ < x < ∞.
Hence,
2 2 n + 1
ϕ0( x) + ... + ϕn
( x)
= const
2
and the Hermite Kernel diverges to ∞ at any ξ = x .
Therefore, while the partial sums of the Hermite Series exist,
their limit does not. That is, due to the singularity at
ξ = x , the
Hermite Series 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 , the Hermite Kernel vanishes, and the integral will be
identically zero, for any function
f ( x ).
To see that the kernel vanishes for any
ξ ≠ x , we apply the
Christoffel Summation Formula, with ξ ≠ x .
n+ 1
ϕn+ 1
ξ ϕn x − ϕn ξ ϕn+
1
0() 0() x + ... +
n() n()
x =
2
ϕ ξ ϕ ϕ ξ ϕ
() () () () x
.
ξ − x
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H. Vic Dannon
We have
1
1 2
− x
ϕ ( )
2
n
x = e H ( )
1 n
x
n
(2 n ! π)
2
=
n
1
1
2
(2 n ! π)
1 2
x 2
2 −x
e e H ( x)
n
By [Szego, p. 105, #5.5.3],
Thus,
and
ϕ
−x
2 2
e H ( x ) ( 1) D n
= − e x .
n
n
( −1)
1 2
x 2
( )
2 n −x
ϕn
x = e D { }
1 x
e ,
n
(2 n ! π)
2
n+
1
n
n+
1
( −1)
1 2
x 2
2 n+ 1 −x
( x) =
e D { }
1 x
e
n+
1
(2 ( n + 1)! π)
2
x
−
n+ 1
ϕn+
1
ξ ϕn x − ϕn ξ ϕn+1
x
2
() () () ()
ξ − x
=
→0,
n→∞
2n
+ 1
n+ 1 −ξ
{ 2 n −x 2 n − ξ
2 n+ 1 −x Dξ
{ e } Dx{ e } Dξ{ e } Dx
{ e
2
}}
1 n+
1 ( −1)
= −
ξ − x
2 n
2 2 πn!( n + 1)!
→0,
n→∞
→
0, as n
→ ∞.
That is, the Hermite Kernel vanishes for any ξ ≠ x .
Plots of the Hermite Sequence confirm that
In the Calculus of Limits,
the Hermite Kernel is either singular or zero
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1.2 Plots of
2
1 − ξ
1
0
ξ
0
+ +
n
π
2 n ! n
ξ
n
e { H ( ) H ( x ) ... H ( ) H ( x )}
In Maple,
223
∑
i = 0
2
1 −x
1
π
i
2 i !
plot( e * HermiteH(,.5)* i HermiteH(, i x), x =− 23..23)
In Maple,
223
2
1 −x
1
∑ − = − 3..23)
i
plot( e * HermiteH( i, 1)* HermiteH( i, x), x 2
i = 0
π
2 i !
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H. Vic Dannon
x 2
e − that suppresses oscillations away from the origin, enhances
them at the origin. Thus, a singularity away from the origin
needs more terms
In Maple,
223
2
1 −x
1
∑
i
π 2 i !
i = 0
plot( e * HermiteH( i, 2) * HermiteH( i, x), x =− 23..23)
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
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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 ,
it spikes at
x = 0 , so that its Infinitesimal Calculus
x =∞
∫
integral is δ( xdx ) = 1,
and
x =−∞
1
δ (0) = < ∞.
dx
Here, we show that in Infinitesimal calculus, the Hermite Kernel
is a hyper-real Delta Function.
egendre
And the Hermite Series L S { f ( x)
associated with a Hyperreal
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, ( 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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negative sign, a family of infinite hyper-reals, a family of
infinite hyper-reals with negative sign, and non-constant
hyper-reals.
8. The hyper-reals are totally ordered, and aligned along a
line: the Hyper-real Line.
9. That line includes the real numbers separated by the nonconstant
hyper-reals. Each real number is the center of an
interval of hyper-reals, that includes no other real number.
10. In particular, zero is separated from any positive real
by the infinitesimals, and from any negative real by the
infinitesimals with negative signs, −dx .
11. Zero is not an infinitesimal, because zero is not strictly
greater than zero.
12. We do not add infinity to the hyper-real line.
13. The infinitesimals, the infinitesimals with negative
signs, the infinite hyper-reals, and the infinite hyper-reals
with negative signs are semi-groups with
respect to addition. Neither set includes zero.
14. The hyper-real line is embedded in , and is not
∞
homeomorphic to the real line. There is no bi-continuous
one-one mapping from the hyper-real onto the real line.
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15. 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 [ 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 =−∞
k =∞
1 −ik( ξ−x
)
10. δξ ( − x)
= e
2π
∫ dk
k =−∞
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H. Vic Dannon
5.
Convergent Series
In [Dan8], we defined convergence of infinite series in
Infinitesimal Calculus
5.1 Sequence Convergence to a finite hyper-real a
a → a iff a − a = infinitesimal .
n
n
5.2 Sequence Convergence to an infinite hyper-real A
a → A iff a
n
represents the infinite hyper-real A.
n
5.3 Series Convergence to a finite hyper-real s
a1 + a2 + ... → s iff a1 + ... + an
− s = infinitesimal .
5.4 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
6.
Hermite Sequence and δξ− ( x)
6.1 Hermite Sequence Definition
If f ( x ) can be expanded in the Hn ( x ) ,
Then,
f ( x) = a H ( x) + a H ( x) + a H ( x) + ...,
0 0 1 1 2 2
x =∞
∫
x =−∞
−x
2
e f( x) H ( x)
dx
n
=
x =∞
∫
−x
2
= f ( xe ) { aH ( x) + aH( x) + aH( x) + ...} H ( xdx )
x =∞
0 0 1 1 2 2
x=∞ x=∞
2 2
−x
−x
0 ∫ 0 n
1 ∫ 1
x=−∞
x=−∞
= a e H ( x) H ( x) dx + a e H ( x) H ( x) dx + ..
n
= 2 n!
πa n
.
0 1
πδ0n
πδ1
n
20! 21!
The Hermite Series partial sums
{ } 0 0
H S f ( x) = a H ( x) + ... + a H ( x )
ermite n n n
ξ=∞
{ n n n }
()
2
1 −ξ
() () ... 1
∫ f ξ e H ξ H x H () ξ H () x dξ.
= + +
ξ=−∞
give rise to the Hermite Sequence
π
0 0 2 n !
n
n
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H. Vic Dannon
2
{ ξ
n n }
1 −ξ
1
π 0 0 2 n !
H n(, ξ x ) = e H () H () x + ... + H () ξ H n()
x .
6.2 Hermite Sequence is a Delta Sequence
For each n = 0,1, 2, 3,...
2
{ ξ
n n }
1 −ξ
1
π 0 0 2 n !
H n( ξ, x ) = e H ( ) H ( x ) + ... + H ( ξ ) H n( x ) ,
1. has the sifting property
ξ=∞
{ ξ
n n
ξ
n }
2
1 −ξ
1
∫ e H H x H H x dξ
ξ=−∞
π
2. is a continuous function
() () + ... + () () = 1
0 0 2 n !
3. peaks for each ξ → x to const ⋅ n + 1
Proof of (1)
ξ=∞
{ ( ξ ) ( ) ... ( ) ( )
n n
ξ
n }
2
1 −ξ
1
∫ e H H x H H x dξ
ξ=−∞
π
+ + =
0 0 2 n !
ξ=∞ ξ=∞
−ξ
∫
2 2
1 1 1
= H 0() x e H 0() ξ d ξ + ... + H n() x e H ()
n
2 n !
n
ξ d ξ
π
π
1 ξ=−∞
1
ξ=−∞
1
By [Spanier, p.222, #24:10:5], for k = 1,2,..., n,
∫
−ξ
ξ=∞
∫
ξ=−∞
−ξ
2
e H ( bξ)
dξ
k
⎧ ⎪
0, k = 1, 3,5,...
⎪⎩
!( 1) , 2, 4,6,...
= ⎨ 1
⎪ 2 n
πn b −
2
k =
Therefore, for k = 1,2,..., n,
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H. Vic Dannon
ξ=∞
2
−ξ
∫ e Hk () ξ dξ
= 0 .
ξ=−∞
Hence,
ξ=∞
{ ξ
n n
ξ
n }
2
1 −ξ
() () ... 1
∫ e H H x + + H () H () x dξ = 1 .
ξ=−∞
π
0 0 2 n !
Proof of (3)
By the Christoffel Summation Formula, [Sansone, p.371],
ϕ () ξ ϕ () x + ... + ϕ () ξ ϕ () x =
0 0
n
n
n + 1 ϕn+ 1() ξ ϕn() x − ϕn() ξ ϕn+
1()
x
,
2
ξ − x
where
For ξ → x ,
1
1 2
− x
ϕ ( )
2
n
x = e H ( )
1 n
x .
n
(2 n ! π)
2
n
+ 1ϕn+ 1() ξ ϕn() x − ϕn() ξ ϕn+
1()
x n + 10
→
2 ξ − x
2 0 .
Applying Bernoulli’s rule to the indeterminate limit,
ϕn+ 1() ξ ϕn() x − ϕn() ξ ϕn+
1()
x
lim
ξ→x
ξ −x
=
Dξϕn+ 1() ξ ϕn() x − Dξϕn() ξ ϕn+
1()
x
lim
ξ→x
D ( ξ −x)
ξ
= lim[ ϕ '( ξ) ϕ ( x) −ϕ '( ξ) ϕ ( x)]
ξ→x
n+ 1 n n n+
1
= ϕ '( x ) ϕ ( x ) −ϕ '( x ) ϕ ( x )
n+ 1 n n n+
1
Therefore,
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H. Vic Dannon
2 2 n + 1
ϕ0( x) + ... + ϕn( x) = [ ϕn+ 1'( x) ϕn( x) −ϕn
'( x) ϕn+
1( x )].
2
Since ϕ ( ), and
1 ( x ) solve the differential equation, [Szego,
n x
p.105, #5.5.2],
ϕ
n +
we have,
1 ⋅ z''( x) + 0 ⋅ z'( x) + (2n + 1 − x ) z( x) = 0 ,
ax ( ) bx ( )
ϕ '( x) ϕ ( x) − ϕ '( x) ϕ ( x) = ( const)
e
n+ 1 n n n+
1
2
−
∫
bx ( )
dx a( x)
=
( const)
e∫
0⋅dx
= const ,
for any −∞ < x < ∞.
Hence,
Therefore, substituting
2
2 2
ϕn
ϕ0 ( x ) + ... + ( x ) = n + 1 const
n
1 1
2 2
H ( x ) = (2 n ! π) e ϕ ( x ),
n
x
1 1
2 2
n
H () ξ = (2 n! π) e ξ
ϕ () ξ
1 ξ
1
0
ξ
0 n
π
2 n ! n
ξ
n
n
e − { H ( ) H ( x) + ... + H ( ) H ( x)}
=
1
2 2
{ ϕ () ξ ϕ () x ... ϕ () ξ ϕ () }
− ( ξ −x
) 2
0 0
= e
+ + x
ξ→x
2 2
{ ϕ0 ( x ) ... ϕn
( x ) }
→ + + = n + 1const.
n
n
2
2
n
n
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H. Vic Dannon
7.
Hermite Kernel and δξ− ( x)
7.1 Hermite Kernel in the Calculus of Limits
The Hermite Series partial sums
ξ=∞
{ } 1 −ξ
f ( x) = f( ξ) e 2
{ H ( ξ) H ( x) + ... +
1 H ( ξ) H ( x)
n
}
∫
H S dξ.
ermite n π 0 0 2 n ! n n
ξ=−∞
Hermite Sequence
give rise to the Hermite Sequence.
The limit of the Hermite Sequence is an infinite series called the
Hermite Kernel
H
2
{ n
}
1 −ξ
1
ermite
ξ
0
ξ
π
0 2 n ! n
ξ
n
( − x) = e H ( ) H ( x) + ... + H ( ) H ( x) + ...
7.2 In the Calculus of Limits, the Hermite Kernel does not have
the sifting property
Proof: for ξ → x ,
2
{ ξ
n n
ξ
n }
1 −ξ
1
π 0 0 2 n !
e H ( ) H ( x) + ... + H ( ) H ( x) + .. = lim n + 1con
st
That is, for ξ → x ,
2
{ ( ξ ) ( ) ... ( ) ( ) ...
n n
ξ
n }
1 ξ
1
π 0 0 2 n !
→ ∞
n→∞
n→∞
e − H H x + + H H x + is singular.
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H. Vic Dannon
7.3 Hyper-real Hermite Kernel in Infinitesimal Calculus
H
2
{ n
}
1 −ξ
1
ermite
ξ
0
ξ
π
0 2 n ! n
ξ
n
( − x) = e H ( ) H ( x) + ... + H ( ) H ( x) + ...
⎧⎪ n , ξ =
=
⎪
⎨ ⎪ 0 , ξ ≠
⎪⎩
x
x
Proof:
H
= δξ ( − x)
.
2
{ n
}
1 −ξ
1
ermite
ξ
0
ξ
π
0 2 n ! n
ξ
n
( − x) = e H ( ) H ( x) + ... + H ( ) H ( x) + ...
⎧⎪ n , ξ =
=
⎪
⎨ ⎪ 0 , ξ ≠
⎪⎩
x
.
x
Denoting by 1
dx
the infinite hyper-real n ,
⎧
0, ξ ≠ x
= ⎪
⎨ ⎪
1
, ξ x
dx = ⎪⎩
= δξ ( −x).
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H. Vic Dannon
8.
Hermite Series and δξ− ( x)
8.1 Hermite Series of a Hyper-real Function
Let f ( x ) be a hyper-real function integrable on ( −∞,
∞) .
Then, for each
n = 0,1, 2, 3,... , the integrals
−x
a = ∫ e f( x) H ( x)
dx.
n
1
n
2 n !
x =∞
π
x =−∞
2
n
exist, with finite, or infinite hyper-real values. The
a n
are the
Hermite Coefficients of
f ( x ).
The Hermite Series associated with f ( x ) is
{ } =
0 0
+
1 1
+
2 2
HermiteS f ( x) a H ( x) a H ( x) a H ( x) + ...
For each x , it may assume finite or infinite hyper-real values.
8.2 H S{ δξ }
Proof:
ermite
( − x) = δξ ( −x)
{ } 0 0 1 1 2 2
HermiteS
δξ− ( x) = a H ( x) + a H ( x) + a H ( x) + ...
where
n
1
n
2 n !
x =∞
2
−x
∫ δξ ( ) Hn( x)
dx
a = e −x
π
x =−∞
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H. Vic Dannon
Therefore,
H S
= e H
n n () ξ .
1
2 n !
π
− ξ
2
{ } 1 −ξ
δξ ( − x) = e 2
{ H ( ξ) H ( x) + ... + 1 H ( ξ) H ( x) + ...
n
}
ermite π 0 0 2 n ! n n
= H ( ξ −x)
, by 7.3,
ermite
= δξ ( −x), by 7.3.
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H. Vic Dannon
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
H S f ( x) = a H ( x) + ... + a H ( x )
ermite n n n
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Gauge Institute Journal
H. Vic Dannon
As n
⎛
⎞ ⎟
() () () ...
⎠
ξ=∞
2
1
−ξ
= f ξe H0 ξ dξ
H0
x +
π ∫
⎜
ξ =−∞
⎟
⎝
ξ=∞
⎛ ξ=∞
⎞ 2
1 1
−ξ
... + f ( ξ) e H ( ) ( )
n
2 n !
n
ξ dξ
Hn
x
π ∫
⎜
⎝ ξ =−∞
⎠⎟
{ n n n }
2
1 −ξ
1
∫ f () ξ e H () ξ H () x ... H () ξ H () x dξ.
= + +
ξ=−∞
π
→∞, the Hermite Sequence
2
0 0 2 n !
{ ( ξ ) ( ) ... ( ) ( )
n n
ξ
n }
1 − ξ
1
0 0
+ +
π
2 n !
becomes the Hermite Kernel,
e H H x H H x
2
{ ( ξ ) ( ) ... ( ) ( ) ...
n n
ξ
n }
1 ξ
1
π 0 0 2 n !
e − H H x + + H H x + ,
By 7.2, the Hermite Kernel diverges to infinity at any ξ = x .
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 ξ = x .
Avoiding the singularity at
ξ = x , by using the Cauchy Principal
Value of the integral does not recover the Theorem, because for
any
ξ ≠ x , the Hermite Kernel vanishes, and the integral is
identically zero, for any function
f ( x ).
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H. Vic Dannon
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, f ( x) = H S{ f( x )}
Proof:
ermite
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H. Vic Dannon
Substituting from 7.3,
2
ξ =∞
∫
fx ( ) = f( ξδξ ) ( −xd
) ξ
ξ =−∞
{ n n n }
1 −ξ
1
π 0 0 2 n !
δξ ( − x ) = e H ( ξ ) H ( x ) + ... + H ( ξ ) H ( x ) + ... ,
ξ=∞
∫
ξ=−∞
2
{ n n n
+ }
1 −ξ
1
π 0 0 2 n !
f ( x) = f( ξ) e H ( ξ) H ( x) + ... + H ( ξ) H ( x) ... dξ
This Hyper-real Integral is the summation,
ξ=∞
∑
ξ=−∞
2
{ + +
n n n
+ }
1 −ξ
1
π 0 0 2 n !
f ( ξ) e H ( ξ) H ( x) ... H ( ξ) H ( x) ... dξ
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
⎛ ξ=∞
⎞ 1
2
−ξ
= f() ξe H0() ξ dξ
H0() x + ...
∫
⎜ π
⎝ ξ=−∞
⎠⎟
a
0 0 1 1 2 2
0
= aH( x) + aH( x) + aH( x) + ...
{ f ( x)
}
= H S .
ermite
⎛
ξ=∞
⎞ 1
2
−ξ
... + f( ξ) e Hn( ξ) dξ
Hn( x) + ...
n ∫
⎜ 2 n ! π
⎝
ξ=−∞
⎠⎟
a
n
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In particular, the Delta Function violates Uspensky’s Conditions
The Hyper-real
δ( x)
and is not integrable in any interval.
, is not defined in the Calculus of Limits,
But by 8.2,
δξ− ( x)
satisfies the Hermite Series Theorem.
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H. Vic Dannon
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[Abramowitz] Abramowitz, M., and Stegun, I., “Handbook of Mathematical
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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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42

Gauge Institute Journal
H. Vic Dannon
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