Under certain assumptions that we will specify further, it is possible to obtain the existence of a secondary measure and even to express it.
For example, if one works in the Hilbert space L^{2}([0, 1], R, ρ)
∀
x
∈
[
0
,
1
]
,
μ
(
x
)
=
ρ
(
x
)
φ
2
(
x
)
4
+
π
2
ρ
2
(
x
)
with
φ
(
x
)
=
lim
ε
→
0
+
2
∫
0
1
(
x
−
t
)
ρ
(
t
)
(
x
−
t
)
2
+
ε
2
d
t
in the general case, or:
φ
(
x
)
=
2
ρ
(
x
)
ln
(
x
1
−
x
)
−
2
∫
0
1
ρ
(
t
)
−
ρ
(
x
)
t
−
x
d
t
when ρ satisfies a Lipschitz condition.
This application φ is called the reducer of ρ.
More generally, μ et ρ are linked by their Stieltjes transformation with the following formula:
S
μ
(
z
)
=
z
−
c
1
−
1
S
ρ
(
z
)
in which c_{1} is the moment of order 1 of the measure ρ.
These secondary measures, and the theory around them, lead to some surprising results, and make it possible to find in an elegant way quite a few traditional formulas of analysis, mainly around the Euler Gamma function, Riemann Zeta function, and Euler's constant.
They also allowed the clarification of integrals and series with a tremendous effectiveness, though it is a priori difficult.
Finally they make it possible to solve integral equations of the form
f
(
x
)
=
∫
0
1
g
(
t
)
−
g
(
x
)
t
−
x
ρ
(
t
)
d
t
where g is the unknown function, and lead to theorems of convergence towards the Chebyshev and Dirac measures.
Let ρ be a measure of positive density on an interval I and admitting moments of any order. We can build a family {P_{n}} of orthogonal polynomials for the inner product induced by ρ. Let us call {Q_{n}} the sequence of the secondary polynomials associated with the family P. Under certain conditions there is a measure for which the family Q is orthogonal. This measure, which we can clarify from ρ is called a secondary measure associated initial measure ρ.
When ρ is a probability density function, a sufficient condition so that μ, while admitting moments of any order can be a secondary measure associated with ρ is that its Stieltjes Transformation is given by an equality of the type:
S
μ
(
z
)
=
a
(
z
−
c
1
−
1
S
ρ
(
z
)
)
,
a is an arbitrary constant and c_{1} indicating the moment of order 1 of ρ.
For a = 1 we obtain the measure known as secondary, remarkable since for n ≥ 1 the norm of the polynomial P_{n} for ρ coincides exactly with the norm of the secondary polynomial associated Q_{n} when using the measure μ.
In this paramount case, and if the space generated by the orthogonal polynomials is dense in L^{2}(I, R, ρ), the operator T_{ρ} defined by
f
(
x
)
↦
∫
I
f
(
t
)
−
f
(
x
)
t
−
x
ρ
(
t
)
d
t
creating the secondary polynomials can be furthered to a linear map connecting space L^{2}(I, R, ρ) to L^{2}(I, R, μ) and becomes isometric if limited to the hyperplane H_{ρ} of the orthogonal functions with P_{0} = 1.
For unspecified functions square integrable for ρ we obtain the more general formula of covariance:
⟨
f
/
g
⟩
ρ
−
⟨
f
/
1
⟩
ρ
×
⟨
g
/
1
⟩
ρ
=
⟨
T
ρ
(
f
)
/
T
ρ
(
g
)
⟩
μ
.
The theory continues by introducing the concept of reducible measure, meaning that the quotient ρ/μ is element of L^{2}(I, R, μ). The following results are then established:
The reducer φ of ρ is an antecedent of ρ/μ for the operator T_{ρ}. (In fact the only antecedent which belongs to H_{ρ}).
For any function square integrable for ρ, there is an equality known as the reducing formula:
⟨
f
/
φ
⟩
ρ
=
⟨
T
ρ
(
f
)
/
1
⟩
ρ
.
The operator
f
↦
φ
×
f
−
T
ρ
(
f
)
defined on the polynomials is prolonged in an isometry S_{ρ} linking the closure of the space of these polynomials in L^{2}(I, R, ρ^{2}μ^{−1}) to the hyperplane H_{ρ} provided with the norm induced by ρ.
Under certain restrictive conditions the operator S_{ρ} acts like the adjoint of T_{ρ} for the inner product induced by ρ.
Finally the two operators are also connected, provided the images in question are defined, by the fundamental formula of composition:
T
ρ
∘
S
ρ
(
f
)
=
ρ
μ
×
(
f
)
.
The Lebesgue measure on the standard interval [0, 1] is obtained by taking the constant density ρ(x) = 1.
The associated orthogonal polynomials are called Legendre polynomials and can be clarified by
P
n
(
x
)
=
d
n
d
x
n
(
x
n
(
1
−
x
)
n
)
.
The norm of P_{n} is worth
n
!
2
n
+
1
.
The recurrence relation in three terms is written:
2
(
2
n
+
1
)
X
P
n
(
X
)
=
−
P
n
+
1
(
X
)
+
(
2
n
+
1
)
P
n
(
X
)
−
n
2
P
n
−
1
(
X
)
.
The reducer of this measure of Lebesgue is given by
φ
(
x
)
=
2
ln
(
x
1
−
x
)
.
The associated secondary measure is then clarified as
μ
(
x
)
=
1
ln
2
(
x
1
−
x
)
+
π
2
.
If we normalize the polynomials of Legendre, the coefficients of Fourier of the reducer φ related to this orthonormal system are null for an even index and are given by
C
n
(
φ
)
=
−
4
2
n
+
1
n
(
n
+
1
)
for an odd index n.
The Laguerre polynomials are linked to the density ρ(x) = e^{−x} on the interval I = [0, ∞). They are clarified by
L
n
(
x
)
=
e
x
n
!
d
n
d
x
n
(
x
n
e
−
x
)
=
∑
k
=
0
n
(
n
k
)
(
−
1
)
k
x
k
k
!
and are normalized.
The reducer associated is defined by
φ
(
x
)
=
2
(
ln
(
x
)
−
∫
0
∞
e
−
t
ln

x
−
t

d
t
)
.
The coefficients of Fourier of the reducer φ related to the Laguerre polynomials are given by
C
n
(
φ
)
=
−
1
n
∑
k
=
0
n
−
1
1
(
n
−
1
k
)
.
This coefficient C_{n}(φ) is no other than the opposite of the sum of the elements of the line of index n in the table of the harmonic triangular numbers of Leibniz.
The Hermite polynomials are linked to the Gaussian density
ρ
(
x
)
=
e
−
x
2
2
2
π
on I = R.
They are clarified by
H
n
(
x
)
=
1
n
!
e
x
2
2
d
n
d
x
n
(
e
−
x
2
2
)
and are normalized.
The reducer associated is defined by
φ
(
x
)
=
−
2
2
π
∫
−
∞
∞
t
e
−
t
2
2
ln

x
−
t

d
t
.
The coefficients of Fourier of the reducer φ related to the system of Hermite polynomials are null for an even index and are given by
C
n
(
φ
)
=
(
−
1
)
n
+
1
2
(
n
−
1
2
)
!
n
!
for an odd index n.
The Chebyshev measure of the second form. This is defined by the density
ρ
(
x
)
=
8
π
x
(
1
−
x
)
on the interval [0, 1].
It is the only one which coincides with its secondary measure normalised on this standard interval. Under certain conditions it occurs as the limit of the sequence of normalized secondary measures of a given density.
Jacobi measure on (0, 1) of density
ρ
(
x
)
=
2
π
1
−
x
x
.
Chebyshev measure on (−1, 1) of the first form of density
ρ
(
x
)
=
1
π
1
−
x
2
.
The secondary measure μ associated with a probability density function ρ has its moment of order 0 given by the formula
d
0
=
c
2
−
c
1
2
,
where c_{1} and c_{2} indicating the respective moments of order 1 and 2 of ρ.
To be able to iterate the process then, one 'normalizes' μ while defining ρ_{1} = μ/d_{0} which becomes in its turn a density of probability called naturally the normalised secondary measure associated with ρ.
We can then create from ρ_{1} a secondary normalised measure ρ_{2}, then defining ρ_{3} from ρ_{2} and so on. We can therefore see a sequence of successive secondary measures, created from ρ_{0} = ρ, is such that ρ_{n+1} that is the secondary normalised measure deduced from ρ_{n}
It is possible to clarify the density ρ_{n} by using the orthogonal polynomials P_{n} for ρ, the secondary polynomials Q_{n} and the reducer associated φ. That gives the formula
ρ
n
(
x
)
=
1
d
0
n
−
1
ρ
(
x
)
(
P
n
−
1
(
x
)
φ
(
x
)
2
−
Q
n
−
1
(
x
)
)
2
+
π
2
ρ
2
(
x
)
P
n
−
1
2
(
x
)
.
The coefficient
d
0
n
−
1
is easily obtained starting from the leading coefficients of the polynomials P_{n−1} and P_{n}. We can also clarify the reducer φ_{n} associated with ρ_{n}, as well as the orthogonal polynomials corresponding to ρ_{n}.
A very beautiful result relates the evolution of these densities when the index tends towards the infinite and the support of the measure is the standard interval [0, 1].
Let
x
P
n
(
x
)
=
t
n
P
n
+
1
(
x
)
+
s
n
P
n
(
x
)
+
t
n
−
1
P
n
−
1
(
x
)
be the classic recurrence relation in three terms. If
lim
n
↦
∞
t
n
=
1
4
,
lim
n
↦
∞
s
n
=
1
2
,
then the sequence {ρ_{n}} converges completely towards the Chebyshev density of the second form
ρ
t
c
h
(
x
)
=
8
π
x
(
1
−
x
)
.
These conditions about limits are checked by a very broad class of traditional densities. A derivation of the sequence of secondary measures and convergence can be found in
One calls two measures thus leading to the same normalised secondary density. It is remarkable that the elements of a given class and having the same moment of order 1 are connected by a homotopy. More precisely, if the density function ρ has its moment of order 1 equal to c_{1}, then these densities equinormal with ρ are given by a formula of the type:
ρ
t
(
x
)
=
t
ρ
(
x
)
(
1
2
(
t
−
1
)
(
x
−
c
1
)
φ
(
x
)
−
t
)
2
+
π
2
ρ
2
(
x
)
(
t
−
1
)
2
(
x
−
c
1
)
2
,
t describing an interval containing ]0, 1].
If μ is the secondary measure of ρ, that of ρ_{t} will be tμ.
The reducer of ρ_{t} is
φ
t
(
x
)
=
2
(
x
−
c
1
)
−
t
G
(
x
)
(
(
x
−
c
1
)
−
t
1
2
G
(
x
)
)
2
+
t
2
π
2
μ
2
(
x
)
by noting G(x) the reducer of μ.
Orthogonal polynomials for the measure ρ_{t} are clarified from n = 1 by the formula
P
n
t
(
x
)
=
t
P
n
(
x
)
+
(
1
−
t
)
(
x
−
c
1
)
Q
n
(
x
)
t
with Q_{n} secondary polynomial associated with P_{n}.
It is remarkable also that, within the meaning of distributions, the limit when t tends towards 0 per higher value of ρ_{t} is the Dirac measure concentrated at c_{1}.
For example, the equinormal densities with the Chebyshev measure of the second form are defined by:
ρ
t
(
x
)
=
2
t
1
−
x
2
π
[
t
2
+
4
(
1
−
t
)
x
2
]
,
with t describing ]0, 2]. The value t = 2 gives the Chebyshev measure of the first form.
In the formulas below G is Catalan's constant, γ is the Euler's constant, β_{2n} is the Bernoulli number of order 2n, H_{2n+1} is the harmonic number of order 2n+1 and Ei is the Exponential integral function.
1
ln
(
p
)
=
1
p
−
1
+
∫
0
∞
1
(
x
+
p
)
(
ln
2
(
x
)
+
π
2
)
d
x
∀
p
>
1
γ
=
∫
0
∞
ln
(
1
+
1
x
)
ln
2
(
x
)
+
π
2
d
x
γ
=
1
2
+
∫
0
∞
(
x
+
1
)
cos
(
π
x
)
¯
x
+
1
d
x
The notation
x
↦
(
x
+
1
)
cos
(
π
x
)
¯
indicating the 2 periodic function coinciding with
x
↦
(
x
+
1
)
cos
(
π
x
)
on (−1, 1).
γ
=
1
2
+
∑
k
=
1
n
β
2
k
2
k
−
β
2
n
ζ
(
2
n
)
∫
1
∞
⌊
t
⌋
cos
(
2
π
t
)
t
−
2
n
−
1
d
t
β
k
=
(
−
1
)
k
k
!
π
Im
(
∫
−
∞
∞
e
x
(
1
+
e
x
)
(
x
−
i
π
)
k
d
x
)
∫
0
1
ln
2
n
(
x
1
−
x
)
d
x
=
(
−
1
)
n
+
1
(
2
2
n
−
2
)
β
2
n
π
2
n
∫
0
1
⋯
∫
0
1
(
∑
k
=
1
2
n
ln
(
t
k
)
∏
i
≠
k
(
t
k
−
t
i
)
)
d
t
1
⋯
d
t
2
n
=
1
2
(
−
1
)
n
+
1
(
2
π
)
2
n
β
2
n
∫
0
∞
e
−
α
x
Γ
(
x
+
1
)
d
x
=
e
e
−
α
−
1
+
∫
0
∞
1
−
e
−
x
(
ln
(
x
)
+
α
)
2
+
π
2
d
x
x
∀
α
∈
R
∑
n
=
1
∞
(
1
n
∑
k
=
0
n
−
1
1
(
n
−
1
k
)
)
2
=
4
9
π
2
=
∫
0
∞
4
(
E
i
(
1
,
−
x
)
+
i
π
)
2
e
−
3
x
d
x
.
23
15
−
ln
(
2
)
=
∑
n
=
0
∞
1575
2
(
n
+
1
)
(
2
n
+
1
)
(
4
n
−
3
)
(
4
n
−
1
)
(
4
n
+
1
)
(
4
n
+
5
)
(
4
n
+
7
)
(
4
n
+
9
)
G
=
∑
k
=
0
∞
(
−
1
)
k
4
k
+
1
(
1
(
4
k
+
3
)
2
+
2
(
4
k
+
2
)
2
+
2
(
4
k
+
1
)
2
)
+
π
8
ln
(
2
)
G
=
π
8
ln
(
2
)
+
∑
n
=
0
∞
(
−
1
)
n
H
2
n
+
1
2
n
+
1
.
If the measure ρ is reducible and let φ be the associated reducer, one has the equality
∫
I
φ
2
(
x
)
ρ
(
x
)
d
x
=
4
π
2
3
∫
I
ρ
3
(
x
)
d
x
.
If the measure ρ is reducible with μ the associated reducer, then if f is square integrable for μ, and if g is square integrable for ρ and is orthogonal with P_{0} = 1 one has equivalence:
f
(
x
)
=
∫
I
g
(
t
)
−
g
(
x
)
t
−
x
ρ
(
t
)
d
t
⇔
g
(
x
)
=
(
x
−
c
1
)
f
(
x
)
−
T
μ
(
f
(
x
)
)
=
φ
(
x
)
μ
(
x
)
ρ
(
x
)
f
(
x
)
−
T
ρ
(
μ
(
x
)
ρ
(
x
)
f
(
x
)
)
c_{1} indicates the moment of order 1 of ρ and T_{ρ} the operator
g
(
x
)
↦
∫
I
g
(
t
)
−
g
(
x
)
t
−
x
ρ
(
t
)
d
t
.
In addition, the sequence of secondary measures has applications in Quantum Mechanics. The sequence gives rise to the socalled sequence of residual spectral densities for specialized PauliFierz Hamiltonians. This also provides a physical interpretation for the sequence of secondary measures.