# Secondary measure

In mathematics, the **secondary measure** associated with a measure of positive density ρ when there is one, is a measure of positive density μ, turning the secondary polynomials associated with the orthogonal polynomials for ρ into an orthogonal system.

## Introduction

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)

with

in the general case, or:

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:

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

where *g* is the unknown function, and lead to theorems of convergence towards the Chebyshev and Dirac measures.

## The broad outlines of the theory

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*} the sequence of the secondary polynomials associated with the family

_{n}*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:

*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*when using the measure μ.

_{n}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

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:

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:

- .

The operator

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:

## Case of the Lebesgue measure and some other examples

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

The norm of *P _{n}* is worth

The recurrence relation in three terms is written:

The reducer of this measure of Lebesgue is given by

The associated secondary measure is then clarified as

- .

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

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

and are normalized.

The reducer associated is defined by

The coefficients of Fourier of the reducer φ related to the Laguerre polynomials are given by

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

on *I* = **R**.

They are clarified by

and are normalized.

The reducer associated is defined by

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

for an odd index *n*.

The Chebyshev measure of the second form. This is defined by the density

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.

### Examples of non-reducible measures

Jacobi measure on (0, 1) of density

Chebyshev measure on (−1, 1) of the first form of density

## Sequence of secondary measures

The secondary measure μ associated with a probability density function ρ has its moment of order 0 given by the formula

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*and the reducer associated φ. That gives the formula

_{n}The coefficient 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

be the classic recurrence relation in three terms. If

then the sequence {ρ_{n}} converges completely towards the Chebyshev density of the second form

- .

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 [1]

### Equinormal measures

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:

*t* describing an interval containing ]0, 1].

If μ is the secondary measure of ρ, that of ρ_{t} will be *t*μ.

The reducer of ρ_{t} is

by noting *G*(*x*) the reducer of μ.

Orthogonal polynomials for the measure ρ_{t} are clarified from *n* = 1 by the formula

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:

with *t* describing ]0, 2]. The value *t* = 2 gives the Chebyshev measure of the first form.

## A few beautiful applications

In the formulas below *G* is Catalan's constant, γ is the Euler's constant, β_{2n} is the Bernoulli number of order 2*n*, *H*_{2n+1} is the harmonic number of order 2*n*+1 and Ei is the Exponential integral function.

The notation indicating the 2 periodic function coinciding with on (−1, 1).

If the measure ρ is reducible and let φ be the associated reducer, one has the equality

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:

*c*_{1} indicates the moment of order 1 of ρ and *T*_{ρ} the operator

In addition, the sequence of secondary measures has applications in Quantum Mechanics. The sequence gives rise to the so-called *sequence of residual spectral densities* for specialized Pauli-Fierz Hamiltonians. This also provides a physical interpretation for the sequence of secondary measures. [1]

## See also

## References

- Mappings of open quantum systems onto chain representations and Markovian embeddings, M. P. Woods, R. Groux, A. W. Chin, S. F. Huelga, M. B. Plenio. https://arxiv.org/abs/1111.5262