# Secondary measure

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Secondary measure

In mathematics, the secondary measure associated with a measure of positive density $ho$ when there is one, is a measure of positive density $mu$, turning the secondary polynomials associated with the orthogonal polynomials for $ho$ 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 $L^2\left( \left[0,1\right] ,R, ho\right)$

: $forall x in \left[0,1\right] , ;mu\left(x\right)=frac\left\{ ho\left(x\right)\right\}\left\{frac\left\{varphi^2\left(x\right)\right\}\left\{4\right\} + pi^2 ho^2\left(x\right)\right\}$

with

: $varphi\left(x\right) = lim_\left\{varepsilon o 0+\right\}2int_0^1frac\left\{\left(x-t\right) ho\left(t\right)\right\}\left\{\left(x-t\right)^2+varepsilon^2\right\} , dt$

in the general case,

or:

: $varphi\left(x\right) = 2 ho\left(x\right) ext\left\{ln\right\}left\left(frac\left\{x\right\}\left\{1-x\right\} ight\right) - 2 int_0^1frac\left\{ ho\left(t\right)- ho\left(x\right)\right\}\left\{t-x\right\} , dt$

when $ho$ satisfy a Lipschitz condition.

This application $varphi$ is called the reducer of $ho.$

More generally, $mu$ et $ho$ are linked by their Stieltjes transformation with the following formula:

: $S_\left\{mu\right\}\left(z\right)=z-c_1-frac\left\{1\right\}\left\{S_\left\{ ho\right\}\left(z\right)\right\}$

in which $c_1$ is the moment of order 1 of the measure $ho$.

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

: $f\left(x\right)=int_0^1frac\left\{g\left(t\right)-g\left(x\right)\right\}\left\{t-x\right\} ho\left(t\right),dt$

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 $ho$ be a measure of positive density on an interval I and admitting moments of any order. We can build a family $\left(P_n\right)_\left\{nin N\right\}$ of orthogonal polynomials for the inner product induced by $ho$. Let us call $\left(Q_n\right)_\left\{n in N\right\}$ 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 $ho$ is called a secondary measure associated initial measure $ho$.

When $ho$ is a probability density function, a sufficient condition so that $mu$ , while admitting moments of any order can be a secondary measure associated with $ho$ is that its Stieltjes Transformation is given by an equality of the type:

: $S_\left\{mu\right\}\left(z\right)=aleft\left(z-c_1-frac\left\{1\right\}\left\{S_\left\{ ho\right\}\left(z\right)\right\} ight\right),$

$a$ is an arbitrary constant and $, c_1$ indicating the moment of order 1 of $ho$.

For $a=1$ we obtain the measure know as secondary, remarkable since for $ngeq1$ the norm of the polynomial $P_n$ for $ho$ coincides exactly with the norm of the secondary polynomial associated $Q_n$ when using the measure $mu$.

In this paramount case, and if the space generated by the orthogonal polynomials is dense in $L^2left\left(I,mathbf R, ho ight\right)$, the operator $T_ ho$ defined by $f\left(x\right) mapsto int_I frac\left\{f\left(t\right)-f\left(x\right)\right\}\left\{t-x\right\} ho \left(t\right)dt$ creating the secondary polynomials can be furthered to a linear map connecting space $L^2left\left(I,mathbf R, ho ight\right)$ to $L^2left\left(I,mathbf R,mu ight\right)$ and becomes isometric if limited to the hyperplane $H_ ho$ of the orthogonal functions with $P_0=1$.

For unspecified functions square integrable for $ho$ we obtain the more general formula of covariance:

: $langle f/g angle_ ho - langle f/1 angle_ ho imes langle g/1 angle_ ho = langle T_ ho\left(f\right)/T_ ho \left(g\right) angle_mu.$

The theory continues by introducing the concept of reducible measure, meaning that the quotient $frac\left\{ ho\right\}\left\{mu\right\}$ is element of $L^2left\left(I,mathbf R,mu ight\right)$. The following results are then established:

The reducer $varphi$ of $ho$ is an antecedent of $frac\left\{ ho\right\}\left\{mu\right\}$ for the operator $T_ ho$. (In fact the only antecedent which belongs to $H_ ho$).

For any function square integrable for $ho$, there is an equality known as the reducing formula: $langle f/varphi angle_ ho = langle T_ ho \left(f\right)/1 angle_ ho$.

The operator $fmapsto \left\{varphi imes f -T_ ho \left(f\right)\right\}$ defined on the polynomials is prolonged in an isometry $S_ ho$ linking the closure of the space of these polynomials in $L^2left\left(I,mathbf R,frac \left\{ ho^2\right\}\left\{mu\right\} ight\right)$ to the hyperplane $H_ ho$ provided with the norm induced by $ho$.

Under certain restrictive conditions the operator $S_ ho$ acts like the adjoint of $T_ ho$ for the inner product induced by $ho$.

Finally the two operators are also connected, provided the images in question are defined, by the fundamental formula of composition:

: $T_ hocirc S_ ho left\left( f ight\right)=frac\left\{ ho\right\}\left\{mu\right\} imes left\left(f ight\right).$

Case of the Lebesgue measure and some other examples

The Lebesgue measure on the standard interval $left \left[0,1 ight\right]$ is obtained by taking the constant density $ho\left(x\right)=1$.

The associated orthogonal polynomials are called Legendre polynomials and can be clarified by $P_n\left(x\right)=frac\left\{d^\left\{\left(n\right)\left\{dx^n\right\}left\left(x^n\left(1-x\right)^n ight\right)$. The norm of $P_n$ is worth $frac\left\{n!\right\}\left\{sqrt\left\{2n+1$. The reoccurrence relation in three terms is written:

: $2\left(2n+1\right)XP_n\left(X\right)=-P_\left\{n+1\right\}\left(X\right)+\left(2n+1\right)P_n\left(X\right)-n^2P_\left\{n-1\right\}\left(X\right).$

The reducer of this measure of Lebesgue is given by $varphi\left(x\right)=2lnleft\left(frac\left\{x\right\}\left\{1-x\right\} ight\right)$. The associated secondary measure is then clarified as : $mu\left(x\right)=frac\left\{1\right\}\left\{ln^2left\left(frac\left\{x\right\}\left\{1-x\right\} ight\right)+pi^2\right\}$.

If we normalize the polynomials of Legendre, the coefficients of Fourier of the reducer $varphi$ related to this orthonormal system are null for an even index and are given by $C_n\left(varphi\right)=-frac\left\{4sqrt\left\{2n+1\left\{n\left(n+1\right)\right\}$ for an odd index $n$.

The Laguerre polynomials are linked to the density $ho\left(x\right)=e^\left\{-x\right\}$ on the interval $I = left \left[0,+infty ight\right)$. They are clarified by

:

and are normalized.

The reducer associated is defined by

: $varphi\left(x\right)=2left \left[ln\left(x\right)-int_0^\left\{+infty\right\}e^\left\{-t\right\}ln|x-t|dt ight\right] .$

The coefficients of Fourier of the reducer $varphi$ related to the Laguerre polynoms are given by

:

This coefficient $C_n\left(varphi\right)$ 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 polynoms are linked to the Gaussian density

: $ho\left(x\right)=frac\left\{e^\left\{-frac\left\{x^2\right\}\left\{2\right\}\left\{sqrt\left\{2pi$ on $I= R.$

They are clarified by

: $H_n\left(x\right)=frac\left\{1\right\}\left\{sqrt\left\{n!e^\left\{frac\left\{x^2\right\}\left\{2frac\left\{d^n\right\}\left\{dx^n\right\}left\left(e^\left\{-frac\left\{x^2\right\}\left\{2 ight\right)$

and are normalized.

The reducer associated is defined by

: $varphi\left(x\right)=-frac\left\{2\right\}\left\{sqrt\left\{2piint_\left\{-infty\right\}^\left\{+infty\right\}te^\left\{-frac\left\{t^2\right\}\left\{2ln|x-t|,dt.$

The coefficients of Fourier of the reducer $varphi$ related to the system of Hermite polynoms are null for an even index and are given by

: $C_n\left(varphi\right)=\left(-1\right)^\left\{frac\left\{n+1\right\}\left\{2frac\left\{left\left(frac\left\{n-1\right\}\left\{2\right\} ight\right)!\right\}\left\{sqrt\left\{n!$

for an odd index $n$.

The Chebyshev measure of the second form. This is defined by the density $ho\left(x\right)=frac\left\{8\right\}\left\{pi\right\}sqrt\left\{x\left(1-x\right)\right\}$ 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 of density $ho\left(x\right)=frac\left\{2\right\}\left\{pi\right\}sqrt\left\{frac\left\{1-x\right\}\left\{x$ on (0, 1).

Chebyshev measure of the first form of density $ho\left(x\right)=frac\left\{1\right\}\left\{pisqrt\left\{1-x^2$ on (−1, 1).

equence of secondary measures

The secondary measure $mu$ associated with a probability density function $ho$ has its moment of order 0 gived by the formula $d_0 =c_2 -\left(c_1\right)^2$ , ($c_1$ and $c_2$ indicating the respective moments of order 1 and 2 of $ho$).

To be able to iterate the process then one 'normalize' $mu$ while defining $ho_1 =frac\left\{mu\right\}\left\{d_0\right\}$ which becomes in its turn a density of probability called naturally the normalised secondary measure associated with $ho$.

We can then create from $ho_1$ a secondary normalised measure $ho_2$, then defining $ho_3$ from $ho_2$ and so on. We can therefore see a sequence of successive secondary measures, created from $ho_0= ho$, is such that $ho_\left\{n+1\right\}$ that is the secondary normalised measure deduced from $ho_\left\{n\right\}$

It is possible to clarify the density $ho_n$ by using the orthogonal polynomials $P_n$ for $ho$, the secondary polynoms $Q_n$ and the reducer associated $varphi$. That gives the formula

: $ho_n\left(x\right)=frac\left\{1\right\}\left\{d_0^\left\{n-1 frac\left\{ ho\left(x\right)\right\}\left\{left\left(P_\left\{n-1\right\}\left(x\right) frac\left\{varphi\left(x\right)\right\}\left\{2\right\}-Q_\left\{n-1\right\}\left(x\right) ight\right)^2 + pi^2 ho^2\left(x\right) P_\left\{n-1\right\}^2\left(x\right)\right\}.$

The coefficient $d_0^\left\{n-1\right\}$ is easily obtained starting from the leading coefficients of the polynomials $P_\left\{n-1\right\}$ and $P_n$. We can also clarify the reducer $varphi_n$ associated with $ho_n$, as well as the orthogonal polynoms corresponding to $ho_n$.

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 $left \left[0,1 ight\right]$.

Let $xP_n \left(x\right)=t_nP_\left\{n+1\right\}\left(x\right)+s_nP_n\left(x\right)+t_\left\{n-1\right\}P_\left\{n-1\right\}\left(x\right)$ be the classic reoccurrence relation in three terms.

If $lim_\left\{n mapsto infty\right\}t_n=frac\left\{1\right\}\left\{4\right\}$ and $lim_\left\{n mapsto infty\right\}s_n=frac\left\{1\right\}\left\{2\right\}$, then the sequence $nmapsto ho_n$ converges completely towards the Chebyshev density of the second form $ho_\left\{tch\right\}\left(x\right)=frac\left\{8\right\}\left\{pi\right\}sqrt\left\{x\left(1-x\right)\right\}$.

" 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 $ho$ has its moment of order 1 equal to $c_1$, then these densities equinormal with $ho$ are given by a formula of the type: $ho_\left\{t\right\}\left(x\right)=frac\left\{t ho\left(x\right)\right\}\left\{left \left[left\left(t-1 ight\right)\left(x-c_1\right)frac\left\{varphileft\left(x ight\right)\right\}\left\{2\right\}-t ight\right] ^2+pi^2 ho^2\left(x\right)\left(t-1\right)^2\left(x-c_1\right)^2\right\}$ , t describing an interval containing] 0, 1] .

If $mu$ is the secondary measure of $ho$,that of $ho_t$ will be $tmu$.

The reducer of $ho_t$ is : $varphi_t\left(x\right)=frac\left\{2left\left(x-c_1 ight\right)-tG\left(x\right)\right\}\left\{left\left(\left(x-c_1\right)-tfrac\left\{G\left(x\right)\right\}\left\{2\right\} ight\right)^2+t^2pi^2mu^2\left(x\right)\right\}$ by noting $G\left(x\right)$ the reducer of $mu$.

Orthogonal polynoms for the measure $ho_t$ are clarified from $n=1$ by the formula

: $P_n^t\left(x\right)=frac\left\{1\right\}\left\{sqrt\left\{tleft \left[tP_n\left(x\right)+\left(1-t\right)\left(x-c_1\right)Q_n\left(x\right) ight\right]$ 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 $ho_t$ is the Dirac measure concentrated at $c_1$.

For example, the equinormal densities with the Chebyshev measure of the second form are defined by: $ho_t\left(x\right)=frac\left\{2tsqrt\left\{1-x^2\left\{pileft \left[t^2+4\left(1-t\right)x^2 ight\right] \right\}$ , with $t$ describing] 0,2] . The value $t$=2 gives the Chebishev measure of the first form.

A few beautiful applications

: $forall p >1 qquadfrac\left\{1\right\}\left\{ln\left(p\right)\right\}=frac\left\{1\right\}\left\{p-1\right\}+int_0^\left\{+infty\right\}frac\left\{dx\right\}\left\{\left(x+p\right)\left(ln^2\left(x\right)+pi^2\right)\right\}.qquad$

: $gamma=int_0^\left\{+infty\right\}frac\left\{ln\left(1+frac\left\{1\right\}\left\{x\right\}\right)dx\right\}\left\{ln^2\left(x\right)+pi^2\right\}qquad$. (with $gamma$ the Euler's constant).

: $gamma=frac\left\{1\right\}\left\{2\right\}+int_0^\left\{+infty\right\}frac\left\{overline \left\{\left(x+1\right)cos\left(pi x\right)\right\} dx\right\}\left\{x+1\right\}$.
(the notation $xmapsto overline \left\{\left(x+1\right)cos\left(pi x\right)\right\}$ indicating the 2 periodic function coinciding with $xmapsto \left(x+1\right) cos\left(pi x\right)$ on (−1, 1)).

:

(with $E$ is the floor function and the Bernoulli number of order $2n$).

:

:

:

: $qquad int_0^\left\{+infty\right\}frac\left\{e^\left\{-alpha x\right\}dx\right\} \left\{Gamma\left(x+1\right)\right\} = e^\left\{e^\left\{-alpha - 1 + int_0^\left\{+infty\right\} frac\left\{1-e^\left\{-x\left\{left \left[\left(ln\left(x\right)+alpha\right)^2+pi^2 ight\right] \right\} frac\left\{dx\right\}\left\{x\right\}.$

(for any real $alpha$)

:

(Ei indicate the integral exponentiel function here).

: $frac\left\{23\right\}\left\{15\right\}-ln\left(2\right) = sum_\left\{n=0\right\}^\left\{n=+infty\right\} frac\left\{1575\right\}\left\{2\left(n+1\right)\left(2n+1\right)\left(4n-3\right)\left(4n-1\right)\left(4n+1\right)\left(4n+5\right)\left(4n+7\right)\left(4n+9\right)\right\}$

: $mbox\left\{Catalan \right\} = sum_\left\{k=0\right\}^\left\{k=+infty\right\} frac\left\{\left(-1\right)^k\right\}\left\{4^\left\{k+1 left\left(frac\left\{1\right\}\left\{\left(4k+3\right)^2\right\}+frac\left\{2\right\}\left\{\left(4k+2\right)^2\right\}+frac\left\{2\right\}\left\{\left(4k+1\right)^2\right\} ight\right)+frac\left\{piln\left(2\right)\right\}\left\{8\right\}$

: $mbox\left\{Catalan\right\} = frac\left\{piln\left(2\right)\right\}\left\{8\right\}+sum_\left\{n=0\right\}^\left\{n=infty\right\}\left(-1\right)^nfrac\left\{H_\left\{2n+1\left\{2n+1\right\}.$

(The Catalan's constant is defined as $sum_\left\{n=0\right\}^\left\{n=infty\right\}frac\left\{\left(-1\right)^n\right\}\left\{\left(2n+1\right)^2\right\}$ and $H_\left\{2n+1\right\}=sum_\left\{k=1\right\}^\left\{k=2n+1\right\}frac\left\{1\right\}\left\{k\right\}$) is the harmonic number of order $2n+1$.

If the measure $ho$ is reducible and let $varphi$ be the associated reducer, one has the equality

: $int_Ivarphi^2\left(x\right) ho\left(x\right) , dx = frac\left\{4pi^2\right\}\left\{3\right\}int_I ho^3\left(x\right) , dx.$

If the measure $ho$ is reducible with $mu$ the associated reducer, then if $f$ is square integrable for $mu$, and if $g$ is sqare integrable for $ho$ and is orthogonal with $P_0=1$ one has equivalence:

: $f\left(x\right)=int_Ifrac\left\{g\left(t\right)-g\left(x\right)\right\}\left\{t-x\right\} ho\left(t\right)dtLeftrightarrow g\left(x\right) = \left(x-c_1\right)f\left(x\right) - T_\left\{mu\right\}\left(f\left(x\right)\right) = frac\left\{varphi\left(x\right)mu\left(x\right)\right\}\left\{ ho\left(x\right)\right\}f\left(x\right)-T_\left\{ ho\right\} left\left(frac\left\{mu\left(x\right)\right\}\left\{ ho\left(x\right)\right\}f\left(x\right) ight\right)$

($c_1$ indicates the moment of order 1 of $ho$ and $T_\left\{ ho\right\}$ the operator $g\left(x\right)mapsto int_Ifrac\left\{g\left(t\right)-g\left(x\right)\right\}\left\{t-x\right\} ho\left(t\right),dt$).

ee also

* Orthogonal polynomials
* Probability

* [http://perso.orange.fr/roland.groux personal page of Roland Groux about the theory of secondary measures]

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