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 $L^2(\; [0,1]\; ,R,\; ho)$

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

with

: $varphi(x)\; =\; lim\_\{varepsilon\; o\; 0+\}2int\_0^1frac\{(x-t)\; ho(t)\}\{(x-t)^2+varepsilon^2\}\; ,\; dt$

in the general case,

or:

: $varphi(x)\; =\; 2\; ho(x)\; ext\{ln\}left(frac\{x\}\{1-x\}\; ight)\; -\; 2\; int\_0^1frac\{\; ho(t)-\; ho(x)\}\{t-x\}\; ,\; dt$

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:

: $S\_\{mu\}(z)=z-c\_1-frac\{1\}\{S\_\{\; ho\}(z)\}$

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

: $f(x)=int\_0^1frac\{g(t)-g(x)\}\{t-x\}\; ho(t),dt$

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 $(P\_n)\_\{nin\; N\}$ of orthogonal polynomials for the inner product induced by $ho$. Let us call $(Q\_n)\_\{n\; in\; N\}$ 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 $ho$ 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:

: $S\_\{mu\}(z)=aleft(z-c\_1-frac\{1\}\{S\_\{\; ho\}(z)\}\; ight),$

$a$ 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 $L^2left(I,mathbf\; R,\; ho\; ight)$, the operator $T\_\; ho$ defined by $f(x)\; mapsto\; int\_I\; frac\{f(t)-f(x)\}\{t-x\}\; ho\; (t)dt$ creating the secondary polynomials can be furthered to a linear map connecting space $L^2left(I,mathbf\; R,\; ho\; ight)$ to $L^2left(I,mathbf\; R,mu\; ight)$ 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:

: $langle\; f/g\; angle\_\; ho\; -\; langle\; f/1\; angle\_\; ho\; imes\; langle\; g/1\; angle\_\; ho\; =\; langle\; T\_\; ho(f)/T\_\; ho\; (g)\; angle\_mu.$

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

The reducer $varphi$ of $ho$ is an antecedent of $frac\{\; ho\}\{mu\}$ 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: $langle\; f/varphi\; angle\_\; ho\; =\; langle\; T\_\; ho\; (f)/1\; angle\_\; ho$.

The operator $fmapsto\; \{varphi\; imes\; f\; -T\_\; ho\; (f)\}$ defined on the polynomials is prolonged in an isometry $S\_\; ho$ linking the closure of the space of these polynomials in $L^2left(I,mathbf\; R,frac\; \{\; ho^2\}\{mu\}\; ight)$ 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:

: $T\_\; hocirc\; S\_\; ho\; left(\; f\; ight)=frac\{\; ho\}\{mu\}\; imes\; left(f\; ight).$

Case of the Lebesgue measure and some other examples

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

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

: $2(2n+1)XP\_n(X)=-P\_\{n+1\}(X)+(2n+1)P\_n(X)-n^2P\_\{n-1\}(X).$

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

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 $C\_n(varphi)=-frac\{4sqrt\{2n+1\{n(n+1)\}$ for an odd index $n$.

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

:

and are normalized.

The reducer associated is defined by

: $varphi(x)=2left\; [ln(x)-int\_0^\{+infty\}e^\{-t\}ln|x-t|dt\; ight]\; .$

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

:

This coefficient $C\_n(varphi)$ 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

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

They are clarified by

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

and are normalized.

The reducer associated is defined by

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

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

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

for an odd index $n$.

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

Examples of non reducible measures.

Jacobi measure of density $ho(x)=frac\{2\}\{pi\}sqrt\{frac\{1-x\}\{x$ on (0, 1).

Chebyshev measure of the first form of density $ho(x)=frac\{1\}\{pisqrt\{1-x^2$ 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 $d\_0\; =c\_2\; -(c\_1)^2$ , ($c\_1$ 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 $ho\_1\; =frac\{mu\}\{d\_0\}$ 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 $ho\_\{n+1\}$ that is the secondary normalised measure deduced from $ho\_\{n\}$

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

: $ho\_n(x)=frac\{1\}\{d\_0^\{n-1\; frac\{\; ho(x)\}\{left(P\_\{n-1\}(x)\; frac\{varphi(x)\}\{2\}-Q\_\{n-1\}(x)\; ight)^2\; +\; pi^2\; ho^2(x)\; P\_\{n-1\}^2(x)\}.$

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 $varphi\_n$ 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 $left\; [0,1\; ight]$.

Let $xP\_n\; (x)=t\_nP\_\{n+1\}(x)+s\_nP\_n(x)+t\_\{n-1\}P\_\{n-1\}(x)$ be the classic reoccurrence relation in three terms.

If $lim\_\{n\; mapsto\; infty\}t\_n=frac\{1\}\{4\}$ and $lim\_\{n\; mapsto\; infty\}s\_n=frac\{1\}\{2\}$, then the sequence $nmapsto\; ho\_n$ converges completely towards the Chebyshev density of the second form $ho\_\{tch\}(x)=frac\{8\}\{pi\}sqrt\{x(1-x)\}$.

These conditions about limits are checked by a very broad class of traditional densities.

" 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: $ho\_\{t\}(x)=frac\{t\; ho(x)\}\{left\; [left(t-1\; ight)(x-c\_1)frac\{varphileft(x\; ight)\}\{2\}-t\; ight]\; ^2+pi^2\; ho^2(x)(t-1)^2(x-c\_1)^2\}$ , 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 : $varphi\_t(x)=frac\{2left(x-c\_1\; ight)-tG(x)\}\{left((x-c\_1)-tfrac\{G(x)\}\{2\}\; ight)^2+t^2pi^2mu^2(x)\}$ by noting $G(x)$ the reducer of $mu$.

Orthogonal polynoms for the measure $ho\_t$ are clarified from $n=1$ by the formula

: $P\_n^t(x)=frac\{1\}\{sqrt\{tleft\; [tP\_n(x)+(1-t)(x-c\_1)Q\_n(x)\; ight]$ 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: $ho\_t(x)=frac\{2tsqrt\{1-x^2\{pileft\; [t^2+4(1-t)x^2\; ight]\; \}$ , with $t$ describing] 0,2] . The value $t$=2 gives the Chebishev measure of the first form.

A few beautiful applications

: $forall\; p\; >1\; qquadfrac\{1\}\{ln(p)\}=frac\{1\}\{p-1\}+int\_0^\{+infty\}frac\{dx\}\{(x+p)(ln^2(x)+pi^2)\}.qquad$

: $gamma=int\_0^\{+infty\}frac\{ln(1+frac\{1\}\{x\})dx\}\{ln^2(x)+pi^2\}qquad$. (with $gamma$ the Euler's constant).

: $gamma=frac\{1\}\{2\}+int\_0^\{+infty\}frac\{overline\; \{(x+1)cos(pi\; x)\}\; dx\}\{x+1\}$.
(the notation $xmapsto\; overline\; \{(x+1)cos(pi\; x)\}$ indicating the 2 periodic function coinciding with $xmapsto\; (x+1)\; cos(pi\; x)$ on (−1, 1)).

:

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

:

:

:

: $qquad\; int\_0^\{+infty\}frac\{e^\{-alpha\; x\}dx\}\; \{Gamma(x+1)\}\; =\; e^\{e^\{-alpha\; -\; 1\; +\; int\_0^\{+infty\}\; frac\{1-e^\{-x\{left\; [(ln(x)+alpha)^2+pi^2\; ight]\; \}\; frac\{dx\}\{x\}.$

(for any real $alpha$)

:

(Ei indicate the integral exponentiel function here).

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

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

: $mbox\{Catalan\}\; =\; frac\{piln(2)\}\{8\}+sum\_\{n=0\}^\{n=infty\}(-1)^nfrac\{H\_\{2n+1\{2n+1\}.$

(The Catalan's constant is defined as $sum\_\{n=0\}^\{n=infty\}frac\{(-1)^n\}\{(2n+1)^2\}$ and $H\_\{2n+1\}=sum\_\{k=1\}^\{k=2n+1\}frac\{1\}\{k\}$) 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

: $int\_Ivarphi^2(x)\; ho(x)\; ,\; dx\; =\; frac\{4pi^2\}\{3\}int\_I\; ho^3(x)\; ,\; dx.$

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:

: $f(x)=int\_Ifrac\{g(t)-g(x)\}\{t-x\}\; ho(t)dtLeftrightarrow\; g(x)\; =\; (x-c\_1)f(x)\; -\; T\_\{mu\}(f(x))\; =\; frac\{varphi(x)mu(x)\}\{\; ho(x)\}f(x)-T\_\{\; ho\}\; left(frac\{mu(x)\}\{\; ho(x)\}f(x)\; ight)$

($c\_1$ indicates the moment of order 1 of $ho$ and $T\_\{\; ho\}$ the operator $g(x)mapsto\; int\_Ifrac\{g(t)-g(x)\}\{t-x\}\; ho(t),dt$).

ee also

* Orthogonal polynomials
* Probability

External links

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