Secondary measure

In mathematics, the secondary measure associated with a measure of positive density $\rho$ when there is one, is a measure of positive density $\mu$ , turning the secondary polynomials associated with the orthogonal polynomials for $\rho$ into an orthogonal system.

1 Introduction
2 The broad outlines of the theory
3 Case of the Lebesgue measure and some other examples
4 Sequence of secondary measures
5 A few beautiful applications
6 See also
7 External links

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

$\forall x \in [0,1], \; \mu(x)=\frac{\rho(x)}{\frac{\varphi^2(x)}{4} %2B \pi^2\rho^2(x)}$

with

$\varphi(x) = \lim_{\varepsilon \to 0%2B} 2\int_0^1\frac{(x-t)\rho(t)}{(x-t)^2%2B\varepsilon^2} \, dt$

in the general case,

or:

$\varphi(x) = 2\rho(x)\text{ln}\left(\frac{x}{1-x}\right) - 2 \int_0^1\frac{\rho(t)-\rho(x)}{t-x} \, dt$

when $\rho$ satisfy a Lipschitz condition.

This application $\varphi$ is called the reducer of $\rho.$

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

$S_{\mu}(z)=z-c_1-\frac{1}{S_{\rho}(z)}$

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

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

$f(x)=\int_0^1\frac{g(t)-g(x)}{t-x}\rho(t)\,dt$

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

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

$S_{\mu}(z)=a\left(z-c_1-\frac{1}{S_{\rho}(z)}\right),$

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

For $a=1$ we obtain the measure known as secondary, remarkable since for $n\geq1$ the norm of the polynomial $P_n$ for $\rho$ 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 $L^2\left(I,\mathbf R,\rho \right)$ , the operator $T_\rho$ defined by $f(x) \mapsto \int_I \frac{f(t)-f(x)}{t-x}\rho (t)dt$ creating the secondary polynomials can be furthered to a linear map connecting space $L^2\left(I,\mathbf R,\rho \right)$ to $L^2\left(I,\mathbf R,\mu \right)$ and becomes isometric if limited to the hyperplane $H_\rho$ of the orthogonal functions with $P_0=1$ .

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

$\langle f/g \rangle_\rho - \langle f/1 \rangle_\rho\times \langle g/1\rangle_\rho = \langle T_\rho(f)/T_\rho (g) \rangle_\mu.$

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

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

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

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

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

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

$T_\rho\circ S_\rho \left( f\right)=\frac{\rho}{\mu}\times \left(f \right).$

Case of the Lebesgue measure and some other examples

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

The associated orthogonal polynomials are called Legendre polynomials and can be clarified by $P_n(x)=\frac{d^{(n)}}{dx^n}\left(x^n(1-x)^n\right)$ . The norm of $P_n$ is worth $\frac{n!}{\sqrt{2n%2B1}}$ . The recurrence relation in three terms is written:

$2(2n%2B1)XP_n(X)=-P_{n%2B1}(X)%2B(2n%2B1)P_n(X)-n^2P_{n-1}(X).$

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

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 $C_n(\varphi)=-\frac{4\sqrt{2n%2B1}}{n(n%2B1)}$ for an odd index $n$ .

The Laguerre polynomials are linked to the density $\rho(x)=e^{-x}$ on the interval $I = \left[0,%2B\infty \right)$ .

They are clarified by

$L_n(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}(x^ne^{-x})=\sum_{k=0}^{k=n}\binom{n}{k}(-1)^k\frac{x^k}{k!}$

and are normalized.

The reducer associated is defined by

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

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

$C_n(\varphi)=-\frac{1}{n}\sum_{k=0}^{k=n-1}\frac{1}{\binom{n-1}{k}}.$

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 polynomials are linked to the Gaussian density

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

They are clarified by

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

and are normalized.

The reducer associated is defined by

$\varphi(x)=-\frac{2}{\sqrt{2\pi}}\int_{-\infty}^{%2B\infty}te^{-\frac{t^2}{2}}\ln|x-t|\,dt.$

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

$C_n(\varphi)=(-1)^{\frac{n%2B1}{2}}\frac{\left(\frac{n-1}{2}\right)!}{\sqrt{n!}}$

for an odd index $n$ .

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

Chebyshev measure of the first form of density $\rho(x)=\frac{1}{\pi\sqrt{1-x^2}}$ on (−1, 1).

Sequence of secondary measures

The secondary measure $\mu$ associated with a probability density function $\rho$ has its moment of order 0 given 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 $\rho$ ).

To be able to iterate the process then, one 'normalizes' $\mu$ while defining $\rho_1 =\frac{\mu}{d_0}$ which becomes in its turn a density of probability called naturally the normalised secondary measure associated with $\rho$ .

We can then create from $\rho_1$ a secondary normalised measure $\rho_2$ , then defining $\rho_3$ from $\rho_2$ and so on. We can therefore see a sequence of successive secondary measures, created from $\rho_0=\rho$ , is such that $\rho_{n%2B1}$ that is the secondary normalised measure deduced from $\rho_{n}$

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

$\rho_n(x)=\frac{1}{d_0^{n-1}} \frac{\rho(x)}{\left(P_{n-1}(x) \frac{\varphi(x)}{2}-Q_{n-1}(x)\right)^2 %2B \pi^2\rho^2(x) P_{n-1}^2(x)}.$

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 $\rho_n$ , as well as the orthogonal polynomials corresponding to $\rho_n$ .

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

Let $xP_n (x)=t_nP_{n%2B1}(x)%2Bs_nP_n(x)%2Bt_{n-1}P_{n-1}(x)$ be the classic recurrence 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 $n\mapsto \rho_n$ converges completely towards the Chebyshev density of the second form $\rho_{tch}(x)=\frac{8}{\pi}\sqrt{x(1-x)}$ .

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

If $\mu$ is the secondary measure of $\rho$ ,that of $\rho_t$ will be $t\mu$ .

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

Orthogonal polynomials for the measure $\rho_t$ are clarified from $n=1$ by the formula

$P_n^t(x)=\frac{1}{\sqrt{t}}\left[tP_n(x)%2B(1-t)(x-c_1)Q_n(x)\right]$ 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 $\rho_t$ is the Dirac measure concentrated at $c_1$ .

For example, the equinormal densities with the Chebyshev measure of the second form are defined by: $\rho_t(x)=\frac{2t\sqrt{1-x^2}}{\pi\left[t^2%2B4(1-t)x^2\right]}$ , with $t$ describing]0,2]. The value $t$ =2 gives the Chebyshev measure of the first form.

A few beautiful applications

$\forall p >1 \qquad\frac{1}{\ln(p)}=\frac{1}{p-1}%2B\int_0^{%2B\infty}\frac{dx}{(x%2Bp)(\ln^2(x)%2B\pi^2)}.\qquad$

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

$\gamma=\frac{1}{2}%2B\int_0^{%2B\infty}\frac{\overline {(x%2B1)\cos(\pi x)} dx}{x%2B1}$ .

(the notation $x\mapsto \overline {(x%2B1)\cos(\pi x)}$ indicating the 2 periodic function coinciding with $x\mapsto (x%2B1) \cos(\pi x)$ on (−1, 1)).

$\gamma = \frac{1}{2} %2B \sum_{k=1}^{k=n} \frac{\beta_{2k}}{2k} - \frac{\beta_{2n}}{\zeta(2n)} \int_1^{%2B\infty} \frac{E(t)\cos(2\pi t)dt}{t^{2n%2B1}}$

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

$\beta_k = \frac{(-1)^kk!}{\pi} Im\left(\int_{-\infty}^{\infty} \frac{e^x \, dx}{(1%2Be^x)(x-i\pi)^k}\right).$

$\int_0^1\ln^{2n}\left(\frac{x}{1-x}\right)\,dx = (-1)^{n%2B1}2(2^{2n-1}-1)\beta_{2n}\pi^{2n}.$

$\int_0^1 \int_0^1\cdots \int_0^1 \left(\sum_{k=1}^{k=2n} \frac{ln(t_k)} {\prod_{i\not=k}(t_k-t_i)}\right) \, dt_1 \, dt_2\cdots dt_{2n} = \frac{(-1)^{n%2B1}(2\pi)^{2n}\beta_{2n}}{2}.$

$\qquad \int_0^{%2B\infty}\frac{e^{-\alpha x}dx} {\Gamma(x%2B1)} = e^{e^{-\alpha}} - 1 %2B \int_0^{%2B\infty} \frac{1-e^{-x}}{\left[(\ln(x)%2B\alpha)^2%2B\pi^2\right]} \frac{dx}{x}.$

(for any real $\alpha$ )

$\sum_{n=1}^{n=%2B\infty} \left(\frac{1}{n}\sum_{k=0}^{k=n-1} \frac{1}{\binom{n-1}{k}}\right)^2 = \frac{4\pi^2}{9}=\int_0^{%2B\infty}4[\mathrm {Ei} (1,-x)%2Bi\pi]^2e^{-3x} \, dx.$

(Ei indicate the integral exponentiel function here).

$\frac{23}{15}-\ln(2) = \sum_{n=0}^{n=%2B\infty} \frac{1575}{2(n%2B1)(2n%2B1)(4n-3)(4n-1)(4n%2B1)(4n%2B5)(4n%2B7)(4n%2B9)}$

$\mbox{Catalan } = \sum_{k=0}^{k=%2B\infty} \frac{(-1)^k}{4^{k%2B1}} \left(\frac{1}{(4k%2B3)^2}%2B\frac{2}{(4k%2B2)^2}%2B\frac{2}{(4k%2B1)^2}\right)%2B\frac{\pi\ln(2)}{8}$

$\mbox{Catalan} = \frac{\pi\ln(2)}{8}%2B\sum_{n=0}^{n=\infty}(-1)^n\frac{H_{2n%2B1}}{2n%2B1}.$

(The Catalan's constant is defined as $\sum_{n=0}^{n=\infty}\frac{(-1)^n}{(2n%2B1)^2}$ and $H_{2n%2B1}=\sum_{k=1}^{k=2n%2B1}\frac{1}{k}$ ) is the harmonic number of order $2n%2B1$ .

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

$\int_I\varphi^2(x)\rho(x) \, dx = \frac{4\pi^2}{3}\int_I\rho^3(x) \, dx.$

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

$f(x)=\int_I\frac{g(t)-g(x)}{t-x}\rho(t)dt\Leftrightarrow g(x) = (x-c_1)f(x) - T_{\mu}(f(x)) = \frac{\varphi(x)\mu(x)}{\rho(x)}f(x)-T_{\rho} \left(\frac{\mu(x)}{\rho(x)}f(x)\right)$

( $c_1$ indicates the moment of order 1 of $\rho$ and $T_{\rho}$ the operator $g(x)\mapsto \int_I\frac{g(t)-g(x)}{t-x}\rho(t)\,dt$ ).

External links

personal page of Roland Groux about the theory of secondary measures