Secondary measure: Wikis

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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.

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

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

with

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

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 $ρ$ satisfy a Lipschitz condition.

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

More generally, $μ$ et $ρ$ 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 $ρ$ .

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

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

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 $ρ$ 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 $ρ$ . 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 $ρ$ 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:

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

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

For $a = 1$ we obtain the measure know as secondary, remarkable since for $n\geq1$ 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 $L^2\left(I,\mathbf R,\rho \right)$ , the operator $T ρ$ 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 $H ρ$ of the orthogonal functions with $P 0 = 1$ .

For unspecified functions square integrable for $ρ$ 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 $ρ$ is an antecedent of $\frac{\rho}{\mu}$ 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: $\langle f/\varphi \rangle_\rho = \langle T_\rho (f)/1 \rangle_\rho$ .

The operator $f\mapsto {\varphi\times f -T_\rho (f)}$ defined on the polynomials is prolonged in an isometry $S ρ$ linking the closure of the space of these polynomials in $L^2\left(I,\mathbf R,\frac {\rho^2}{\mu}\right)$ to the hyperplane $H ρ$ 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:

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

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 $ρ(x) = 1$ .

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 $\frac{n!}{\sqrt{2n+1}}$ . 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 $\varphi(x)=2\ln\left(\frac{x}{1-x}\right)$ . The associated secondary measure is then clarified as : $\mu(x)=\frac{1}{\ln^2\left(\frac{x}{1-x}\right)+\pi^2}$ .

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

The Laguerre polynomials are linked to the density $ρ(x) = e - x$ on the interval $I = \left[0,+\infty \right)$ .

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

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

The coefficients of Fourier of the reducer $\varphi$ related to the Laguerre polynoms 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 polynoms 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

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

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

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

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 $μ$ associated with a probability density function $ρ$ has its moment of order 0 given by the formula $d 0 = c 2 - (c 1) 2$ , ( $c 1$ 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 $\rho_1 =\frac{\mu}{d_0}$ 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 polynoms $Q n$ and the reducer associated $\varphi$ . That gives the formula

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

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 $ρ n$ , as well as the orthogonal polynoms 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 $\left[0,1\right]$ .

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 $\lim_{n \mapsto \infty}t_n=\frac{1}{4}$ 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 $ρ$ has its moment of order 1 equal to $c 1$ , then these densities equinormal with $ρ$ are given by a formula of the type: $\rho_{t}(x)=\frac{t\rho(x)}{\left[\left(t-1\right)(x-c_1)\frac{\varphi\left(x\right)}{2}-t\right]^2+\pi^2\rho^2(x)(t-1)^2(x-c_1)^2}$ , t describing an interval containing]0, 1].

If $μ$ is the secondary measure of $ρ$ ,that of $ρ t$ will be $t μ$ .

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

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

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

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

A few beautiful applications

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

$\gamma=\int_0^{+\infty}\frac{\ln(1+\frac{1}{x})dx}{\ln^2(x)+\pi^2}\qquad$ . (with

γ

the Euler's constant).

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

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

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

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

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

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

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

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

(for any real $α$ )

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

(Ei indicate the integral exponentiel function here).

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

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

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

(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 $2 n + 1$ .

If the measure $ρ$ 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 $ρ$ is reducible with $μ$ the associated reducer, then if $f$ is square integrable for $μ$ , and if $g$ is sqare integrable for $ρ$ and is orthogonal with $P 0 = 1$ one has equivalence:

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