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 L²([0, 1], R, ρ)

[math]\displaystyle{ \forall x \in [0,1], \qquad \mu(x)=\frac{\rho(x)}{\frac{\varphi^2(x)}{4} + \pi^2\rho^2(x)} }[/math]

with

[math]\displaystyle{ \varphi(x) = \lim_{\varepsilon \to 0^+} 2\int_0^1\frac{(x-t)\rho(t)}{(x-t)^2+\varepsilon^2} \, dt }[/math]

in the general case, or:

[math]\displaystyle{ \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 }[/math]

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:

[math]\displaystyle{ S_{\mu}(z)=z-c_1-\frac{1}{S_{\rho}(z)} }[/math]

in which c₁ 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

[math]\displaystyle{ f(x)=\int_0^1\frac{g(t)-g(x)}{t-x}\rho(t)\,dt }[/math]

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_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:

[math]\displaystyle{ S_{\mu}(z)=a\left(z-c_1-\frac{1}{S_{\rho}(z)}\right), }[/math]

a is an arbitrary constant and c₁ 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_n when using the measure μ.

In this paramount case, and if the space generated by the orthogonal polynomials is dense in L²(I, R, ρ), the operator T_ρ defined by

[math]\displaystyle{ f(x) \mapsto \int_I \frac{f(t)-f(x)}{t-x}\rho (t)dt }[/math]

creating the secondary polynomials can be furthered to a linear map connecting space L²(I, R, ρ) to L²(I, R, μ) and becomes isometric if limited to the hyperplane H_ρ of the orthogonal functions with P₀ = 1.

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

[math]\displaystyle{ \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. }[/math]

The theory continues by introducing the concept of reducible measure, meaning that the quotient ρ/μ is element of L²(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:

[math]\displaystyle{ \langle f/\varphi \rangle_\rho = \langle T_\rho (f)/1 \rangle_\rho }[/math].

The operator

[math]\displaystyle{ f\mapsto \varphi\times f -T_\rho (f) }[/math]

defined on the polynomials is prolonged in an isometry S_ρ linking the closure of the space of these polynomials in L²(I, R, ρ²μ⁻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:

[math]\displaystyle{ T_\rho\circ S_\rho \left( f\right)=\frac{\rho}{\mu}\times (f). }[/math]

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

[math]\displaystyle{ P_n(x)=\frac{d^n}{dx^n}\left(x^n(1-x)^n\right). }[/math]

The norm of P_n is worth

[math]\displaystyle{ \frac{n!}{\sqrt{2n+1}}. }[/math]

The recurrence relation in three terms is written:

[math]\displaystyle{ 2(2n+1)XP_n(X)=-P_{n+1}(X)+(2n+1)P_n(X)-n^2P_{n-1}(X). }[/math]

The reducer of this measure of Lebesgue is given by

[math]\displaystyle{ \varphi(x)=2\ln\left(\frac{x}{1-x}\right). }[/math]

The associated secondary measure is then clarified as

[math]\displaystyle{ \mu(x)=\frac{1}{\ln^2\left(\frac{x}{1-x}\right)+\pi^2} }[/math].

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

[math]\displaystyle{ C_n(\varphi)=-\frac{4\sqrt{2n+1}}{n(n+1)} }[/math]

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

[math]\displaystyle{ L_n(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}(x^ne^{-x})=\sum_{k=0}^{n}\binom{n}{k}(-1)^k\frac{x^k}{k!} }[/math]

and are normalized.

The reducer associated is defined by

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

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

[math]\displaystyle{ C_n(\varphi)=-\frac{1}{n}\sum_{k=0}^{n-1}\frac{1}{\binom{n-1}{k}}. }[/math]

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

[math]\displaystyle{ \rho(x)=\frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}} }[/math]

on I = R.

They are clarified by

[math]\displaystyle{ H_n(x)=\frac{1}{\sqrt{n!}}e^{\frac{x^2}{2}}\frac{d^n}{dx^n}\left(e^{-\frac{x^2}{2}}\right) }[/math]

and are normalized.

The reducer associated is defined by

[math]\displaystyle{ \varphi(x)=-\frac{2}{\sqrt{2\pi}}\int_{-\infty}^{\infty}te^{-\frac{t^2}{2}}\ln|x-t|\,dt. }[/math]

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

[math]\displaystyle{ C_n(\varphi)=(-1)^{\frac{n+1}{2}}\frac{\left(\frac{n-1}{2}\right)!}{\sqrt{n!}} }[/math]

for an odd index n.

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

[math]\displaystyle{ \rho(x)=\frac{8}{\pi}\sqrt{x(1-x)} }[/math]

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

[math]\displaystyle{ \rho(x)=\frac{2}{\pi}\sqrt{\frac{1-x}{x}}. }[/math]

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

[math]\displaystyle{ \rho(x)=\frac{1}{\pi\sqrt{1-x^2}}. }[/math]

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

[math]\displaystyle{ d_0 = c_2 -c_1^2, }[/math]

where c₁ and c₂ indicating the respective moments of order 1 and 2 of ρ.

To be able to iterate the process then, one 'normalizes' μ while defining ρ₁ = μ/d₀ which becomes in its turn a density of probability called naturally the normalised secondary measure associated with ρ.

We can then create from ρ₁ a secondary normalised measure ρ₂, then defining ρ₃ from ρ₂ and so on. We can therefore see a sequence of successive secondary measures, created from ρ₀ = 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_n and the reducer associated φ. That gives the formula

[math]\displaystyle{ \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)}. }[/math]

The coefficient [math]\displaystyle{ d_0^{n-1} }[/math] 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

[math]\displaystyle{ xP_n (x)=t_nP_{n+1}(x)+s_nP_n(x)+t_{n-1}P_{n-1}(x) }[/math]

be the classic recurrence relation in three terms. If

[math]\displaystyle{ \lim_{n \mapsto \infty} t_n=\tfrac{1}{4}, \quad \lim_{n \mapsto \infty} s_n =\tfrac{1}{2}, }[/math]

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

[math]\displaystyle{ \rho_{tch}(x)=\frac{8}{\pi}\sqrt{x(1-x)} }[/math].

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

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 c₁, then these densities equinormal with ρ are given by a formula of the type:

[math]\displaystyle{ \rho_{t}(x)=\frac{t\rho(x)}{\left (\tfrac{1}{2}(t-1)(x-c_1)\varphi(x)-t\right )^2+\pi^2\rho^2(x)(t-1)^2(x-c_1)^2}, }[/math]

t describing an interval containing ]0, 1].

If μ is the secondary measure of ρ, that of ρ_t will be tμ.

The reducer of ρ_t is

[math]\displaystyle{ \varphi_t(x)=\frac{2 (x-c_1)-tG(x)}{\left((x-c_1)-t\tfrac{1}{2}G(x) \right)^2+t^2\pi^2\mu^2(x)} }[/math]

by noting G(x) the reducer of μ.

Orthogonal polynomials for the measure ρ_t are clarified from n = 1 by the formula

[math]\displaystyle{ P_n^t(x)=\frac{tP_n(x)+(1-t)(x-c_1)Q_n(x)}{\sqrt{t}} }[/math]

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 c₁.

For example, the equinormal densities with the Chebyshev measure of the second form are defined by:

[math]\displaystyle{ \rho_t(x)=\frac{2t\sqrt{1-x^2}}{\pi\left[t^2+4(1-t)x^2\right]}, }[/math]

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 2n, H_2n+1 is the harmonic number of order 2n+1 and Ei is the Exponential integral function.

[math]\displaystyle{ \frac{1}{\ln(p)} = \frac{1}{p-1}+\int_0^{\infty}\frac{1}{(x+p)(\ln^2(x)+\pi^2)} dx \qquad \qquad \forall p \gt 1 }[/math]

[math]\displaystyle{ \gamma = \int_0^{\infty}\frac{\ln(1+\frac{1}{x})}{\ln^2(x)+\pi^2} dx }[/math]

[math]\displaystyle{ \gamma = \frac{1}{2}+\int_0^{\infty} \frac{\overline {(x+1)\cos(\pi x)}}{x+1} dx }[/math]

The notation [math]\displaystyle{ x\mapsto \overline {(x+1)\cos(\pi x)} }[/math] indicating the 2 periodic function coinciding with [math]\displaystyle{ x\mapsto (x+1) \cos(\pi x) }[/math] on (−1, 1).

[math]\displaystyle{ \gamma = \frac{1}{2} + \sum_{k=1}^{n} \frac{\beta_{2k}}{2k} - \frac{\beta_{2n}}{\zeta(2n)}\int_1^{\infty} \lfloor t \rfloor \cos(2\pi t) t^{-2n-1} dt }[/math]

[math]\displaystyle{ \beta_k = \frac{(-1)^kk!}{\pi} \text{Im} \left(\int_{-\infty}^{\infty} \frac{e^x}{(1+e^x)(x-i\pi)^k} dx \right) }[/math]

[math]\displaystyle{ \int_0^1\ln^{2n}\left(\frac{x}{1-x}\right)\,dx = (-1)^{n+1}(2^{2n}-2)\beta_{2n}\pi^{2n} }[/math]

[math]\displaystyle{ \int_0^1\cdots \int_0^1 \left(\sum_{k=1}^{2n} \frac{\ln(t_k)}{\prod_{i \neq k}(t_k-t_i)}\right)\, dt_1 \cdots dt_{2n}=\tfrac{1}{2}(-1)^{n+1}(2\pi)^{2n}\beta_{2n} }[/math]

[math]\displaystyle{ \int_0^{\infty}\frac{e^{-\alpha x}}{\Gamma(x+1)}dx = e^{e^{-\alpha}}-1+\int_0^{\infty} \frac{1-e^{-x}}{(\ln(x)+\alpha)^2+\pi^2}\frac{dx}{x} \qquad \qquad \forall \alpha \in \mathbf{R} }[/math]

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

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

[math]\displaystyle{ G= \sum_{k=0}^{\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}{8}\ln(2) }[/math]

[math]\displaystyle{ G= \frac{\pi}{8}\ln(2)+\sum_{n=0}^{\infty}(-1)^n\frac{H_{2n+1}}{2n+1}. }[/math]

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

[math]\displaystyle{ \int_I\varphi^2(x)\rho(x) \, dx = \frac{4\pi^2}{3}\int_I\rho^3(x) \, dx. }[/math]

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 P₀ = 1 one has equivalence:

[math]\displaystyle{ 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) }[/math]

c₁ indicates the moment of order 1 of ρ and T_ρ the operator

[math]\displaystyle{ g(x)\mapsto \int_I\frac{g(t)-g(x)}{t-x}\rho(t)\,dt. }[/math]

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.

See also

• Orthogonal polynomials
• Probability

External links

• personal page of Roland Groux about the theory of secondary measures