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)

\forall x \in [0,1], \qquad \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 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:

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

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

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} 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:

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

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

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 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:

\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 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:

\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 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:

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

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

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(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)=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 φ 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 = [0, ∞). They are clarified by

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!}

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 φ related to the Laguerre polynomials are given by

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

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

\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 φ related to the system of Hermite polynomials 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 on (0, 1) of density

\rho(x)=\frac{2}{\pi}\sqrt{\frac{1-x}{x}}.

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

\rho(x)=\frac{1}{\pi\sqrt{1-x^2}}.

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,

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

\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 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

xP_n (x)=t_nP_{n+1}(x)+s_nP_n(x)+t_{n-1}P_{n-1}(x)

be the classic recurrence relation in three terms. If

\lim_{n \mapsto \infty} t_n=\tfrac{1}{4}, \quad \lim_{n \mapsto \infty} s_n =\tfrac{1}{2},

then the sequence {ρ_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. A derivation of the sequence of secondary measures and convergence can be found in ^[1]

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:

\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},

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

\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)}

by noting G(x) the reducer of μ.

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

P_n^t(x)=\frac{tP_n(x)+(1-t)(x-c_1)Q_n(x)}{\sqrt{t}}

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:

\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 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.

\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 > 1

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

\gamma ={\frac {1}{2}}+\int _{0}^{\infty }{\frac {\overline {(x+1)\cos(\pi x)}}{x+1}}dx

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}^{n}{\frac {\beta _{2k}}{2k}}-{\frac {\beta _{2n}}{\zeta (2n)}}\int _{1}^{\infty }\lfloor t\rfloor \cos(2\pi t)t^{-2n-1}dt

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

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

\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}

\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}

\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.

{\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)}}

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)

G={\frac {\pi }{8}}\ln(2)+\sum _{n=0}^{\infty }(-1)^{n}{\frac {H_{2n+1}}{2n+1}}.

If the measure ρ is reducible and let φ 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 square integrable for ρ and is orthogonal with P₀ = 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)

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

g(x)\mapsto \int_I\frac{g(t)-g(x)}{t-x}\rho(t)\,dt.

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. ^[1]

References

1 2 Mappings of open quantum systems onto chain representations and Markovian embeddings, M. P. Woods, R. Groux, A. W. Chin, S. F. Huelga, M. B. Plenio.

http://arxiv.org/abs/1111.5262

External links

personal page of Roland Groux about the theory of secondary measures

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