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)

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

with

${\displaystyle \varphi (x)=\lim _{\varepsilon \to 0^{+}}2\int _{0}^{1}{\frac {(x-t)\rho (t)}{(x-t)^{2}+\varepsilon ^{2}}}\,dt}$

in the general case, or:

${\displaystyle \varphi (x)=2\rho (x){\text{ln}}\left({\frac {x}{1-x}}\right)-2\int _{0}^{1}{\frac {\rho (t)-\rho (x)}{t-x}}\,dt}$

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:

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

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

${\displaystyle f(x)=\int _{0}^{1}{\frac {g(t)-g(x)}{t-x}}\rho (t)\,dt}$

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:

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

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

${\displaystyle f(x)\mapsto \int _{I}{\frac {f(t)-f(x)}{t-x}}\rho (t)dt}$

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:

${\displaystyle \langle f/g\rangle _{\rho }-\langle f/1\rangle _{\rho }\times \langle g/1\rangle _{\rho }=\langle T_{\rho }(f)/T_{\rho }(g)\rangle _{\mu }.}$

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:

${\displaystyle \langle f/\varphi \rangle _{\rho }=\langle T_{\rho }(f)/1\rangle _{\rho }}$.

The operator

${\displaystyle f\mapsto \varphi \times f-T_{\rho }(f)}$

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:

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

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

${\displaystyle P_{n}(x)={\frac {d^{n}}{dx^{n}}}\left(x^{n}(1-x)^{n}\right).}$

The norm of P_n is worth

${\displaystyle {\frac {n!}{\sqrt {2n+1}}}.}$

The recurrence relation in three terms is written:

${\displaystyle 2(2n+1)XP_{n}(X)=-P_{n+1}(X)+(2n+1)P_{n}(X)-n^{2}P_{n-1}(X).}$

The reducer of this measure of Lebesgue is given by

${\displaystyle \varphi (x)=2\ln \left({\frac {x}{1-x}}\right).}$

The associated secondary measure is then clarified as

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

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

${\displaystyle C_{n}(\varphi )=-{\frac {4{\sqrt {2n+1}}}{n(n+1)}}}$

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

${\displaystyle L_{n}(x)={\frac {e^{x}}{n!}}{\frac {d^{n}}{dx^{n}}}(x^{n}e^{-x})=\sum _{k=0}^{n}{\binom {n}{k}}(-1)^{k}{\frac {x^{k}}{k!}}}$

and are normalized.

The reducer associated is defined by

${\displaystyle \varphi (x)=2\left(\ln(x)-\int _{0}^{\infty }e^{-t}\ln |x-t|dt\right).}$

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

${\displaystyle C_{n}(\varphi )=-{\frac {1}{n}}\sum _{k=0}^{n-1}{\frac {1}{\binom {n-1}{k}}}.}$

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

${\displaystyle \rho (x)={\frac {e^{-{\frac {x^{2}}{2}}}}{\sqrt {2\pi }}}}$

on I = R.

They are clarified by

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

and are normalized.

The reducer associated is defined by

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

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

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

for an odd index n.

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

${\displaystyle \rho (x)={\frac {8}{\pi }}{\sqrt {x(1-x)}}}$

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

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

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

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

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

${\displaystyle d_{0}=c_{2}-c_{1}^{2},}$

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

${\displaystyle \rho _{n}(x)={\frac {1}{d_{0}^{n-1}}}{\frac {\rho (x)}{\left(P_{n-1}(x){\frac {\varphi (x)}{2}}-Q_{n-1}(x)\right)^{2}+\pi ^{2}\rho ^{2}(x)P_{n-1}^{2}(x)}}.}$

The coefficient ${\displaystyle d_{0}^{n-1}}$ 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

${\displaystyle xP_{n}(x)=t_{n}P_{n+1}(x)+s_{n}P_{n}(x)+t_{n-1}P_{n-1}(x)}$

be the classic recurrence relation in three terms. If

${\displaystyle \lim _{n\mapsto \infty }t_{n}={\tfrac {1}{4}},\quad \lim _{n\mapsto \infty }s_{n}={\tfrac {1}{2}},}$

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

${\displaystyle \rho _{tch}(x)={\frac {8}{\pi }}{\sqrt {x(1-x)}}}$.

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:

${\displaystyle \rho _{t}(x)={\frac {t\rho (x)}{\left({\tfrac {1}{2}}(t-1)(x-c_{1})\varphi (x)-t\right)^{2}+\pi ^{2}\rho ^{2}(x)(t-1)^{2}(x-c_{1})^{2}}},}$

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

${\displaystyle \varphi _{t}(x)={\frac {2(x-c_{1})-tG(x)}{\left((x-c_{1})-t{\tfrac {1}{2}}G(x)\right)^{2}+t^{2}\pi ^{2}\mu ^{2}(x)}}}$

by noting G(x) the reducer of μ.

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

${\displaystyle P_{n}^{t}(x)={\frac {tP_{n}(x)+(1-t)(x-c_{1})Q_{n}(x)}{\sqrt {t}}}}$

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:

${\displaystyle \rho _{t}(x)={\frac {2t{\sqrt {1-x^{2}}}}{\pi \left[t^{2}+4(1-t)x^{2}\right]}},}$

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.

${\displaystyle {\frac {1}{\ln(p)}}={\frac {1}{p-1}}+\int _{0}^{\infty }{\frac {1}{(x+p)(\ln ^{2}(x)+\pi ^{2})}}dx\qquad \qquad \forall p>1}$

${\displaystyle \gamma =\int _{0}^{\infty }{\frac {\ln(1+{\frac {1}{x}})}{\ln ^{2}(x)+\pi ^{2}}}dx}$

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

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

${\displaystyle \gamma ={\frac {1}{2}}+\sum _{k=1}^{n}{\frac {\beta _{2k}}{2k}}-{\frac {\beta _{2n}}{\zeta (2n)}}\int _{1}^{\infty }\lfloor t\rfloor \cos(2\pi t)t^{-2n-1}dt}$

${\displaystyle \beta _{k}={\frac {(-1)^{k}k!}{\pi }}{\text{Im}}\left(\int _{-\infty }^{\infty }{\frac {e^{x}}{(1+e^{x})(x-i\pi )^{k}}}dx\right)}$

${\displaystyle \int _{0}^{1}\ln ^{2n}\left({\frac {x}{1-x}}\right)\,dx=(-1)^{n+1}(2^{2n}-2)\beta _{2n}\pi ^{2n}}$

${\displaystyle \int _{0}^{1}\cdots \int _{0}^{1}\left(\sum _{k=1}^{2n}{\frac {\ln(t_{k})}{\prod _{i\neq k}(t_{k}-t_{i})}}\right)\,dt_{1}\cdots dt_{2n}={\tfrac {1}{2}}(-1)^{n+1}(2\pi )^{2n}\beta _{2n}}$

${\displaystyle \int _{0}^{\infty }{\frac {e^{-\alpha x}}{\Gamma (x+1)}}dx=e^{e^{-\alpha }}-1+\int _{0}^{\infty }{\frac {1-e^{-x}}{(\ln(x)+\alpha )^{2}+\pi ^{2}}}{\frac {dx}{x}}\qquad \qquad \forall \alpha \in \mathbf {R} }$

${\displaystyle \sum _{n=1}^{\infty }\left({\frac {1}{n}}\sum _{k=0}^{n-1}{\frac {1}{\binom {n-1}{k}}}\right)^{2}={\tfrac {4}{9}}\pi ^{2}=\int _{0}^{\infty }4\left(\mathrm {Ei} (1,-x)+i\pi \right)^{2}e^{-3x}\,dx.}$

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

${\displaystyle G=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{4^{k+1}}}\left({\frac {1}{(4k+3)^{2}}}+{\frac {2}{(4k+2)^{2}}}+{\frac {2}{(4k+1)^{2}}}\right)+{\frac {\pi }{8}}\ln(2)}$

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

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

${\displaystyle \int _{I}\varphi ^{2}(x)\rho (x)\,dx={\frac {4\pi ^{2}}{3}}\int _{I}\rho ^{3}(x)\,dx.}$

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:

${\displaystyle f(x)=\int _{I}{\frac {g(t)-g(x)}{t-x}}\rho (t)dt\Leftrightarrow g(x)=(x-c_{1})f(x)-T_{\mu }(f(x))={\frac {\varphi (x)\mu (x)}{\rho (x)}}f(x)-T_{\rho }\left({\frac {\mu (x)}{\rho (x)}}f(x)\right)}$

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

${\displaystyle g(x)\mapsto \int _{I}{\frac {g(t)-g(x)}{t-x}}\rho (t)\,dt.}$

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]

See also

• Orthogonal polynomials
• Probability

References

1. 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. https://arxiv.org/abs/1111.5262

External links

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