Secondary measure


Secondary measure

In mathematics, the secondary measure associated with a measure of positive density ho when there is one, is a measure of positive density mu, turning the secondary polynomials associated with the orthogonal polynomials for ho 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 L^2( [0,1] ,R, ho)

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

with

: varphi(x) = lim_{varepsilon o 0+}2int_0^1frac{(x-t) ho(t)}{(x-t)^2+varepsilon^2} , dt

in the general case,

or:

: varphi(x) = 2 ho(x) ext{ln}left(frac{x}{1-x} ight) - 2 int_0^1frac{ ho(t)- ho(x)}{t-x} , dt

when ho satisfy a Lipschitz condition.

This application varphi is called the reducer of ho.

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

: S_{mu}(z)=z-c_1-frac{1}{S_{ ho}(z)}

in which c_1 is the moment of order 1 of the measure ho .

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_0^1frac{g(t)-g(x)}{t-x} ho(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 ho be a measure of positive density on an interval I and admitting moments of any order. We can build a family (P_n)_{nin N} of orthogonal polynomials for the inner product induced by ho. 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 ho is called a secondary measure associated initial measure ho.

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

: S_{mu}(z)=aleft(z-c_1-frac{1}{S_{ ho}(z)} ight),

a is an arbitrary constant and , c_1 indicating the moment of order 1 of ho.

For a=1 we obtain the measure know as secondary, remarkable since for ngeq1 the norm of the polynomial P_n for ho 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^2left(I,mathbf R, ho ight), the operator T_ ho defined by f(x) mapsto int_I frac{f(t)-f(x)}{t-x} ho (t)dt creating the secondary polynomials can be furthered to a linear map connecting space L^2left(I,mathbf R, ho ight) to L^2left(I,mathbf R,mu ight) and becomes isometric if limited to the hyperplane H_ ho of the orthogonal functions with P_0=1.

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

: langle f/g angle_ ho - langle f/1 angle_ ho imes langle g/1 angle_ ho = langle T_ ho(f)/T_ ho (g) angle_mu.

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

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

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

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

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

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

: T_ hocirc S_ ho left( f ight)=frac{ ho}{mu} imes left(f ight).

Case of the Lebesgue measure and some other examples

The Lebesgue measure on the standard interval left [0,1 ight] is obtained by taking the constant density ho(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 ight). The norm of P_n is worth frac{n!}{sqrt{2n+1. The reoccurrence 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)=2lnleft(frac{x}{1-x} ight). The associated secondary measure is then clarified as : mu(x)=frac{1}{ln^2left(frac{x}{1-x} ight)+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{4sqrt{2n+1{n(n+1)} for an odd index n.

The Laguerre polynomials are linked to the density ho(x)=e^{-x} on the interval I = left [0,+infty ight). They are clarified by

: L_n(x)=frac{e^x}{n!}frac{d^n}{dx^n}(x^ne^{-x})=sum_{k=0}^{k=n}inom{n}{k}(-1)^kfrac{x^k}{k!}

and are normalized.

The reducer associated is defined by

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

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}{inom{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

: ho(x)=frac{e^{-frac{x^2}{2}{sqrt{2pi on I= R.

They are clarified by

: H_n(x)=frac{1}{sqrt{n!e^{frac{x^2}{2frac{d^n}{dx^n}left(e^{-frac{x^2}{2 ight)

and are normalized.

The reducer associated is defined by

: varphi(x)=-frac{2}{sqrt{2piint_{-infty}^{+infty}te^{-frac{t^2}{2ln|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}{2frac{left(frac{n-1}{2} ight)!}{sqrt{n!

for an odd index n.

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

Chebyshev measure of the first form of density ho(x)=frac{1}{pisqrt{1-x^2 on (−1, 1).

equence of secondary measures

The secondary measure mu associated with a probability density function ho has its moment of order 0 gived 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 ho).

To be able to iterate the process then one 'normalize' mu while defining ho_1 =frac{mu}{d_0} which becomes in its turn a density of probability called naturally the normalised secondary measure associated with ho.

We can then create from ho_1 a secondary normalised measure ho_2, then defining ho_3 from ho_2 and so on. We can therefore see a sequence of successive secondary measures, created from ho_0= ho, is such that ho_{n+1} that is the secondary normalised measure deduced from ho_{n}

It is possible to clarify the density ho_n by using the orthogonal polynomials P_n for ho, the secondary polynoms Q_n and the reducer associated varphi. That gives the formula

: ho_n(x)=frac{1}{d_0^{n-1 frac{ ho(x)}{left(P_{n-1}(x) frac{varphi(x)}{2}-Q_{n-1}(x) ight)^2 + pi^2 ho^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 ho_n, as well as the orthogonal polynoms corresponding to ho_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 ight] .

Let xP_n (x)=t_nP_{n+1}(x)+s_nP_n(x)+t_{n-1}P_{n-1}(x) be the classic reoccurrence 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 nmapsto ho_n converges completely towards the Chebyshev density of the second form ho_{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 ho has its moment of order 1 equal to c_1, then these densities equinormal with ho are given by a formula of the type: ho_{t}(x)=frac{t ho(x)}{left [left(t-1 ight)(x-c_1)frac{varphileft(x ight)}{2}-t ight] ^2+pi^2 ho^2(x)(t-1)^2(x-c_1)^2} , t describing an interval containing] 0, 1] .

If mu is the secondary measure of ho,that of ho_t will be tmu.

The reducer of ho_t is : varphi_t(x)=frac{2left(x-c_1 ight)-tG(x)}{left((x-c_1)-tfrac{G(x)}{2} ight)^2+t^2pi^2mu^2(x)} by noting G(x) the reducer of mu.

Orthogonal polynoms for the measure ho_t are clarified from n=1 by the formula

: P_n^t(x)=frac{1}{sqrt{tleft [tP_n(x)+(1-t)(x-c_1)Q_n(x) ight] 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 ho_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: ho_t(x)=frac{2tsqrt{1-x^2{pileft [t^2+4(1-t)x^2 ight] } , with t describing] 0,2] . The value t=2 gives the Chebishev measure of the first form.

A few beautiful applications

: forall p >1 qquadfrac{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 gamma the Euler's constant).

: gamma=frac{1}{2}+int_0^{+infty}frac{overline {(x+1)cos(pi x)} dx}{x+1}.
(the notation xmapsto overline {(x+1)cos(pi x)} indicating the 2 periodic function coinciding with xmapsto (x+1) cos(pi x) on (−1, 1)).

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

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

: eta_k = frac{(-1)^kk!}{pi} Imleft(int_{-infty}^{infty} frac{e^x , dx}{(1+e^x)(x-ipi)^k} ight).

: int_0^1ln^{2n}left(frac{x}{1-x} ight),dx = (-1)^{n+1}2(2^{2n-1}-1)eta_{2n}pi^{2n}.

: int_0^1 int_0^1cdots int_0^1 left(sum_{k=1}^{k=2n} frac{ln(t_k)} {prod_{i ot=k}(t_k-t_i)} ight) , dt_1 , dt_2cdots dt_{2n} = frac{(-1)^{n+1}(2pi)^{2n}eta_{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 ight] } frac{dx}{x}.

(for any real alpha)

: sum_{n=1}^{n=+infty} left(frac{1}{n}sum_{k=0}^{k=n-1} frac{1}{inom{n-1}{k ight)^2 = frac{4pi^2}{9}=int_0^{+infty}4 [mathrm {Ei} (1,-x)+ipi] ^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} ight)+frac{piln(2)}{8}

: mbox{Catalan} = frac{piln(2)}{8}+sum_{n=0}^{n=infty}(-1)^nfrac{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 2n+1.

If the measure ho is reducible and let varphi be the associated reducer, one has the equality

: int_Ivarphi^2(x) ho(x) , dx = frac{4pi^2}{3}int_I ho^3(x) , dx.

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

: f(x)=int_Ifrac{g(t)-g(x)}{t-x} ho(t)dtLeftrightarrow g(x) = (x-c_1)f(x) - T_{mu}(f(x)) = frac{varphi(x)mu(x)}{ ho(x)}f(x)-T_{ ho} left(frac{mu(x)}{ ho(x)}f(x) ight)

(c_1 indicates the moment of order 1 of ho and T_{ ho} the operator g(x)mapsto int_Ifrac{g(t)-g(x)}{t-x} ho(t),dt).

ee also

* Orthogonal polynomials
* Probability

External links

* [http://perso.orange.fr/roland.groux personal page of Roland Groux about the theory of secondary measures]


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