Secondary measure - AbsoluteAstronomy.com

In mathematics, the secondary measure associated with a measure

Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

of positive density

Density

The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

when there is one, is a measure of positive density

, turning the secondary polynomials associated with the orthogonal polynomials

Orthogonal polynomials

In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...

for

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

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

with

in the general case,

or:

when

satisfy a Lipschitz

Lipschitz

- People :Lipschitz is a surname, which may be derived from the Polish city of Głubczyce , and may refer to:* Daniel Lipšic, Minister of Interior in Slovakia* Dr...

condition.

This application

is called the reducer of

More generally,

are linked by their Stieltjes transformation with the following formula:

in which

is the moment

Moment

- Science, engineering, and mathematics :* Moment , used in probability theory and statistics* Moment , several related concepts, including:** Angular momentum or moment of momentum, the rotational analog of momentum...

of order 1 of the measure

.

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

Gamma function

In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...

, Riemann Zeta function, and Euler's constant

Euler–Mascheroni constant

The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....

.

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

where

is the unknown function, and lead to theorems of convergence towards the Chebyshev and Dirac measures.

The broad outlines of the theory

Let

be a measure of positive density

Density

on an interval I and admitting moments of any order. We can build a family

of orthogonal polynomials

Orthogonal polynomials

for the inner product induced by

.
Let us call

the sequence of the secondary polynomials associated with the family

.
Under certain conditions there is a measure for which the family Q is orthogonal. This measure, which we can clarify from

is called a secondary measure associated initial measure

.

When

is a probability density function

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

, a sufficient condition so that

, while admitting moments of any order can be a secondary measure associated with

is that its Stieltjes Transformation is given by an equality of the type:

is an arbitrary constant and

indicating the moment of order 1 of

.

For

we obtain the measure known as secondary, remarkable since for

the norm

Norm (mathematics)

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

of the polynomial

for

coincides exactly with the norm of the secondary polynomial associated

when using the measure

.

In this paramount case, and if the space generated by the orthogonal polynomials is dense in

, the operator

defined by

creating the secondary polynomials can be furthered to a linear map connecting space

and becomes isometric if limited to the hyperplane

Hyperplane

A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

of the orthogonal functions with

.

For unspecified functions square integrable for

we obtain the more general formula of covariance

Covariance

In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.- Definition :...

The theory continues by introducing the concept of reducible measure, meaning that the quotient

is element of

. The following results are then established:

The reducer

is an antecedent of

for the operator

. (In fact the only antecedent which belongs to

).

For any function square integrable for

, there is an equality known as the reducing formula:

.

The operator

defined on the polynomials is prolonged in an isometry

Isometry

In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

linking the closure

Closure (topology)

In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...

of the space of these polynomials in

to the hyperplane

Hyperplane

A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

provided with the norm induced by

.

Under certain restrictive conditions the operator

acts like the adjoint

Adjoint

In mathematics, the term adjoint applies in several situations. Several of these share a similar formalism: if A is adjoint to B, then there is typically some formula of the type = .Specifically, adjoint may mean:...

for the inner product induced by

.

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

Case of the Lebesgue measure and some other examples

The Lebesgue measure on the standard interval

is obtained by taking the constant density

.

The associated orthogonal polynomials

Orthogonal polynomials

are called Legendre polynomials and can be clarified by

. The norm

Norm (mathematics)

is worth

. The recurrence relation in three terms is written:

The reducer of this measure of Lebesgue is given by

. The associated secondary measure is then clarified as :

.

If we normalize the polynomials of Legendre, the coefficients of Fourier

Fourier

Fourier most commonly refers to Joseph Fourier , French mathematician and physicist, or the mathematics, physics, and engineering terms named in his honor for his work on the concepts underlying them:In mathematics:...

of the reducer

related to this orthonormal system are null for an even index and are given by

for an odd index

.

The Laguerre polynomials

Laguerre polynomials

In mathematics, the Laguerre polynomials, named after Edmond Laguerre ,are the canonical solutions of Laguerre's equation:x\,y + \,y' + n\,y = 0\,which is a second-order linear differential equation....

are linked to the density

on the interval

.

They are clarified by

and are normalized.

The reducer associated is defined by

The coefficients of Fourier of the reducer

related to the Laguerre polynomials are given by

This coefficient

is no other than the opposite of the sum of the elements of the line of index

in the table of the harmonic triangular numbers of Leibniz.

The Hermite polynomials are linked to the Gaussian density

They are clarified by

and are normalized.

The reducer associated is defined by

The coefficients of Fourier

Fourier

of the reducer

related to the system of Hermite polynomials are null for an even index and are given by

for an odd index

.

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

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

Jacobi

Jacobi may refer to:People with the surname Jacobi:*Jacobi Other:* Jacobi Medical Center, New York* Jacobi sum, a type of character sum in mathematics* Jacobi method, a method for diagonalization of matrices in mathematics...

measure of density

on (0, 1).

Chebyshev measure of the first form of density

on (−1, 1).

Sequence of secondary measures

The secondary measure

associated with a probability density function

Probability density function

has its moment of order 0 given by the formula

, (

and

indicating the respective moments of order 1 and 2 of

).

To be able to iterate the process then, one 'normalizes'

while defining

which becomes in its turn a density of probability called naturally the normalised secondary measure associated with

.

We can then create from

a secondary normalised measure

, then defining

from

and so on. We can therefore see a sequence of successive secondary measures, created from

, is such that

that is the secondary normalised measure deduced from

It is possible to clarify the density

by using the orthogonal polynomials

Orthogonal polynomials

for

, the secondary polynomials

and the reducer associated

. That gives the formula

The coefficient

is easily obtained starting from the leading coefficients of the polynomials

and

. We can also clarify the reducer

associated with

, as well as the orthogonal polynomials corresponding to

.

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

.

Let

be the classic recurrence relation in three terms.

If

and

, then the sequence

converges completely towards the Chebyshev density of the second form

.

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

has its moment of order 1 equal to

, then these densities equinormal with

are given by a formula of the type:

, t describing an interval containing]0, 1].

If

is the secondary measure of

,that of

will be

.

The reducer of

is :

by noting

the reducer of

.

Orthogonal polynomials for the measure

are clarified from

by the formula

with

secondary polynomial associated with

It is remarkable also that, within the meaning of distributions, the limit when

tends towards 0 per higher value of

is the Dirac measure concentrated at

.

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

, with

describing]0,2]. The value

=2 gives the Chebyshev measure of the first form.

A few beautiful applications

. (with

the Euler's constant

Euler–Mascheroni constant

The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....

(the notation

indicating the 2 periodic function coinciding with

on (−1, 1)).

(with

is the floor function and

the Bernoulli number

Bernoulli number

In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers....

of order