In mathematics, the secondary measure associated with a
measure of positive
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
ρ when there is one, is a measure of positive density μ, turning the
secondary polynomials associated with the
orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.
The most widely used orthogonal polynomials are the class ...
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 ''L''
2(
, 1
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
R, ρ)
:
with
:
in the general case, or:
:
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:
:
in which ''c''
1 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
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
, Riemann
Zeta function
In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function
: \zeta(s) = \sum_^\infty \frac 1 .
Zeta functions include:
* Airy zeta function, related to the zeros of the Airy function
* ...
, and
Euler's constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural ...
.
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 ''g'' is the unknown function, and lead to theorems of convergence towards the
Chebyshev
Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics.
Chebyshe ...
and
Dirac measure
In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element ''x'' or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields.
...
s.
The broad outlines of the theory
Let ρ be a measure of positive
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
on an interval I and admitting moments of any order. We can build a family of
orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.
The most widely used orthogonal polynomials are the class ...
for the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
induced by ρ. Let us call 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
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
, 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:
:
''a'' is an arbitrary constant and ''c''
1 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
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
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
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in ''L''
2(''I'', R, ρ), the
operator ''T''
ρ defined by
:
creating the secondary polynomials can be furthered to a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
connecting space ''L''
2(''I'', R, ρ) to ''L''
2(''I'', R, μ) and becomes isometric if limited to the
hyperplane ''H''
ρ of the orthogonal functions with ''P''
0 = 1.
For unspecified functions
square integrable
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value i ...
for ρ we obtain the more general formula of
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the ...
:
:
The theory continues by introducing the concept of reducible measure, meaning that the quotient ρ/μ is element of ''L''
2(''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:
:
.
The operator
:
defined on the polynomials is prolonged in an
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
''S''
ρ linking the
closure of the space of these polynomials in ''L''
2(''I'', R, ρ
2μ
−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
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
:(''Ax'', ''y'') = (''x'', ''By'').
Specifically, adjoin ...
of ''T''
ρ for the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
induced by ρ.
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
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
measure on the standard interval
, 1
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
is obtained by taking the constant density ρ(''x'') = 1.
The associated
orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.
The most widely used orthogonal polynomials are the class ...
are called
Legendre polynomials
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applica ...
and can be clarified by
:
The
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
of ''P
n'' 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 of the reducer φ related to this orthonormal system are null for an even index and are given by
:
for an odd index ''n''.
The
Laguerre polynomials In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation:
xy'' + (1 - x)y' + ny = 0
which is a second-order linear differential equation. This equation has nonsingular solutions on ...
are linked to the density ρ(''x'') = ''e
−x'' on the interval ''I'' = [0, ∞). They are clarified by
:
and are normalized.
The reducer associated is defined by
:
The coefficients of Fourier of the reducer φ related to the Laguerre polynomials are given by
:
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 Gottfried Wilhelm Leibniz, Leibniz.
The Hermite polynomials are linked to the Gaussian density
:
on ''I'' = R.
They are clarified by
:
and are normalized.
The reducer associated is defined by
:
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
:
for an odd index ''n''.
The
Chebyshev
Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics.
Chebyshe ...
measure of the second form. This is defined by the density
:
on the interval
, 1
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
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
:
Chebyshev measure on (−1, 1) of the first form of density
:
Sequence of secondary measures
The secondary measure μ associated with a
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
ρ has its moment of order 0 given by the formula
:
where ''c''
1 and ''c''
2 indicating the respective moments of order 1 and 2 of ρ.
To be able to iterate the process then, one 'normalizes' μ while defining ρ
1 = μ/''d''
0 which becomes in its turn a density of probability called naturally the normalised secondary measure associated with ρ.
We can then create from ρ
1 a secondary normalised measure ρ
2, then defining ρ
3 from ρ
2 and so on. We can therefore see a sequence of successive secondary measures, created from ρ
0 = 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
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.
The most widely used orthogonal polynomials are the class ...
''P
n'' for ρ, the secondary polynomials ''Q
n'' and the reducer associated φ. That gives the formula
:
The coefficient
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
, 1
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
Let
:
be the classic recurrence relation in three terms. If
:
then the sequence converges completely towards the
Chebyshev
Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics.
Chebyshe ...
density of the second form
:
.
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 ''c''
1, then these densities equinormal with ρ are given by a formula of the type:
:
''t'' describing an interval containing ]0, 1].
If μ is the secondary measure of ρ, that of ρ
''t'' will be ''t''μ.
The reducer of ρ
''t'' is
:
by noting ''G''(''x'') the reducer of μ.
Orthogonal polynomials for the measure ρ
''t'' are clarified from ''n'' = 1 by the formula
:
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 ''c''
1.
For example, the equinormal densities with the Chebyshev measure of the second form are defined by:
:
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
In mathematics, Catalan's constant , is defined by
: G = \beta(2) = \sum_^ \frac = \frac - \frac + \frac - \frac + \frac - \cdots,
where is the Dirichlet beta function. Its numerical value is approximately
:
It is not known whether is irra ...
, γ is the
Euler's constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural ...
, β
2''n'' is the
Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
of order 2''n'', ''H''
2''n''+1 is the
harmonic number
In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers:
H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac.
Starting from , the sequence of harmonic numbers begins:
1, \frac, \frac, \frac, \frac, \dot ...
of order 2''n''+1 and Ei is the
Exponential integral
In mathematics, the exponential integral Ei is a special function on the complex plane.
It is defined as one particular definite integral of the ratio between an exponential function and its argument.
Definitions
For real non-zero values of  ...
function.
:
:
:
The notation
indicating the 2 periodic function coinciding with
on (−1, 1).
:
:
:
:
:
:
:
:
:
If the measure ρ is reducible and let φ be the associated reducer, one has the equality
:
If the measure ρ is reducible with μ the associated reducer, then if ''f'' is
square integrable
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value i ...
for μ, and if ''g'' is square integrable for ρ and is orthogonal with ''P''
0 = 1 one has equivalence:
:
''c''
1 indicates the moment of order 1 of ρ and ''T''
ρ the operator
:
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
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.
The most widely used orthogonal polynomials are the class ...
*
Probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
References
{{Reflist,
refs=
[ 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
Measures (measure theory)