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 and Dirac measures.
Let ρ be a measure of positive density on an interval I and admitting moments of any order.
From this, a family {Pn} of orthogonal polynomials for the inner product induced by ρ can be created.
In this paramount case, and if the space generated by the orthogonal polynomials is dense in L2(I, R, ρ), the operator Tρ defined by 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 Hρ of the orthogonal functions with P0 = 1.
For unspecified functions square integrable for ρ a more general formula of covariance may be obtained: 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: Finally the two operators are also connected, provided the images in question are defined, by the fundamental formula of composition: 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 The norm of Pn 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 are linked to the density ρ(x) = e−x on the interval I = [0, ∞).
The reducer associated is defined by The coefficients of Fourier of the reducer φ related to the Laguerre polynomials are given by This coefficient Cn(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.
Under certain conditions it occurs as the limit of the sequence of normalized secondary measures of a given density.
This process can be iterated by 'normalizing' μ while defining ρ1 = μ/d0 which becomes in its turn a density of probability called naturally the normalised secondary measure associated with ρ.
It is remarkable also that, within the meaning of distributions, the limit when t tends towards 0 per higher value of ρt is the Dirac measure concentrated at c1.
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 for μ, and if g is square integrable for ρ and is orthogonal with P0 = 1, the following equivalence holds: c1 indicates the moment of order 1 of ρ and Tρ the operator In addition, the sequence of secondary measures has applications in Quantum Mechanics, where it gives rise to the sequence of residual spectral densities for specialized Pauli-Fierz Hamiltonians.