Secondary Measure - The Broad Outlines of The Theory

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:

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

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:

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:

.

The operator

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: