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 L2(, 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 c1 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, 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
where g is the unknown function, and lead to theorems of convergence towards the Chebyshev and Dirac measures.
Read more about this topic: Secondary Measure
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