Secondary Measure - Introduction

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.