Stieltjes transformation

In mathematics, the Stieltjes transformation "S"ρ("z") of a measure of density ρ on a real interval "I" is the function of the complex variable "z" defined outside "I" by the formula
Under certain conditions we can reconstitute the density function ρ starting from its Stieltjes transformation thanks to the inverse formula of Stieltjes-Perron. For example, if the density ρ is continuous throughout "I", one will have inside this interval
Connections with moments of measures.
If the measure of density ρ has moments of any order defined for each integer by the equality
then the Stieltjes transformation of ρ admits for each integer "n" the asymptotic expansion in the neighbourhood of infinity given by
Under certain conditions the complete expansion as a Laurent series can be obtained:
Relationships to orthogonal polynomials.
The correspondence formula_6 defines an inner product on the space of continuous functions on the interval "I".
If {"Pn"} is a sequence of orthogonal polynomials for this product, we can create the sequence of associated secondary polynomials by the formula
It appears that formula_8 is a Padé approximation of "S"ρ("z") in a neighbourhood of infinity, in the sense that
Since these two sequences of polynomials satisfy the same recurrence relation in three terms, we can develop a continued fraction for the Stieltjes transformation whose successive convergents are the fractions "Fn"("z").
The Stieltjes transformation can also be used to construct from the density ρ an effective measure for transforming the secondary polynomials into an orthogonal system. (For more details see the article secondary measure.)