In mathematics, the Stieltjes transformation Sρ(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
Under certain conditions the complete expansion as a Laurent series can be obtained:
Relationships to orthogonal polynomials
It appears that is a Padé approximation of Sρ(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.)
- H. S. Wall (1948). Analytic Theory of Continued Fractions. D. Van Nostrand Company Inc.