Stieltjes transformation

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 measuresEdit

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 polynomialsEdit

The correspondence   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   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.)

See alsoEdit

ReferencesEdit

  • H. S. Wall (1948). Analytic Theory of Continued Fractions. D. Van Nostrand Company Inc.