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

S_{\rho}(z)=\int_I\frac{\rho(t)\,dt}{z-t}, \qquad t \in I \subset \mathbb{R}, \; z \in \mathbb{C} \backslash \mathbb{R}

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

\rho(x)=\underset{\varepsilon\rightarrow 0^+}{\text{lim}}\frac{S_{\rho}(x-i\varepsilon)-S_{\rho}(x+i\varepsilon)}{2i\pi}.

Connections with moments of measures

Main article: moment problem

If the measure of density ρ has moments of any order defined for each integer by the equality

m_{n}=\int_I t^n\,\rho(t)\,dt,

then the Stieltjes transformation of ρ admits for each integer n the asymptotic expansion in the neighbourhood of infinity given by

S_{\rho}(z)=\sum_{k=0}^{n}\frac{m_k}{z^{k+1}}+o\left(\frac{1}{z^{n+1}}\right).

Under certain conditions the complete expansion as a Laurent series can be obtained:

S_{\rho}(z)=\sum_{n=0}^{\infty}\frac{m_n}{z^{n+1}}.

Relationships to orthogonal polynomials

The correspondence (f,g)\mapsto \int_I f(t)g(t)\rho(t)\,dt 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

Q_n(x)=\int_I \frac{P_n (t)-P_n (x)}{t-x}\rho (t)\,dt.

It appears that F_n(z)=\frac{Q_n(z)}{P_n(z)} is a Padé approximation of Sρ(z) in a neighbourhood of infinity, in the sense that

S_\rho(z)-\frac{Q_n(z)}{P_n(z)}=O\left(\frac{1}{z^{2n}}\right).

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 also

References