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

$$S_{\rho}(z)=\int_I\frac{\rho(t)\,dt}{z-t}, \qquad z \in \mathbb{C} \setminus I.$$

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

$$\rho(x)=\lim_{\varepsilon \to 0^+} \frac{S_{\rho}(x-i\varepsilon)-S_{\rho}(x+i\varepsilon)}{2i\pi}.$$

Connections with moments of measures
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 ${P_{n}}$ 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 $S_{ρ}(z)$ 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 $F_{n}(z)$.

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.)