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 [math]\displaystyle{ S_{\rho}(z)=\int_I\frac{\rho(t)\,dt}{z-t}, \qquad z \in \mathbb{C} \setminus I. }[/math]
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
[math]\displaystyle{ \rho(x)=\lim_{\varepsilon \to 0^+} \frac{S_{\rho}(x-i\varepsilon)-S_{\rho}(x+i\varepsilon)}{2i\pi}. }[/math]
Connections with moments of measures
If the measure of density ρ has moments of any order defined for each integer by the equality [math]\displaystyle{ m_{n}=\int_I t^n\,\rho(t)\,dt, }[/math]
then the Stieltjes transformation of ρ admits for each integer n the asymptotic expansion in the neighbourhood of infinity given by [math]\displaystyle{ S_{\rho}(z)=\sum_{k=0}^{n}\frac{m_k}{z^{k+1}}+o\left(\frac{1}{z^{n+1}}\right). }[/math]
Under certain conditions the complete expansion as a Laurent series can be obtained: [math]\displaystyle{ S_{\rho}(z) = \sum_{n=0}^{\infty}\frac{m_n}{z^{n+1}}. }[/math]
Relationships to orthogonal polynomials
The correspondence [math]\displaystyle{ (f,g) \mapsto \int_I f(t) g(t) \rho(t) \, dt }[/math] 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 [math]\displaystyle{ Q_n(x)=\int_I \frac{P_n (t)-P_n (x)}{t-x}\rho (t)\,dt. }[/math]
It appears that [math]\displaystyle{ F_n(z) = \frac{Q_n(z)}{P_n(z)} }[/math] is a Padé approximation of Sρ(z) in a neighbourhood of infinity, in the sense that [math]\displaystyle{ S_\rho(z)-\frac{Q_n(z)}{P_n(z)}=O\left(\frac{1}{z^{2n}}\right). }[/math]
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
- H. S. Wall (1948). Analytic Theory of Continued Fractions. D. Van Nostrand Company Inc..
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