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 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

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.)

References

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

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Stieltjes transformation

Connections with moments of measures

Relationships to orthogonal polynomials

See also

References