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

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 ${\textstyle (f,g)\mapsto \int _{I}f(t)g(t)\rho (t)\,dt}$ 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 ${\textstyle F_{n}(z)={\frac {Q_{n}(z)}{P_{n}(z)}}}$ 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.)

See also

• Orthogonal polynomials
• Secondary polynomials
• Secondary measure

References

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