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

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

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

Recall from basic calculus that $${\displaystyle \int _{-\infty }^{\infty }{\frac {1}{x^{2}+1}}dx=\lim _{x\to \infty }\arctan x-\lim _{x\to -\infty }\arctan x={\tfrac {\pi }{2}}-(-{\tfrac {\pi }{2}})=\pi {\text{.}}}$$ Hence ${\displaystyle f(x)={\tfrac {1}{\pi }}(x^{2}+1)^{-1}}$ is the probability density function of a distribution—a Cauchy distribution. Via the change of variables ${\displaystyle x=(t-t_{0})/\varepsilon }$ we get the full family of Cauchy distributions: $${\displaystyle 1=\int _{-\infty }^{\infty }{\frac {1/\pi }{x^{2}+1}}dx=\int _{-\infty }^{\infty }{\frac {1/\pi }{({\frac {t-t_{0}}{\varepsilon }})^{2}+1}}{\frac {dx}{dt}}dt=\int _{-\infty }^{\infty }{\frac {\varepsilon /\pi }{(t-t_{0})^{2}+\varepsilon ^{2}}}dt}$$ As ${\displaystyle \varepsilon \to 0^{+}}$, these tend to a Dirac distribution with the mass at ${\displaystyle t_{0}}$. Integrating any function ${\displaystyle \rho (t)}$ against that would pick out the value ${\displaystyle \rho (t_{0})}$. Rather integrating $${\displaystyle \int _{-\infty }^{\infty }{\frac {\varepsilon /\pi }{(t-t_{0})^{2}+\varepsilon ^{2}}}\rho (t)\,dt}$$ for some ${\displaystyle \varepsilon >0}$ instead produces the value at ${\displaystyle t_{0}}$ for some smoothed variant of ${\displaystyle \rho }$—the smaller the value of ${\displaystyle \varepsilon }$, the less smoothing is applied. Used in this way, the factor ${\displaystyle {\frac {\varepsilon /\pi }{(t-t_{0})^{2}+\varepsilon ^{2}}}}$ is also known as the Poisson kernel (for the half-plane).¹

The denominator ${\displaystyle (t-t_{0})^{2}+\varepsilon ^{2}}$ has no real zeroes, but it has two complex zeroes ${\displaystyle t=t_{0}\pm i\varepsilon }$, and thus there is a partial fraction decomposition $${\displaystyle {\frac {\varepsilon /\pi }{(t-t_{0})^{2}+\varepsilon ^{2}}}={\frac {1/2\pi i}{t-(t_{0}+i\varepsilon )}}-{\frac {1/2\pi i}{t-(t_{0}-i\varepsilon )}}}$$ Hence for any measure ${\displaystyle \mu }$, $${\displaystyle \int _{\mathbb {R} }{\frac {\varepsilon /\pi }{(t-x)^{2}+\varepsilon ^{2}}}d\mu (t)={\frac {1}{2\pi i}}\int _{\mathbb {R} }\left({\frac {1}{t-(x+i\varepsilon )}}-{\frac {1}{t-(x-i\varepsilon )}}\right)d\mu (t)={\frac {S_{\mu }(x+i\varepsilon )-S_{\mu }(x-i\varepsilon )}{2\pi i}}}$$ If the measure ${\displaystyle \mu }$ is absolutely continuous (with respect to the Lebesgue measure) at ${\displaystyle x}$ then as ${\displaystyle \varepsilon \to 0^{+}}$ that integral tends to the density at ${\displaystyle x}$. If instead the measure has a point mass at ${\displaystyle x}$, then the limit as ${\displaystyle \varepsilon \to 0^{+}}$ of the integral diverges, and the Stieltjes transform ${\displaystyle S_{\mu }}$ has a pole at ${\displaystyle x}$.

If the measure of density ρ has moments of any order defined for each integer by the equality $${\displaystyle m_{n}=\int _{I}t^{n}\,\rho (t)\,dt,}$$

then the Stieltjes transformation of ρ admits for each integer n the asymptotic expansion in the neighbourhood of infinity given by $${\displaystyle S_{\rho }(z)=\sum _{k=0}^{n}{\frac {m_{k}}{z^{k+1}}}+o\left({\frac {1}{z^{n+1}}}\right).}$$

Under certain conditions the complete expansion as a Laurent series can be obtained: $${\displaystyle S_{\rho }(z)=\sum _{n=0}^{\infty }{\frac {m_{n}}{z^{n+1}}}.}$$

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 $${\displaystyle Q_{n}(x)=\int _{I}{\frac {P_{n}(t)-P_{n}(x)}{t-x}}\rho (t)\,dt.}$$

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 $${\displaystyle S_{\rho }(z)-{\frac {Q_{n}(z)}{P_{n}(z)}}=O\left({\frac {1}{z^{2n+1}}}\right).}$$

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.)