Stieltjes transformation

Short description: Mathematical 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

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

Inverse formula

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

Derivation of formula

Recall from basic calculus that $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{1}{x^2+1}dx=\underset{x\to \mathrm{\infty }}{\lim }\arctan x-\underset{x\to -\mathrm{\infty }}{\lim }\arctan x=\frac{\pi }{2}-(-\frac{\pi }{2})=\pi \text{.}$$ Hence $f(x)=\frac{1}{\pi }(x^2+1{)}^{-1}$ is the probability density function of a distribution—a Cauchy distribution. Via the change of variables $x=(t-t_0)/\epsilon$ we get the full family of Cauchy distributions: $$1=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{1/\pi }{x^2+1}dx=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{1/\pi }{(\frac{t-t_0}{\epsilon }{)}^2+1}\frac{dx}{dt}dt=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\epsilon /\pi }{(t-t_0{)}^2+{\epsilon }^2}dt$$ As $\epsilon \to 0^{+}$, these tend to a Dirac distribution with the mass at $t_0$. Integrating any function $\rho (t)$ against that would pick out the value $\rho (t_0)$. Rather integrating $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\epsilon /\pi }{(t-t_0{)}^2+{\epsilon }^2}\rho (t)dt$$ for some $\epsilon >0$ instead produces the value at $t_0$ for some smoothed variant of $\rho$—the smaller the value of $\epsilon$, the less smoothing is applied. Used in this way, the factor $\frac{\epsilon /\pi }{(t-t_0{)}^2+{\epsilon }^2}$ is also known as the Poisson kernel (for the half-plane).

The denominator $(t-t_0{)}^2+{\epsilon }^2$ has no real zeroes, but it has two complex zeroes $t=t_0\pm i\epsilon$, and thus there is a partial fraction decomposition $$\frac{\epsilon /\pi }{(t-t_0{)}^2+{\epsilon }^2}=\frac{1/2\pi i}{t-(t_0+i\epsilon )}-\frac{1/2\pi i}{t-(t_0-i\epsilon )}$$ Hence for any measure $\mu$, $$\underset{\mathbb{ℝ}}{\int }\frac{\epsilon /\pi }{(t-x{)}^2+{\epsilon }^2}d\mu (t)=\frac{1}{2\pi i}\underset{\mathbb{ℝ}}{\int }(\frac{1}{t-(x+i\epsilon )}-\frac{1}{t-(x-i\epsilon )})d\mu (t)=\frac{S_{\mu }(x+i\epsilon )-S_{\mu }(x-i\epsilon )}{2\pi i}$$ If the measure $\mu$ is absolutely continuous (with respect to the Lebesgue measure) at $x$ then as $\epsilon \to 0^{+}$ that integral tends to the density at $x$. If instead the measure has a point mass at $x$, then the limit as $\epsilon \to 0^{+}$ of the integral diverges, and the Stieltjes transform $S_{\mu }$ has a pole at $x$.

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 $$m_n=\underset{I}{\int }{t}^n\rho (t)dt,$$

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: $$S_{\rho }(z)=\sum _{n=0}^{\mathrm{\infty }}\frac{m_n}{z^{n+1}}.$$

Relationships to orthogonal polynomials

The correspondence $(f,g)\mapsto \underset{I}{\int }f(t)g(t)\rho (t)dt$ 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 $$Q_n(x)=\underset{I}{\int }\frac{P_n(t)-P_n(x)}{t-x}\rho (t)dt.$$

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

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.)

See also

• Orthogonal polynomials
• Secondary polynomials
• Secondary measure
• Hilbert transform

References

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