Stieltjes transformation

Template:Short description Template:No footnotes In mathematics, the Stieltjes transformation Template:Math of a measure of density Template:Math on a real interval Template:Mvar is the function of the complex variable Template:Mvar defined outside Template:Mvar by the formula

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

Inverse formula

Under certain conditions we can reconstitute the density function Template:Math starting from its Stieltjes transformation thanks to the inverse formula of Stieltjes–Perron. For example, if the density Template:Math is continuous throughout Template:Mvar, 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 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1}{x^2+1}dx=\underset{x\to \mathrm{\infty }}{lim}\arctan x-\underset{x\to -\mathrm{\infty }}{lim}\arctan x=\frac{\pi }{2}-(-\frac{\pi }{2})=\pi \text{.}$$ 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: $$1={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1/\pi }{x^2+1}dx={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{1/\pi }{(\frac{t-t_0}{\epsilon }{)}^2+1}\frac{dx}{dt}dt={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\epsilon /\pi }{(t-t_0{)}^2+{\epsilon }^2}dt$$ 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 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\epsilon /\pi }{(t-t_0{)}^2+{\epsilon }^2}\rho (t)dt$$ 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 $\mu$, $${\int }_{ℝ}\frac{\epsilon /\pi }{(t-x{)}^2+{\epsilon }^2}d\mu (t)=\frac{1}{2\pi i}{\int }_{ℝ}(\frac{1}{t-(x+i\epsilon )}-\frac{1}{t-(x-i\epsilon )})d\mu (t)=\frac{S_{\mu }(x+i\epsilon )-S_{\mu }(x-i\epsilon )}{2\pi i}$$ 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

If the measure of density Template:Math 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 Template:Math admits for each integer Template:Mvar 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 $(f,g)\mapsto {\int }_{I}f(t)g(t)\rho (t)dt$ defines an inner product on the space of continuous functions on the interval Template:Mvar.

If Template:Math 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 Template:Math 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 Template:Math.

The Stieltjes transformation can also be used to construct from the density Template:Math 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

• Template:Cite book