Stieltjes transformation

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

Derivation of formula

Recall from basic calculus that \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 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 x = (t - t_0) / \varepsilon we get the full family of Cauchy distributions: 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 \varepsilon \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_{-\infty}^\infty \frac{ \varepsilon/\pi }{ (t-t_0)^2 + \varepsilon^2 } \rho(t) , dt for some \varepsilon 0 instead produces the value at t_0 for some smoothed variant of \rho —the smaller the value of \varepsilon , the less smoothing is applied. Used in this way, the factor \frac{ \varepsilon/\pi }{ (t-t_0)^2 + \varepsilon^2 } is also known as the Poisson kernel (for the half-plane).

The denominator (t-t_0)^2 + \varepsilon^2 has no real zeroes, but it has two complex zeroes t = t_0 \pm i\varepsilon , and thus there is a partial fraction decomposition \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 \mu , \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 \mu is absolutely continuous (with respect to the Lebesgue measure) at x then as \varepsilon \to 0^+ that integral tends to the density at x . If instead the measure has a point mass at x , then the limit as \varepsilon \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}=\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 {Pn} 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+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 Fn(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

References

1. (2021). "Computing Spectral Measures and Spectral Types". Communications in Mathematical Physics.