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 $$S_{\rho }(z)=\underset{I}{\int }\frac{\rho (t)dt}{t-z},z\in \mathbb{ℂ}\setminus I.$$ 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 $$\rho (x)=\underset{\epsilon \to 0^{+}}{\lim }\frac{S_{\rho }(x-i\epsilon )-S_{\rho }(x+i\epsilon )}{2i\pi }.$$

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 $$m_n=\underset{I}{\int }{t}^n\rho (t)dt,$$ 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 $$S_{\rho }(z)-\frac{Q_n(z)}{P_n(z)}=O\left(\frac{1}{z^{2n}}\right).$$ 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.