Stieltjes transformation

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In mathematics, the Stieltjes transformation Sρ(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)=\int _{I}{\frac {\rho (t)\,dt}{z-t}}.

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 {\varepsilon \rightarrow 0^{+}}{{\text{lim}}}}{\frac {S_{{\rho }}(x-i\varepsilon )-S_{{\rho }}(x+i\varepsilon )}{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}}=\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}}^{{k=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}}^{{n=\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 Sρ(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 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.)

See also

References

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