Stieltjes transformation


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_{ ho}(z)=int_Ifrac{ ho(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

: ho(x)=underset{varepsilon ightarrow 0^+}{ ext{limfrac{S_{ ho}(x-ivarepsilon)-S_{ ho}(x+ivarepsilon)}{2ipi}.

Links with the moments of the measure

If the measure of density ρ has moments of any order defined for each integer by the equality

:c_{n}=int_I t^n, ho(t),dt,

then the Stieltjes transformation of ρ admits for each integer "n" the asymptotic expansion in the neighbourhood of infinity given by

:S_{ ho}(z)=sum_{k=0}^{k=n}frac{c_k}{z^{k+1+oleft(frac{1}{z^{n+1 ight).

Under certain conditions the complete expansion as a Laurent series can be obtained: :S_{ ho}(z)=sum_{n=0}^{n=infty}frac{c_n}{z^{n+1.

Relationships to the orthogonal polynomials

The correspondence (f,g)mapsto int_I f(t)g(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} ho (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_ ho(z)-frac{Q_n(z)}{P_n(z)}=Oleft(frac{1}{z^{2n ight).

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.)

ee also

* Orthogonal polynomials
* Secondary polynomials
* Secondary measure

References

*cite book|author = H. S. Wall|title = Analytic Theory of Continued Fractions|publisher = D. Van Nostrand Company Inc.|year = 1948


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