# Stieltjes transformation

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Stieltjes transformation

In mathematics, the Stieltjes transformation "S"&rho;("z") of a measure of density &rho; on a real interval "I" is the function of the complex variable "z" defined outside "I" by the formula

:$S_\left\{ ho\right\}\left(z\right)=int_Ifrac\left\{ ho\left(t\right),dt\right\}\left\{z-t\right\}.$

Under certain conditions we can reconstitute the density function &rho; starting from its Stieltjes transformation thanks to the inverse formula of Stieltjes-Perron. For example, if the density &rho; is continuous throughout "I", one will have inside this interval

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

Links with the moments of the measure

If the measure of density &rho; has moments of any order defined for each integer by the equality

:$c_\left\{n\right\}=int_I t^n, ho\left(t\right),dt,$

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

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

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

Relationships to the orthogonal polynomials

The correspondence $\left(f,g\right)mapsto int_I f\left(t\right)g\left(t\right),dt$ 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

: $Q_n\left(x\right)=int_I frac\left\{P_n \left(t\right)-P_n \left(x\right)\right\}\left\{t-x\right\} ho \left(t\right),dt.$

It appears that $F_n\left(z\right)=frac\left\{Q_n\left(z\right)\right\}\left\{P_n\left(z\right)\right\}$ is a Padé approximation of "S"&rho;("z") in a neighbourhood of infinity, in the sense that

: $S_ ho\left(z\right)-frac\left\{Q_n\left(z\right)\right\}\left\{P_n\left(z\right)\right\}=Oleft\left(frac\left\{1\right\}\left\{z^\left\{2n ight\right).$

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 &rho; 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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