Stieltjes transformation

Stieltjes transformation: translation

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 {"P_n"} is a sequence of orthogonal polynomials for this product, we can create the sequence of associated secondary polynomials by the formula

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

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 "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.)

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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