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 {"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