Stieltjes transformation explained

In mathematics, the Stieltjes transformation of a measure of density on a real interval is the function of the complex variable defined outside by the formula

$$S\_(z)=\backslash int\_I\backslash frac,\; \backslash qquad\; z\; \backslash in\; \backslash mathbb\; \backslash setminus\; I.$$

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

$$\backslash rho(x)=\backslash lim\_\; \backslash frac.$$

Connections with moments of measures

See main article: Moment problem.

If the measure of density has moments of any order defined for each integer by the equality$$m\_=\backslash int\_I\; t^n\backslash ,\backslash rho(t)\backslash ,dt,$$

then the Stieltjes transformation of admits for each integer the asymptotic expansion in the neighbourhood of infinity given by$$S\_(z)=\backslash sum\_^\backslash frac+o\backslash left(\backslash frac\backslash right).$$

Under certain conditions the complete expansion as a Laurent series can be obtained:$$S\_(z)\; =\; \backslash sum\_^\backslash frac.$$

Relationships to orthogonal polynomials

The correspondence $(f,g)\; \backslash mapsto\; \backslash int\_I\; f(t)\; g(t)\; \backslash rho(t)\; \backslash ,\; dt$ defines an inner product on the space of continuous functions on the interval .

If is a sequence of orthogonal polynomials for this product, we can create the sequence of associated secondary polynomials by the formula$$Q\_n(x)=\backslash int\_I\; \backslash frac\backslash rho\; (t)\backslash ,dt.$$

It appears that $F\_n(z)\; =\; \backslash frac$ is a Padé approximation of in a neighbourhood of infinity, in the sense that$$S\_\backslash rho(z)-\backslash frac=O\backslash left(\backslash frac\backslash right).$$

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 .

The Stieltjes transformation can also be used to construct from the density an effective measure for transforming the secondary polynomials into an orthogonal system. (For more details see the article secondary measure.)

See also

• Orthogonal polynomials
• Secondary polynomials
• Secondary measure

References

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