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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)=\int_I\frac, \qquad z \in \mathbb \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 \rho(x)=\lim_ \frac.


Connections with moments of measures

If the measure of density has moments of any order defined for each integer by the equality m_=\int_I t^n\,\rho(t)\,dt, then the Stieltjes transformation of admits for each integer the asymptotic expansion in the neighbourhood of infinity given by S_(z)=\sum_^\frac+o\left(\frac\right). Under certain conditions the complete expansion as a Laurent series can be obtained: S_(z) = \sum_^\frac.


Relationships to orthogonal polynomials

The correspondence (f,g) \mapsto \int_I f(t) g(t) \rho(t) \, dt defines an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on the space of continuous functions on the interval . If is a sequence of
orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the class ...
for this product, we can create the sequence of associated
secondary polynomials In mathematics, the secondary polynomials \ associated with a sequence \ of polynomials orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to ...
by the formula Q_n(x)=\int_I \frac\rho (t)\,dt. It appears that F_n(z) = \frac is a Padé approximation of in a neighbourhood of infinity, in the sense that S_\rho(z)-\frac=O\left(\frac\right). Since these two sequences of polynomials satisfy the same recurrence relation in three terms, we can develop a
continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
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 In mathematics, the secondary measure associated with a measure of positive density ρ when there is one, is a measure of positive density μ, turning the secondary polynomials associated with the orthogonal polynomials for ρ into an orthogonal ...
.)


See also

*
Orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the class ...
*
Secondary polynomials In mathematics, the secondary polynomials \ associated with a sequence \ of polynomials orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to ...
*
Secondary measure In mathematics, the secondary measure associated with a measure of positive density ρ when there is one, is a measure of positive density μ, turning the secondary polynomials associated with the orthogonal polynomials for ρ into an orthogonal ...


References

*{{cite book, author = H. S. Wall, title = Analytic Theory of Continued Fractions, publisher = D. Van Nostrand Company Inc., year = 1948 Integral transforms Continued fractions