mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates Limit of a function#Limits at infinity, tends to infinity. In pro ...

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 In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansio ...

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, ofte ...

on the space of

continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

s 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 In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geom ...

for this product, we can create the sequence of associated secondary polynomials 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 A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...

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.)

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

Connections with moments of measures

Relationships to orthogonal polynomials

See also

References