Stieltjes transformation
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
Inverse formula
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 intervalDerivation of formula
Recall from basic calculus thatHence is the probability density function of a distribution—a Cauchy distribution. Via the change of variables we get the full family of Cauchy distributions:
As, these tend to a Dirac distribution with the mass at. Integrating any function against that would pick out the value. Rather integrating
for some instead produces the value at for some smoothed variant of —the smaller the value of, the less smoothing is applied. Used in this way, the factor is also known as the Poisson kernel.
The denominator has no real zeroes, but it has two complex zeroes, and thus there is a partial fraction decomposition
Hence for any measure,
If the measure is absolutely continuous at then as that integral tends to the density at. If instead the measure has a point mass at, then the limit as of the integral diverges, and the Stieltjes transform has a pole at.
Connections with moments of measures
If the measure of density has moments of any order defined for each integer by the equalitythen the Stieltjes transformation of admits for each integer the asymptotic expansion in the neighbourhood of infinity given by
Under certain conditions the complete expansion as a Laurent series can be obtained:
Relationships to orthogonal polynomials
The correspondence 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
It appears that is a Padé approximation of in a neighbourhood of infinity, in the sense that
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.