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Stieltjes transformation

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In mathematics, the Stieltjes transformation Sρ(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

Contents

S ρ ( z ) = I ρ ( t ) d t z t , t I R , z C R

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

ρ ( x ) = lim ε 0 + S ρ ( x i ε ) S ρ ( x + i ε ) 2 i π .

Connections with moments of measures

If the measure of density ρ has moments of any order defined for each integer by the equality

m n = I t n ρ ( t ) d t ,

then the Stieltjes transformation of ρ admits for each integer n the asymptotic expansion in the neighbourhood of infinity given by

S ρ ( z ) = k = 0 n m k z k + 1 + o ( 1 z n + 1 ) .

Under certain conditions the complete expansion as a Laurent series can be obtained:

S ρ ( z ) = n = 0 m n z n + 1 .

Relationships to orthogonal polynomials

The correspondence ( f , g ) I f ( t ) g ( t ) ρ ( t ) d t defines an inner product on the space of continuous functions on the interval I.

If {Pn} is a sequence of orthogonal polynomials for this product, we can create the sequence of associated secondary polynomials by the formula

Q n ( x ) = I P n ( t ) P n ( x ) t x ρ ( t ) d t .

It appears that F n ( z ) = Q n ( z ) P n ( z ) is a Padé approximation of Sρ(z) in a neighbourhood of infinity, in the sense that

S ρ ( z ) Q n ( z ) P n ( z ) = O ( 1 z 2 n ) .

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 Fn(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.)

References

Stieltjes transformation Wikipedia