Stieltjes transformation

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

S ρ ( z ) = ∫ I ρ ( t ) d t z − t , z ∈ C ∖ I . {\displaystyle S_{\rho }(z)=\int _{I}{\frac {\rho (t)\,dt}{z-t}},\qquad z\in \mathbb {C} \setminus I.}

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 . {\displaystyle \rho (x)=\lim _{\varepsilon \to 0^{+}}{\frac {S_{\rho }(x-i\varepsilon )-S_{\rho }(x+i\varepsilon )}{2i\pi }}.}

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 , {\displaystyle m_{n}=\int _{I}t^{n}\,\rho (t)\,dt,}

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 ) . {\displaystyle S_{\rho }(z)=\sum _{k=0}^{n}{\frac {m_{k}}{z^{k+1}}}+o\left({\frac {1}{z^{n+1}}}\right).}

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

S ρ ( z ) = ∑ n = 0 ∞ m n z n + 1 . {\displaystyle S_{\rho }(z)=\sum _{n=0}^{\infty }{\frac {m_{n}}{z^{n+1}}}.}

Relationships to orthogonal polynomials

The correspondence ( f , g ) ↦ ∫ I f ( t ) g ( t ) ρ ( t ) d t {\textstyle (f,g)\mapsto \int _{I}f(t)g(t)\rho (t)\,dt} 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 . {\displaystyle Q_{n}(x)=\int _{I}{\frac {P_{n}(t)-P_{n}(x)}{t-x}}\rho (t)\,dt.}

It appears that F n ( z ) = Q n ( z ) P n ( z ) {\textstyle F_{n}(z)={\frac {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 ) . {\displaystyle S_{\rho }(z)-{\frac {Q_{n}(z)}{P_{n}(z)}}=O\left({\frac {1}{z^{2n}}}\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 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.)

See also

  • Orthogonal polynomials
  • Secondary polynomials
  • Secondary measure

References

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