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| In mathematics, the '''Stieltjes transformation''' ''S''<sub>ρ</sub>(''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
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| :<math>S_{\rho}(z)=\int_I\frac{\rho(t)\,dt}{z-t}.</math>
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| 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
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| :<math>\rho(x)=\underset{\varepsilon\rightarrow 0^+}{\text{lim}}\frac{S_{\rho}(x-i\varepsilon)-S_{\rho}(x+i\varepsilon)}{2i\pi}.</math>
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| ==Connections with moments of measures==
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| {{Main|moment problem}}
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| If the measure of density ρ has [[moment (mathematics)|moments]] of any order defined for each integer by the equality
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| :<math>m_{n}=\int_I t^n\,\rho(t)\,dt,</math>
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| then the [[Stieltjes]] transformation of ρ admits for each integer ''n'' the [[Asymptotic analysis|asymptotic]] expansion in the neighbourhood of infinity given by
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| :<math>S_{\rho}(z)=\sum_{k=0}^{k=n}\frac{m_k}{z^{k+1}}+o\left(\frac{1}{z^{n+1}}\right).</math>
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| Under certain conditions the complete expansion as a [[Laurent series]] can be obtained:
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| :<math>S_{\rho}(z)=\sum_{n=0}^{n=\infty}\frac{m_n}{z^{n+1}}.</math>
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| ==Relationships to orthogonal polynomials==
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| The correspondence <math>(f,g)\mapsto \int_I f(t)g(t)\rho(t)\,dt</math> defines an [[inner product]] on the space of [[continuous function]]s on the interval ''I''.
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| If {''P<sub>n</sub>''} is a sequence of [[orthogonal polynomials]] for this product, we can create the sequence of associated [[secondary polynomials]] by the formula
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| : <math>Q_n(x)=\int_I \frac{P_n (t)-P_n (x)}{t-x}\rho (t)\,dt.</math> | |
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| It appears that <math>F_n(z)=\frac{Q_n(z)}{P_n(z)}</math> is a [[Padé approximation]] of ''S''<sub>ρ</sub>(''z'') in a neighbourhood of infinity, in the sense that
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| : <math>S_\rho(z)-\frac{Q_n(z)}{P_n(z)}=O\left(\frac{1}{z^{2n}}\right).</math>
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| Since these two sequences of polynomials satisfy the same recurrence relation in three terms, we can develop a [[generalized continued fraction|continued fraction]] for the Stieltjes transformation whose successive [[Convergent (continued fraction)|convergents]] are the fractions ''F<sub>n</sub>''(''z'').
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| 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]].)
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| ==See also== | |
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| * [[Orthogonal polynomials]]
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| * [[Secondary polynomials]]
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| * [[Secondary measure]]
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| ==References==
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| *{{cite book|author = H. S. Wall|title = Analytic Theory of Continued Fractions|publisher = D. Van Nostrand Company Inc.|year = 1948}}
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| [[Category:Integral transforms]]
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| [[Category:Continued fractions]]
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