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Difference between revisions of "Stieltjes transform"

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$$ \tag{* }
 
$$ \tag{* }
F( x)  =  \int\limits _ { 0 } ^  \infty   f(
+
F( x)  =  \int\limits _ { 0 } ^  \infty \frac{f(t)}{x+t} dt.
\frac{x)}{x+}
 
dt.
 
 
$$
 
$$
  
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\frac{e}{n}
 
\frac{e}{n}
 
  \right )  ^ {2n} [ x
 
  \right )  ^ {2n} [ x
  ^ {2n} F ^ { ( n) } ( x)]  ^ {(} n)  =  f( x)
+
  ^ {2n} F ^ { ( n) } ( x)]  ^ {(n)} =  f( x)
 
$$
 
$$
  
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$$  
 
$$  
F( x)  =  \int\limits _ { 0 } ^  \infty   f( t)  
+
F( x)  =  \int\limits _ { 0 } ^  \infty
\frac{dt}{( x+ t)  ^  \rho  }
+
\frac{f(t)}{( x+ t)  ^  \rho  } dt
 
  ,
 
  ,
 
$$
 
$$
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$$  
 
$$  
 
K( x, t)  =  \left \{
 
K( x, t)  =  \left \{
 +
 +
\begin{array}{ll}
 +
 +
\frac{ \mathop{\rm ln}  x / t }{x-t} ,  & t \neq x,  \\
 +
 +
\frac{1}{x} ,  & t = x.  \\
 +
\end{array}
 +
 +
\right .$$
  
 
Stieltjes transforms are also introduced for generalized functions. The transform (*) was studied by Th.J. Stieltjes (1894–1895).
 
Stieltjes transforms are also introduced for generalized functions. The transform (*) was studied by Th.J. Stieltjes (1894–1895).

Latest revision as of 14:00, 28 June 2020


The integral transform

$$ \tag{* } F( x) = \int\limits _ { 0 } ^ \infty \frac{f(t)}{x+t} dt. $$

The Stieltjes transform arises in the iteration of the Laplace transform and is also a particular case of a convolution transform.

One of the inversion formulas is as follows: If the function $ f( t) \sqrt t $ is continuous and bounded on $ ( 0, \infty ) $, then

$$ \lim\limits _ {n \rightarrow \infty } \frac{(- 1) ^ {n} }{2 \pi } \left ( \frac{e}{n} \right ) ^ {2n} [ x ^ {2n} F ^ { ( n) } ( x)] ^ {(n)} = f( x) $$

for $ x \in ( 0, \infty ) $.

The generalized Stieltjes transform is

$$ F( x) = \int\limits _ { 0 } ^ \infty \frac{f(t)}{( x+ t) ^ \rho } dt , $$

where $ \rho $ is a complex number.

The integrated Stieltjes transform is

$$ F( x) = \int\limits _ { 0 } ^ \infty K( x, t) f( t) dt, $$

where

$$ K( x, t) = \left \{ \begin{array}{ll} \frac{ \mathop{\rm ln} x / t }{x-t} , & t \neq x, \\ \frac{1}{x} , & t = x. \\ \end{array} \right .$$

Stieltjes transforms are also introduced for generalized functions. The transform (*) was studied by Th.J. Stieltjes (1894–1895).

References

[1] D.V. Widder, "The Laplace transform" , Princeton Univ. Press (1972)
[2] R.P. Boas, D.V. Widder, "The iterated Stieltjes transform" Trans. Amer. Math. Soc. , 45 (1939) pp. 1–72
[3] E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948)
[4] Y.A. Brychkov, A.P. Prudnikov, "Integral transforms of generalized functions" , Gordon & Breach (1989) (Translated from Russian)
How to Cite This Entry:
Stieltjes transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stieltjes_transform&oldid=48841
This article was adapted from an original article by Yu.A. BrychkovA.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article