Namespaces
Variants
Actions

Difference between revisions of "Borel transform"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
m (gather refs)
 
(2 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 +
<!--
 +
b0171901.png
 +
$#A+1 = 27 n = 0
 +
$#C+1 = 27 : ~/encyclopedia/old_files/data/B017/B.0107190 Borel transform
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
An integral transform of the type
 
An integral transform of the type
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b0171901.png" /></td> </tr></table>
+
$$
 +
\gamma (t)  = \int\limits _ { 0 } ^  \infty 
 +
f(z)e  ^ {-zt}  dz,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b0171902.png" /> is an entire function of exponential type. The Borel transform is a special case of the [[Laplace transform|Laplace transform]]. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b0171903.png" /> is called the Borel transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b0171904.png" />. If
+
where $  f(z) $
 +
is an entire [[function of exponential type]]. The Borel transform is a special case of the [[Laplace transform|Laplace transform]]. The function $  \gamma (t) $
 +
is called the Borel transform of $  f(z) $.  
 +
If
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b0171905.png" /></td> </tr></table>
+
$$
 +
f(z)  = \sum _ { n=0 } ^  \infty 
 +
 
 +
\frac{a _ {n} }{n!}
 +
z  ^ {n} ,
 +
$$
  
 
then
 
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b0171906.png" /></td> </tr></table>
+
$$
 +
\gamma (t)  = \sum _ { v=0 } ^  \infty 
 +
a _ {v} t ^ {-(v+1) } ;
 +
$$
  
the series converges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b0171907.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b0171908.png" /> is the type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b0171909.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719010.png" /> be the smallest closed convex set containing all the singularities of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719011.png" />; let
+
the series converges for $  | t | > \sigma $,  
 +
where $  \sigma $
 +
is the type of $  f(z) $.  
 +
Let $  \overline{D}\; $
 +
be the smallest closed convex set containing all the singularities of the function $  \gamma (t) $;  
 +
let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719012.png" /></td> </tr></table>
+
$$
 +
K( \phi )  = \max _ {z \in \overline{D}\; } \
 +
\mathop{\rm Re} (ze ^ {-i \phi } )
 +
$$
  
be the supporting function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719013.png" />; and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719014.png" /> be the growth indicator function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719015.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719016.png" />. If in a Borel transform the integration takes place over a ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719017.png" />, the corresponding integral will converge in the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719018.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719019.png" /> be a closed contour surrounding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719020.png" />; then
+
be the supporting function of $  \overline{D}\; $;  
 +
and let $  h ( \phi ) $
 +
be the growth indicator function of $  f(z) $;  
 +
then $  K( \phi ) = h( - \phi ) $.  
 +
If in a Borel transform the integration takes place over a ray $  \mathop{\rm arg}  z = \phi $,  
 +
the corresponding integral will converge in the half-plane $  x  \cos  \phi + y  \sin  \phi > K ( - \phi ) $.  
 +
Let $  C $
 +
be a closed contour surrounding $  \overline{D}\; $;  
 +
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719021.png" /></td> </tr></table>
+
$$
 +
f(z)  =
 +
\frac{1}{2 \pi i }
  
If additional conditions are imposed, other representations may be deduced from this formula. Thus, consider the class of entire functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719022.png" /> of exponential type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719023.png" /> for which
+
\int\limits _ { C } \gamma (t) e  ^ {zt}  dt.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719024.png" /></td> </tr></table>
+
If additional conditions are imposed, other representations may be deduced from this formula. Thus, consider the class of entire functions  $  f(z) $
 +
of exponential type  $  \leq  \sigma $
 +
for which
  
This class is identical with the class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719025.png" /> that can be represented as
+
$$
 +
\int\limits _ {- \infty } ^  \infty 
 +
| f(x) |  ^ {2}  dx  < \infty .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719026.png" /></td> </tr></table>
+
This class is identical with the class of functions  $  f(z) $
 +
that can be represented as
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719027.png" />.
+
$$
 +
f(z)  = \
  
====References====
+
\frac{1}{\sqrt {2 \pi } }
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Borel, "Leçons sur les series divergentes" , Gauthier-Villars (1928) {{MR|}} {{ZBL|54.0223.01}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.M. Dzhrbashyan, "Integral transforms and representation of functions in the complex domain" , Moscow (1966) (In Russian)</TD></TR></table>
 
  
 +
\int\limits _ {- \sigma } ^  \sigma 
 +
e  ^ {izt} \phi (t)  dt,
 +
$$
  
 +
where  $  \phi (t) \in {L _ {2} } ( - \sigma , \sigma ) $.
  
 
====Comments====
 
====Comments====
The statement at the end of the article above is called the [[Paley–Wiener theorem|Paley–Wiener theorem]].
+
The statement at the end of the article above is called the [[Paley–Wiener theorem]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.P. Boas, "Entire functions" , Acad. Press (1954) {{MR|0068627}} {{ZBL|0058.30201}} </TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> E. Borel, "Leçons sur les séries divergentes" , Gauthier-Villars (1928) {{ZBL|54.0223.01}} </TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top"> M.M. Dzhrbashyan, "Integral transforms and representation of functions in the complex domain" , Moscow (1966) (In Russian)</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> R.P. Boas, "Entire functions" , Acad. Press (1954) {{MR|0068627}} {{ZBL|0058.30201}} </TD></TR>
 +
</table>

Latest revision as of 17:35, 11 November 2023


An integral transform of the type

$$ \gamma (t) = \int\limits _ { 0 } ^ \infty f(z)e ^ {-zt} dz, $$

where $ f(z) $ is an entire function of exponential type. The Borel transform is a special case of the Laplace transform. The function $ \gamma (t) $ is called the Borel transform of $ f(z) $. If

$$ f(z) = \sum _ { n=0 } ^ \infty \frac{a _ {n} }{n!} z ^ {n} , $$

then

$$ \gamma (t) = \sum _ { v=0 } ^ \infty a _ {v} t ^ {-(v+1) } ; $$

the series converges for $ | t | > \sigma $, where $ \sigma $ is the type of $ f(z) $. Let $ \overline{D}\; $ be the smallest closed convex set containing all the singularities of the function $ \gamma (t) $; let

$$ K( \phi ) = \max _ {z \in \overline{D}\; } \ \mathop{\rm Re} (ze ^ {-i \phi } ) $$

be the supporting function of $ \overline{D}\; $; and let $ h ( \phi ) $ be the growth indicator function of $ f(z) $; then $ K( \phi ) = h( - \phi ) $. If in a Borel transform the integration takes place over a ray $ \mathop{\rm arg} z = \phi $, the corresponding integral will converge in the half-plane $ x \cos \phi + y \sin \phi > K ( - \phi ) $. Let $ C $ be a closed contour surrounding $ \overline{D}\; $; then

$$ f(z) = \frac{1}{2 \pi i } \int\limits _ { C } \gamma (t) e ^ {zt} dt. $$

If additional conditions are imposed, other representations may be deduced from this formula. Thus, consider the class of entire functions $ f(z) $ of exponential type $ \leq \sigma $ for which

$$ \int\limits _ {- \infty } ^ \infty | f(x) | ^ {2} dx < \infty . $$

This class is identical with the class of functions $ f(z) $ that can be represented as

$$ f(z) = \ \frac{1}{\sqrt {2 \pi } } \int\limits _ {- \sigma } ^ \sigma e ^ {izt} \phi (t) dt, $$

where $ \phi (t) \in {L _ {2} } ( - \sigma , \sigma ) $.

Comments

The statement at the end of the article above is called the Paley–Wiener theorem.

References

[1] E. Borel, "Leçons sur les séries divergentes" , Gauthier-Villars (1928) Zbl 54.0223.01
[2] M.M. Dzhrbashyan, "Integral transforms and representation of functions in the complex domain" , Moscow (1966) (In Russian)
[a1] R.P. Boas, "Entire functions" , Acad. Press (1954) MR0068627 Zbl 0058.30201
How to Cite This Entry:
Borel transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel_transform&oldid=24385
This article was adapted from an original article by A.F. Leont'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article