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The basic form of the theory of regular variation, a subject initiated in 1930 by the Yugoslav mathematician J. Karamata.
 
The basic form of the theory of regular variation, a subject initiated in 1930 by the Yugoslav mathematician J. Karamata.
  
 
Viewed from a modern perspective, Karamata theory is the study of asymptotic relations of the form
 
Viewed from a modern perspective, Karamata theory is the study of asymptotic relations of the form
  
<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/k/k110/k110030/k1100301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
{
 +
\frac{f ( \lambda x ) }{f ( x ) }
 +
} \rightarrow g ( \lambda ) \in ( 0, \infty )  ( x \rightarrow \infty ) ,  \forall \lambda > 0,
 +
$$
  
together with their consequences and ramifications. The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k1100302.png" /> is particularly important; measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k1100303.png" /> satisfying (a1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k1100304.png" /> are called slowly varying; such slowly varying functions are often written <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k1100305.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k1100306.png" /> (for  "lente" ).
+
together with their consequences and ramifications. The case $  g \equiv 1 $
 +
is particularly important; measurable functions $  f $
 +
satisfying (a1) with $  g \equiv 1 $
 +
are called slowly varying; such slowly varying functions are often written $  L $
 +
or $  {\mathcal l} $(
 +
for  "lente" ).
  
 
Many useful and interesting properties are implied by such relations. For instance:
 
Many useful and interesting properties are implied by such relations. For instance:
  
i) The uniform convergence theorem: for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k1100307.png" /> slowly varying, (a1) holds uniformly on compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k1100308.png" />-sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k1100309.png" />. There is a topological analogue, with measurability replaced by the [[Baire property|Baire property]].
+
i) The uniform convergence theorem: for $  f $
 +
slowly varying, (a1) holds uniformly on compact $  \lambda $-
 +
sets in $  ( 0, \infty ) $.  
 +
There is a topological analogue, with measurability replaced by the [[Baire property|Baire property]].
  
ii) The representation theorem: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003010.png" /> is slowly varying if and only if, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003011.png" /> large enough, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003012.png" /> is of the form
+
ii) The representation theorem: $  f $
 +
is slowly varying if and only if, for $  x $
 +
large enough, $  f $
 +
is of the form
  
<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/k/k110/k110030/k11003013.png" /></td> </tr></table>
+
$$
 +
f ( x ) = c ( x ) { \mathop{\rm exp} } \left ( \int\limits _ { a } ^ { x }  {\epsilon ( u ) }  { {
 +
\frac{du }{u}
 +
} } \right ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003015.png" /> are measurable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003017.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003018.png" />.
+
where $  c ( \cdot ) $,  
 +
$  \epsilon ( \cdot ) $
 +
are measurable, $  c ( x ) \rightarrow c \in ( 0, \infty ) $,
 +
$  \epsilon ( x ) \rightarrow 0 $
 +
as $  x \rightarrow \infty $.
  
iii) The characterization theorem: for measurable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003020.png" /> in (a1) must be of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003021.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003022.png" />, called the index of regular variation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003023.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003024.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003025.png" /> slowly varying (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003026.png" />).
+
iii) The characterization theorem: for measurable $  f $,  
 +
$  g ( \lambda ) $
 +
in (a1) must be of the form $  g ( \lambda ) \equiv \lambda  ^  \rho  $
 +
for some $  \rho \in \mathbf R $,  
 +
called the index of regular variation: $  f \in R _  \rho  $.  
 +
Then $  f ( x ) = x  ^  \rho  {\mathcal l} ( x ) $
 +
with $  {\mathcal l} $
 +
slowly varying ( $  {\mathcal l} \in R _ {0} $).
  
iv) Karamata's theorem: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003028.png" />, then
+
iv) Karamata's theorem: if $  f \in R _  \rho  $
 +
and $  \sigma > - ( \rho + 1 ) $,  
 +
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/k/k110/k110030/k11003029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a2 }
 +
{
 +
\frac{x ^ {\rho + 1 } f ( x ) }{\int\limits _ { a } ^ { x }  {t  ^  \sigma  f ( t ) }  {dt } }
 +
} \rightarrow \sigma + \rho + 1  ( x \rightarrow \infty ) .
 +
$$
  
(That is, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003031.png" /> "behaves asymptotically like a constant"  under integration.) Conversely, (a2) implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003032.png" />.
+
(That is, the $  {\mathcal l} $
 +
in $  f ( x ) = x  ^  \rho {\mathcal l} ( x ) $"
 +
behaves asymptotically like a constant"  under integration.) Conversely, (a2) implies $  f \in R _  \rho  $.
  
Perhaps the most important application of Karamata theory to analysis is Karamata's Tauberian theorem (or the Hardy–Littlewood–Karamata theorem): if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003033.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003034.png" />) is increasing, with Laplace–Stieltjes transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003035.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003036.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003037.png" />) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003039.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003040.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003041.png" />.
+
Perhaps the most important application of Karamata theory to analysis is Karamata's Tauberian theorem (or the Hardy–Littlewood–Karamata theorem): if $  f \in R _  \rho  $(
 +
$  \rho \geq  0 $)  
 +
is increasing, with Laplace–Stieltjes transform $  {\widehat{f}  } ( s ) = \int _ {0}  ^  \infty  {e ^ {- sx } }  {df ( x ) } $,  
 +
then $  f ( x ) \sim c { {x  ^  \rho  {\mathcal l} ( x ) } / {\Gamma ( 1 + \rho ) } } $(
 +
$  x \rightarrow \infty $)  
 +
with $  c \geq  0 $,  
 +
$  {\mathcal l} \in R _ {0} $
 +
if and only if $  {\widehat{f}  } ( s ) \sim c s ^ {- \rho } {\mathcal l} ( {1 / s } ) $
 +
$  ( s \downarrow 0 ) $.
  
 
For details, background and references on these and other results, see e.g. [[#References|[a1]]], Chap. 1.
 
For details, background and references on these and other results, see e.g. [[#References|[a1]]], Chap. 1.
  
The union over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003042.png" /> of the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003043.png" /> gives the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003044.png" /> of regularly varying functions. This is contained in the larger class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003045.png" /> of extended regularly varying functions, itself included in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003046.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003048.png" />-regularly varying functions: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003049.png" />. Just as a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003050.png" /> has an index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003051.png" /> of regular variation, and then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003052.png" />, so a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003053.png" /> has a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003054.png" /> of upper and lower Karamata indices (and these are equal, to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003055.png" /> say, if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003056.png" />), and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003057.png" /> has a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003058.png" /> of upper and lower Matuszewska indices. These larger classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110030/k11003060.png" /> have analogues of the results above; for instance, uniform convergence and representation theorems. For details, see e.g. [[#References|[a1]]], Chap. 2.
+
The union over all $  \rho \in \mathbf R $
 +
of the classes $  R _  \rho  $
 +
gives the class $  R $
 +
of regularly varying functions. This is contained in the larger class $  ER $
 +
of extended regularly varying functions, itself included in the class $  OR $
 +
of $  O $-
 +
regularly varying functions: $  R \subset  ER \subset  OR $.  
 +
Just as a function $  f \in R $
 +
has an index $  \rho $
 +
of regular variation, and then $  f \in R _  \rho  $,  
 +
so a function $  f \in ER $
 +
has a pair $  c ( f ) , d ( f ) $
 +
of upper and lower Karamata indices (and these are equal, to $  \rho $
 +
say, if and only if $  f \in R _  \rho  $),  
 +
and a function $  f \in OR $
 +
has a pair $  \alpha ( f ) , \beta ( f ) $
 +
of upper and lower Matuszewska indices. These larger classes $  ER $,  
 +
$  OR $
 +
have analogues of the results above; for instance, uniform convergence and representation theorems. For details, see e.g. [[#References|[a1]]], Chap. 2.
  
 
Karamata theory may be regarded as the  "first-order"  theory of regular variation. There is a corresponding  "second-order"  theory: [[De Haan theory|de Haan theory]] [[#References|[a1]]], Chap. 3.
 
Karamata theory may be regarded as the  "first-order"  theory of regular variation. There is a corresponding  "second-order"  theory: [[De Haan theory|de Haan theory]] [[#References|[a1]]], Chap. 3.
Line 36: Line 112:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N.H. Bingham,   C.M. Goldie,   J.L. Teugels,   "Regular variation" , ''Encycl. Math. Appl.'' , '''27''' , Cambridge Univ. Press (1989) (Edition: Second)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W. Feller,   "An introduction to probability theory and its applications" , '''2''' , Springer (1976) (Edition: Second)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N.H. Bingham, C.M. Goldie, J.L. Teugels, "Regular variation", ''Encycl. Math. Appl.'', '''27''', Cambridge Univ. Press (1989) (Edition: Second)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its applications"]], '''2''', Springer (1976) (Edition: Second)</TD></TR></table>

Latest revision as of 22:14, 5 June 2020


The basic form of the theory of regular variation, a subject initiated in 1930 by the Yugoslav mathematician J. Karamata.

Viewed from a modern perspective, Karamata theory is the study of asymptotic relations of the form

$$ \tag{a1 } { \frac{f ( \lambda x ) }{f ( x ) } } \rightarrow g ( \lambda ) \in ( 0, \infty ) ( x \rightarrow \infty ) , \forall \lambda > 0, $$

together with their consequences and ramifications. The case $ g \equiv 1 $ is particularly important; measurable functions $ f $ satisfying (a1) with $ g \equiv 1 $ are called slowly varying; such slowly varying functions are often written $ L $ or $ {\mathcal l} $( for "lente" ).

Many useful and interesting properties are implied by such relations. For instance:

i) The uniform convergence theorem: for $ f $ slowly varying, (a1) holds uniformly on compact $ \lambda $- sets in $ ( 0, \infty ) $. There is a topological analogue, with measurability replaced by the Baire property.

ii) The representation theorem: $ f $ is slowly varying if and only if, for $ x $ large enough, $ f $ is of the form

$$ f ( x ) = c ( x ) { \mathop{\rm exp} } \left ( \int\limits _ { a } ^ { x } {\epsilon ( u ) } { { \frac{du }{u} } } \right ) , $$

where $ c ( \cdot ) $, $ \epsilon ( \cdot ) $ are measurable, $ c ( x ) \rightarrow c \in ( 0, \infty ) $, $ \epsilon ( x ) \rightarrow 0 $ as $ x \rightarrow \infty $.

iii) The characterization theorem: for measurable $ f $, $ g ( \lambda ) $ in (a1) must be of the form $ g ( \lambda ) \equiv \lambda ^ \rho $ for some $ \rho \in \mathbf R $, called the index of regular variation: $ f \in R _ \rho $. Then $ f ( x ) = x ^ \rho {\mathcal l} ( x ) $ with $ {\mathcal l} $ slowly varying ( $ {\mathcal l} \in R _ {0} $).

iv) Karamata's theorem: if $ f \in R _ \rho $ and $ \sigma > - ( \rho + 1 ) $, then

$$ \tag{a2 } { \frac{x ^ {\rho + 1 } f ( x ) }{\int\limits _ { a } ^ { x } {t ^ \sigma f ( t ) } {dt } } } \rightarrow \sigma + \rho + 1 ( x \rightarrow \infty ) . $$

(That is, the $ {\mathcal l} $ in $ f ( x ) = x ^ \rho {\mathcal l} ( x ) $" behaves asymptotically like a constant" under integration.) Conversely, (a2) implies $ f \in R _ \rho $.

Perhaps the most important application of Karamata theory to analysis is Karamata's Tauberian theorem (or the Hardy–Littlewood–Karamata theorem): if $ f \in R _ \rho $( $ \rho \geq 0 $) is increasing, with Laplace–Stieltjes transform $ {\widehat{f} } ( s ) = \int _ {0} ^ \infty {e ^ {- sx } } {df ( x ) } $, then $ f ( x ) \sim c { {x ^ \rho {\mathcal l} ( x ) } / {\Gamma ( 1 + \rho ) } } $( $ x \rightarrow \infty $) with $ c \geq 0 $, $ {\mathcal l} \in R _ {0} $ if and only if $ {\widehat{f} } ( s ) \sim c s ^ {- \rho } {\mathcal l} ( {1 / s } ) $ $ ( s \downarrow 0 ) $.

For details, background and references on these and other results, see e.g. [a1], Chap. 1.

The union over all $ \rho \in \mathbf R $ of the classes $ R _ \rho $ gives the class $ R $ of regularly varying functions. This is contained in the larger class $ ER $ of extended regularly varying functions, itself included in the class $ OR $ of $ O $- regularly varying functions: $ R \subset ER \subset OR $. Just as a function $ f \in R $ has an index $ \rho $ of regular variation, and then $ f \in R _ \rho $, so a function $ f \in ER $ has a pair $ c ( f ) , d ( f ) $ of upper and lower Karamata indices (and these are equal, to $ \rho $ say, if and only if $ f \in R _ \rho $), and a function $ f \in OR $ has a pair $ \alpha ( f ) , \beta ( f ) $ of upper and lower Matuszewska indices. These larger classes $ ER $, $ OR $ have analogues of the results above; for instance, uniform convergence and representation theorems. For details, see e.g. [a1], Chap. 2.

Karamata theory may be regarded as the "first-order" theory of regular variation. There is a corresponding "second-order" theory: de Haan theory [a1], Chap. 3.

Karamata theory has found extensive use in several areas of analysis, such as Abelian, Tauberian and Mercerian theorems ([a1], Chap. 4, 5; cf. also Tauberian theorems; Mercer theorem; Abel theorem) and the Levin–Pfluger theory of completely regular growth of entire functions ([a1], Chap. 6; cf. also Entire function), and is also useful in asymptotic questions in analytic number theory [a1], Chap. 7. It has been widely used also in probability theory, following the work of W. Feller [a2]; [a1], Chap. 8.

References

[a1] N.H. Bingham, C.M. Goldie, J.L. Teugels, "Regular variation", Encycl. Math. Appl., 27, Cambridge Univ. Press (1989) (Edition: Second)
[a2] W. Feller, "An introduction to probability theory and its applications", 2, Springer (1976) (Edition: Second)
How to Cite This Entry:
Karamata theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Karamata_theory&oldid=14907
This article was adapted from an original article by N.H. Bingham (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article