# Karamata theory

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=47478
This article was adapted from an original article by N.H. Bingham (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article