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)