View source for 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

<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>

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" ).

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]].

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

<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>

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" />.

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" />).

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

<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>

(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" />.

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" />.

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.

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

====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>