{{Short description|Function in mathematics}} In real analysis, a branch of mathematics, a '''slowly varying function''' is a function of a real variable whose behaviour at infinity is in some sense similar to the behaviour of a function converging at infinity. Similarly, a '''regularly varying function''' is a function of a real variable whose behaviour at infinity is similar to the behaviour of a power law function (like a polynomial) near infinity. These classes of functions were both introduced by Jovan Karamata,<ref name="GaSe" >See {{harv|Galambos|Seneta|1973}}</ref><ref name="BiGoTe">See {{harv|Bingham|Goldie|Teugels|1987}}.</ref> and have found several important applications, for example in probability theory and extreme value theory.
== Basic definitions ==
{{EquationRef|1|Definition 1}}. A measurable function {{math|''L'' : (0, +∞) → (0, +∞)}} is called ''slowly varying'' (at infinity) if for all {{math|''a'' > 0}}, :<math>\lim_{x \to \infty} \frac{L(ax)}{L(x)}=1.</math>
{{EquationRef|2|Definition 2}}. Let {{math|''L'' : (0, +∞) → (0, +∞)}}. Then {{math|''L''}} is a regularly varying function if and only if <math>\forall a > 0, g_L(a) = \lim_{x \to \infty} \frac{L(ax)}{L(x)} \in \mathbb{R}^{+}</math>. In particular, the limit must be finite.
These definitions are due to Jovan Karamata.<ref name="GaSe" /><ref name="BiGoTe" />
== Basic properties ==
Regularly varying functions have some important properties:<ref name="GaSe" /> a partial list of them is reported below. More extensive analyses of the properties characterizing regular variation are presented in the monograph by {{harvtxt|Bingham|Goldie|Teugels|1987}}.
===Uniformity of the limiting behaviour=== {{EquationRef|3|Theorem 1}}. The limit in {{EquationNote|1|definitions 1}} and {{EquationNote|2|2}} is uniform if {{mvar|''a''}} is restricted to a compact interval.
===Karamata's characterization theorem=== {{EquationRef|4|Theorem 2}}. Every regularly varying function {{math|''f'' : (0, +∞) → (0, +∞)}} is of the form :<math>f(x)=x^\beta L(x)</math> where *{{mvar|β}} is a real number, *{{mvar|L}} is a slowly varying function. '''Note'''. This implies that the function {{math|''g''(''a'')}} in {{EquationNote|2|definition 2}} has necessarily to be of the following form :<math>g(a)=a^\rho</math> where the real number {{mvar|''ρ''}} is called the ''index of regular variation''.
===Karamata representation theorem=== {{EquationRef|5|Theorem 3}}. A function {{mvar|''L''}} is slowly varying if and only if there exists {{math|''B'' > 0}} such that for all {{math|''x'' ≥ ''B''}} the function can be written in the form
:<math>L(x) = \exp \left( \eta(x) + \int_B^x \frac{\varepsilon(t)}{t} \,dt \right)</math>
where *{{math|''η''(''x'')}} is a bounded measurable function of a real variable converging to a finite number as {{mvar|''x''}} goes to infinity *{{math|''ε''(''x'')}} is a bounded measurable function of a real variable converging to zero as {{mvar|''x''}} goes to infinity.
== Examples == * If {{mvar|''L''}} is a measurable function and has a limit ::<math>\lim_{x \to \infty} L(x) = b \in (0,\infty),</math> :then {{mvar|''L''}} is a slowly varying function. * For any {{math|''β'' ∈ '''R'''}}, the function {{math|''L''(''x'') {{=}} log<sup>{{hairsp}}''β''</sup>{{hairsp}}''x''}} is slowly varying. * The function {{math|''L''(''x'') {{=}} ''x''}} is not slowly varying, nor is {{math|''L''(''x'') {{=}} ''x''<sup>{{hairsp}}''β''</sup>}} for any real {{math|''β ''≠ 0}}. However, these functions are regularly varying.
==See also== *Analytic number theory *Hardy–Littlewood tauberian theorem and its treatment by Karamata
==Notes== {{reflist|30em}}
==References== * {{SpringerEOM|title=Karamata theory|oldid=25937|first=N.H.|last=Bingham}} * {{Citation | last=Bingham | first=N. H. | last2=Goldie | first2=C. M. | last3=Teugels | first3=J. L. | title=Regular Variation | place=Cambridge | publisher=Cambridge University Press | series=Encyclopedia of Mathematics and its Applications | volume=27 | year=1987 | edition= | url=https://archive.org/details/regularvariation0000bing | doi= | isbn=0-521-30787-2 | mr=0898871 | zbl=0617.26001 | url-access=registration }} * {{Citation |author2-link=Eugene Seneta | last1=Galambos | first1=J. | last2=Seneta | first2=E. | title=Regularly Varying Sequences | year=1973 | journal=Proceedings of the American Mathematical Society | issn=0002-9939 | volume=41 | issue=1 | pages=110–116 | doi=10.2307/2038824 | jstor=2038824| doi-access=free }}.
Category:Real analysis Category:Tauberian theorems Category:Types of functions