%0 Journal Article
%T The Role of Asymptotic Mean in the Geometric Theory of Asymptotic Expansions in the Real Domain
%A Antonio Granata
%J Advances in Pure Mathematics
%P 100-119
%@ 2160-0384
%D 2015
%I Scientific Research Publishing
%R 10.4236/apm.2015.52013
%X <p style=\"text-align:justify;\">
	<span style=\"font-size:12px;line-height:102%;font-family:Verdana;\"><span style=\"line-height:1.5;\">We
call ¡°asymptotic mean¡± (at +¡Þ) of a real-valued function </span><img src=\"Edit_5b9bff7a-06a8-4d37-a49c-61176513d961.bmp\" alt=\"\" /></span><span></span><span style=\"font-size:12px;line-height:102%;font-family:Verdana;\"><span style=\"line-height:1.5;\"> the number, supposed to exist, </span><img src=\"Edit_e1eeeddc-d84b-4e30-997a-d605fe05ef2e.bmp\" alt=\"\" /></span><span style=\"font-size:12px;line-height:102%;font-family:Verdana;\"><span style=\"line-height:1.5;\">,
and highlight its role in the geometric theory of asymptotic expansions in the
real domain of type (*) </span><img src=\"Edit_b5f12166-ac0a-416d-907d-b51d37443095.bmp\" alt=\"\" /></span><span style=\"font-size:12px;line-height:1.5;font-family:Verdana;\"> where the comparison functions </span><span><img src=\"Edit_35a5770f-1735-4388-a795-2f3eefccc0b1.bmp\" width=\"91\" height=\"23\" alt=\"\" /></span><span style=\"font-size:10pt;\" \"=\"\"><span style=\"line-height:1.5;font-family:Verdana;font-size:12px;\">,
forming an asymptotic scale at +¡Þ, belong to one of the </span><span style=\"line-height:1.5;font-family:Verdana;font-size:12px;\">three classes having a definite ¡°type of variation¡± at +¡Þ, slow, regular
or rapid. For regularly </span><span style=\"line-height:1.5;font-family:Verdana;font-size:12px;\">varying comparison functions we can characterize
the existence of an asymptotic expansion (*) by the nice property that a
certain quantity </span><span style=\"font-size:12px;line-height:1.5;\">F£¨t)</span></span><span style=\"font-size:10.0pt;line-height:102%;font-family:;\" \"=\"\"><span style=\"font-family:Verdana;font-size:12px;line-height:1.5;\"> has an asymptotic mean at +¡Þ. This quantity is
defined via a linear differential operator in </span><i><span style=\"font-family:Verdana;font-size:12px;line-height:1.5;\">f</span></i><span style=\"font-family:Verdana;font-size:12px;\"><span style=\"line-height:1.5;\"> and admits of a remarkable geometric interpretation as it
measures the ordinate of the point wherein that special curve </span><img src=\"Edit_65048ab3-3d9f-4f4d-babf-1347e02f7445.bmp\" alt=\"\" /></span></span><span style=\"font-size:10.0pt;line-height:102%;font-family:;\" \"=\"\"><span style=\"font-family:Verdana;font-size:12px;line-height:1.5;\">,
which has a contact of order </span><i><span style=\"font-family:Verdana;font-size:12px;line-height:1.5;\">n</span></i><span style=\"font-family:Verdana;font-size:12px;line-height:1.5;\"> -
%K Asymptotic Expansions
%K Formal Differentiation of Asymptotic Expansions
%K Regularly-Varying and Rapidly-Varying Functions
%K Asymptotic Mean
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=54251