This presentation is in two parts. First, we recall the definition of two types of asymptotic expansions known as nested form and nested expansions. This theory makes it possible to adapt the asymptotic scale to the function under expansion and is based on the theory of Hardy fields [1]. Next, we suggest a reformulation of nested forms in terms of generalized products called star products, and a prospective theory of multivariate Hardy fields called partial Hardy fields.





g 

(ln  x)<g 

(x)=g 

(  x^{d}  )  =g 

(  e_{k}  (  l_{k}^{d}(x)  )  )  <g 

(exp x). 
g_{0}  (  x^{1}  )  <g_{0}(ln x)<g_{0}(x) but g_{1}(ln x)<g_{1}(x)=g_{1}  (  x^{1}  )  . 

~ 


, and hence  L_{p}(x)f 

(x) ~ 


. 
f=e_{s}  (  l_{m}^{d}(x)f  )  where s,m³0, dÎR^{+} and g_{1}(f)<g_{1}(l_{m}(x)). 
e_{1}  (  l_{2}^{2}(x)e_{2}  (  l_{5}^{1/3}(x)(2+o(1))  )  )  , and e_{1}^{1} 
æ ç ç è 
x 

l_{1}(x)e_{2} 
æ ç ç è 
l 

(x)(13+o(1)) 
ö ÷ ÷ ø 
ö ÷ ÷ ø 
. 



f(x,y)x® +¥  ~ e 

æ ç ç è 
l 

(x)e 

æ ç ç è 
··· e 

æ ç ç è 
l 

(x)(f(y)+o(1)) 
ö ÷ ÷ ø 
··· 
ö ÷ ÷ ø 
ö ÷ ÷ ø 
, 
This document was translated from L^{A}T_{E}X by H^{E}V^{E}A.