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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r0809401.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r0809402.png" /> be a sequence of real numbers (indexed by the non-negative integers). A regularization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r0809403.png" /> is a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r0809404.png" /> obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r0809405.png" /> by replacing certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r0809406.png" /> which are  "excessively high"  with respect to the others by suitable lower values. An important application of regularized sequences is to the problem of equivalence of classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r0809408.png" />-functions; that is, the problem of when two sequences of constants determine the same [[Quasi-analytic class|quasi-analytic class]] of functions. The answers tend to be given in the form that the two sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r0809409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094010.png" /> determine the same quasi-analytic class if suitably regularized sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094012.png" /> are the same, cf. [[#References|[a1]]], [[#References|[a2]]].
+
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Some important regularization procedures are as follows. A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094013.png" /> of real numbers is called a convex sequence if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094014.png" /> is convex, i.e. if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094015.png" />,
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/r/r080/r080940/r08094016.png" /></td> </tr></table>
+
Let  $  a _ {n} $,
 +
$  n = 0, 1 \dots $
 +
be a sequence of real numbers (indexed by the non-negative integers). A regularization of  $  \{ a _ {n} \} $
 +
is a sequence  $  \{ a _ {n}  ^ {( r)} \} $
 +
obtained from  $  \{ a _ {n} \} $
 +
by replacing certain  $  a _ {n} $
 +
which are  "excessively high" with respect to the others by suitable lower values. An important application of regularized sequences is to the problem of equivalence of classes of  $  C  ^  \infty  $-
 +
functions; that is, the problem of when two sequences of constants determine the same [[Quasi-analytic class|quasi-analytic class]] of functions. The answers tend to be given in the form that the two sequences  $  \{ M _ {n} \} $
 +
and  $  \{ L _ {n} \} $
 +
determine the same quasi-analytic class if suitably regularized sequences  $  \{ M _ {n}  ^ {( r)} \} $
 +
and  $  \{ L _ {n}  ^ {( r)} \} $
 +
are the same, cf. [[#References|[a1]]], [[#References|[a2]]].
  
that is, if the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094017.png" /> is located below or on the segment in the plane joining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094019.png" /> (cf. [[Convex function (of a real variable)|Convex function (of a real variable)]]).
+
Some important regularization procedures are as follows. A sequence  $  \{ a _ {n} \} $
 +
of real numbers is called a convex sequence if the function  $  n \mapsto a _ {n} $
 +
is convex, i.e. if for all  $  0\leq  r < i < s $,
  
The convex regularization, or Newton regularization, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094021.png" /> is the largest convex minorant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094022.png" /> (cf. [[Majorant and minorant|Majorant and minorant]], 1)).
+
$$
 +
a _ {i}  \leq 
 +
\frac{i- r }{s- r }
 +
a _ {s} + 
 +
\frac{s-i}{s-r} a _ {r} ;
 +
$$
  
The log-convex regularization, or convex regularization by means of the logarithm, of a sequence of positive numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094023.png" /> is the sequence of positive numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094024.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094025.png" /> is the convex regularization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094026.png" />. It is defined by the relations
+
that is, if the point  $  ( i, a _ {i} ) $
 +
is located below or on the segment in the plane joining  $  ( r, a _ {r} ) $
 +
and  $  ( s, a _ {s} ) $ (cf. [[Convex function (of a real variable)|Convex function (of a real variable)]]).
  
<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/r/r080/r080940/r08094027.png" /></td> </tr></table>
+
The convex regularization, or Newton regularization,  $  \{ a _ {n}  ^ {( c)} \} $
 +
of  $  \{ a _ {n} \} $
 +
is the largest convex minorant of  $  \{ a _ {n} \} $ (cf. [[Majorant and minorant|Majorant and minorant]], 1)).
  
The exponential regularization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094029.png" /> is defined by the relations
+
The log-convex regularization, or convex regularization by means of the logarithm, of a sequence of positive numbers  $  \{ a _ {n} \} $
 +
is the sequence of positive numbers  $  \{ a _ {n}  ^ {( lc)} \} $
 +
such that  $  \{  \mathop{\rm log}  a _ {n}  ^ {( lc)} \} $
 +
is the convex regularization of $  \{  \mathop{\rm log}  a _ {n} \} $.  
 +
It is defined by the relations
  
<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/r/r080/r080940/r08094030.png" /></td> </tr></table>
+
$$
 +
T _ {a} ( r)  = \sup _ { n>0 } 
 +
\frac{r  ^ {n} }{a _ {n} }
 +
,\ \
 +
a _ {n}  ^ {( lc)}  = \sup _ { r>0
 +
\frac{r  ^ {n} }{T _ {a} ( r) }
 +
.
 +
$$
  
The Newton regularization of a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094031.png" /> is very much related to the Newton polygon of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094032.png" /> (this explains the name "Newton regularization" , cf. also [[Newton diagram|Newton diagram]], which discusses the context in which the Newton polygon first arose). For a finite sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094033.png" />, its Newton polygon is the highest convex polygonal line in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094034.png" /> joining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094035.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094036.png" />, i.e. it is the polygonal line consisting of the segments joining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094037.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094039.png" />. Thus, the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094040.png" /> is the ordinate of the point of the Newton polygon of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094041.png" /> with abscissa <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094042.png" />.
+
The exponential regularization $  \{ a _ {n}  ^ {( e)} \} $
 +
of  $  \{ a _ {n} \} $
 +
is defined by the relations
  
An example of this for the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094044.png" />, with convex regularization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094045.png" />, is given in Fig. a.
+
$$
 +
S _ {a} ( r)  =  = \max _ {n \leq  r } 
 +
\frac{r  ^ {n} }{a _ {n} }
 +
\  ( r \geq  1) ,\ \
 +
a _ {n}  ^ {( e)}  =  \sup _ {r \geq  n } 
 +
\frac{r  ^ {n}
 +
}{S _ {a} ( r) }
 +
.
 +
$$
 +
 
 +
The Newton regularization of a sequence  $  \{ a _ {n} \} $
 +
is very much related to the Newton polygon of $  \{ a _ {n} \} $
 +
(this explains the name  "Newton regularization" , cf. also [[Newton diagram|Newton diagram]], which discusses the context in which the Newton polygon first arose). For a finite sequence  $  \{ a _ {n} \} _ {n=0}  ^ {N} $,
 +
its Newton polygon is the highest convex polygonal line in  $  \mathbf R  ^ {2} $
 +
joining  $  ( 0, a _ {0} ) $
 +
to  $  ( N, a _ {N} ) $,
 +
i.e. it is the polygonal line consisting of the segments joining  $  ( i, a _ {i}  ^ {( c)} ) $
 +
to  $  ( i+ 1 , a _ {i+ 1}  ^ {( c)} ) $,  
 +
$  i= 0 \dots N- 1 $.  
 +
Thus, the number  $  a _ {i}  ^ {( c)} $
 +
is the ordinate of the point of the Newton polygon of  $  \{ a _ {i} \} $
 +
with abscissa  $  i $.
 +
 
 +
An example of this for the sequence  $  ( 1, 1, - 2, 1, - 4/3, 1/3, 0) $,
 +
$  N= 6 $,  
 +
with convex regularization $  ( 1, - 1/2, - 2, - 5/3, - 4/3, - 2/3, 0) $,  
 +
is given in Fig. a.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r080940a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r080940a.gif" />
Line 25: Line 94:
 
Figure: r080940a
 
Figure: r080940a
  
To avoid certain pathologies (like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094046.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094047.png" />), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094048.png" /> be bounded from below. The Newton polygon of this sequence is defined as the limit of the Newton polygons of the finite sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094049.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094050.png" />. It remains true that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094051.png" /> is determined by the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094052.png" /> lies on the Newton polygon of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094053.png" />.
+
To avoid certain pathologies (like $  a _ {i}  ^ {( c)} = - \infty $
 +
for all $  i > 0 $),  
 +
let $  \{ a _ {n} \} _ {n=0}  ^  \infty  $
 +
be bounded from below. The Newton polygon of this sequence is defined as the limit of the Newton polygons of the finite sequences $  \{ a _ {n} \} _ {n=0}  ^ {N} $
 +
as $  N \rightarrow \infty $.  
 +
It remains true that $  a _ {i}  ^ {( c)} $
 +
is determined by the condition that $  ( i, a _ {i}  ^ {( c)} ) $
 +
lies on the Newton polygon of $  \{ a _ {n} \} _ {n=0}  ^  \infty  $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094054.png" /> be a non-Archimedean valued field with [[Valuation|valuation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094055.png" /> (cf. also [[Norm on a field|Norm on a field]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094056.png" /> be a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094057.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094058.png" />. The Newton polygon of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094059.png" /> is the Newton polygon of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094060.png" />. It carries immediate information on the valuations of the roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094061.png" /> (in a complete algebraic closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094062.png" />). Indeed, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094063.png" /> is the slope of a segment of the Newton polygon of (abscissa) length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094064.png" />, then there are precisely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094065.png" /> roots of valuation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094066.png" /> (counted with multiplicities); an analogous result holds for roots of power series (this is related to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094067.png" />-adic Weierstrass preparation theorem, cf. (the editorial comments to) [[Weierstrass theorem|Weierstrass theorem]], and [[#References|[a3]]]).
+
Let $  K $
 +
be a non-Archimedean valued field with [[Valuation|valuation]] $  v $
 +
(cf. also [[Norm on a field|Norm on a field]]). Let $  1+ a _ {1} X + \dots + a _ {N} X  ^ {N} = f( X) $
 +
be a polynomial of degree $  N $
 +
over $  K $.  
 +
The Newton polygon of the polynomial $  f( X) $
 +
is the Newton polygon of the sequence $  ( v( 1), v( a _ {1} ) \dots v ( a _ {N} )) $.  
 +
It carries immediate information on the valuations of the roots of $  f( X) $
 +
(in a complete algebraic closure of $  K  $).  
 +
Indeed, if $  \lambda $
 +
is the slope of a segment of the Newton polygon of (abscissa) length r $,  
 +
then there are precisely r $
 +
roots of valuation $  - \lambda $
 +
(counted with multiplicities); an analogous result holds for roots of power series (this is related to a $  p $-adic Weierstrass preparation theorem, cf. (the editorial comments to) [[Weierstrass theorem|Weierstrass theorem]], and [[#References|[a3]]]).
  
The  Newton polygon of a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094068.png" /> can be obtained geometrically as follows. For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094069.png" /> one considers the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094070.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094071.png" /> through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094072.png" /> of slope <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094073.png" />; it is given by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094074.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094075.png" /> be the [[Supergraph|supergraph]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094076.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094077.png" /> be the graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094079.png" />. Then the Newton polygon is the lower boundary of the convex set
+
The  Newton polygon of a sequence $  \{ a _ {n} \} $
 +
can be obtained geometrically as follows. For all $  t, c \in ( - \infty , \infty ) $
 +
one considers the line $  l( t, c) $
 +
in $  \mathbf R  ^ {2} $
 +
through $  ( 0, c) $
 +
of slope $  t $;  
 +
it is given by the equation $  y= tx+ c $.  
 +
Let $  U( t, c)= \{ {( x, y) } : {x\geq  0,  y\geq  tx+ c } \} $
 +
be the [[Supergraph|supergraph]] of $  l( t, c) $.  
 +
Let $  A $
 +
be the graph of $  \{ a _ {n} \} $,
 +
$  A= \{ {( i, a _ {i} ) } : {i= 0 , 1 ,\dots } \} $.  
 +
Then the Newton polygon is the lower boundary of the convex set
  
<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/r/r080/r080940/r08094080.png" /></td> </tr></table>
+
$$
 +
\bigcap _ {A \subset  U( t,c) } U( t, c).
 +
$$
  
As noted, the Newton regularization (convex regularization) of a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094081.png" /> is determined by its Newton polygon. This construction has been generalized. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094082.png" /> be a non-decreasing function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094083.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094084.png" />. Let
+
As noted, the Newton regularization (convex regularization) of a sequence $  \{ a _ {n} \} $
 +
is determined by its Newton polygon. This construction has been generalized. Let $  \omega ( t) $
 +
be a non-decreasing function of $  t $
 +
with values in $  [ 0, \infty ] $.  
 +
Let
  
<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/r/r080/r080940/r08094085.png" /></td> </tr></table>
+
$$
 +
U  ^  \omega  ( t, c)  = U( t, c) \cup \{ {( x, y) } : {x > \omega ( t) } \}
 +
.
 +
$$
  
 
The lower boundary of
 
The lower boundary of
  
<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/r/r080/r080940/r08094086.png" /></td> </tr></table>
+
$$
 +
\bigcap _ {A \subset  U  ^  \omega  ( t,c) } U  ^  \omega  ( t, c)
 +
$$
  
now defines the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094087.png" />-regularized sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094088.png" />. Newton and exponential regularization correspond to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094089.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094090.png" />, respectively.
+
now defines the $  \omega $-regularized sequence $  \{ a _ {n} ^ {( \omega ) } \} $.  
 +
Newton and exponential regularization correspond to $  \omega ( t) \equiv \infty $
 +
and $  \omega ( t) = \mathop{\rm exp} ( t) $,  
 +
respectively.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Mandelbrojt,  "Séries adhérentes, régularisations des suites, applications" , Gauthier-Villars  (1952)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.A. Siddigi,  "On the equivalence of classes of infinitely differentiable functions"  ''Soviet J. Contemp. Math. Anal. Arm. Acad. Sci.'' , '''19''' :  1  (1984)  pp. 18–29  ''Izv. Akad. Nauk Arm.SSR Mat.'' , '''19''' :  1  (1984)  pp. 19–30</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Koblitz,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094091.png" />-adic numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080940/r08094092.png" />-adic analysis, and zeta-functions" , Springer  (1977)  pp. Chapt. IV, §3–4</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Mandelbrojt,  "Séries adhérentes, régularisations des suites, applications" , Gauthier-Villars  (1952)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  J.A. Siddigi,  "On the equivalence of classes of infinitely differentiable functions"  ''Soviet J. Contemp. Math. Anal. Arm. Acad. Sci.'' , '''19''' :  1  (1984)  pp. 18–29  ''Izv. Akad. Nauk Arm.SSR Mat.'' , '''19''' :  1  (1984)  pp. 19–30</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Koblitz,  "$p$-adic numbers, $p$-adic analysis, and zeta-functions" , Springer  (1977)  pp. Chapt. IV, §3–4</TD></TR>
 +
</table>

Latest revision as of 12:49, 12 March 2021


Let $ a _ {n} $, $ n = 0, 1 \dots $ be a sequence of real numbers (indexed by the non-negative integers). A regularization of $ \{ a _ {n} \} $ is a sequence $ \{ a _ {n} ^ {( r)} \} $ obtained from $ \{ a _ {n} \} $ by replacing certain $ a _ {n} $ which are "excessively high" with respect to the others by suitable lower values. An important application of regularized sequences is to the problem of equivalence of classes of $ C ^ \infty $- functions; that is, the problem of when two sequences of constants determine the same quasi-analytic class of functions. The answers tend to be given in the form that the two sequences $ \{ M _ {n} \} $ and $ \{ L _ {n} \} $ determine the same quasi-analytic class if suitably regularized sequences $ \{ M _ {n} ^ {( r)} \} $ and $ \{ L _ {n} ^ {( r)} \} $ are the same, cf. [a1], [a2].

Some important regularization procedures are as follows. A sequence $ \{ a _ {n} \} $ of real numbers is called a convex sequence if the function $ n \mapsto a _ {n} $ is convex, i.e. if for all $ 0\leq r < i < s $,

$$ a _ {i} \leq \frac{i- r }{s- r } a _ {s} + \frac{s-i}{s-r} a _ {r} ; $$

that is, if the point $ ( i, a _ {i} ) $ is located below or on the segment in the plane joining $ ( r, a _ {r} ) $ and $ ( s, a _ {s} ) $ (cf. Convex function (of a real variable)).

The convex regularization, or Newton regularization, $ \{ a _ {n} ^ {( c)} \} $ of $ \{ a _ {n} \} $ is the largest convex minorant of $ \{ a _ {n} \} $ (cf. Majorant and minorant, 1)).

The log-convex regularization, or convex regularization by means of the logarithm, of a sequence of positive numbers $ \{ a _ {n} \} $ is the sequence of positive numbers $ \{ a _ {n} ^ {( lc)} \} $ such that $ \{ \mathop{\rm log} a _ {n} ^ {( lc)} \} $ is the convex regularization of $ \{ \mathop{\rm log} a _ {n} \} $. It is defined by the relations

$$ T _ {a} ( r) = \sup _ { n>0 } \frac{r ^ {n} }{a _ {n} } ,\ \ a _ {n} ^ {( lc)} = \sup _ { r>0 } \frac{r ^ {n} }{T _ {a} ( r) } . $$

The exponential regularization $ \{ a _ {n} ^ {( e)} \} $ of $ \{ a _ {n} \} $ is defined by the relations

$$ S _ {a} ( r) = = \max _ {n \leq r } \frac{r ^ {n} }{a _ {n} } \ ( r \geq 1) ,\ \ a _ {n} ^ {( e)} = \sup _ {r \geq n } \frac{r ^ {n} }{S _ {a} ( r) } . $$

The Newton regularization of a sequence $ \{ a _ {n} \} $ is very much related to the Newton polygon of $ \{ a _ {n} \} $ (this explains the name "Newton regularization" , cf. also Newton diagram, which discusses the context in which the Newton polygon first arose). For a finite sequence $ \{ a _ {n} \} _ {n=0} ^ {N} $, its Newton polygon is the highest convex polygonal line in $ \mathbf R ^ {2} $ joining $ ( 0, a _ {0} ) $ to $ ( N, a _ {N} ) $, i.e. it is the polygonal line consisting of the segments joining $ ( i, a _ {i} ^ {( c)} ) $ to $ ( i+ 1 , a _ {i+ 1} ^ {( c)} ) $, $ i= 0 \dots N- 1 $. Thus, the number $ a _ {i} ^ {( c)} $ is the ordinate of the point of the Newton polygon of $ \{ a _ {i} \} $ with abscissa $ i $.

An example of this for the sequence $ ( 1, 1, - 2, 1, - 4/3, 1/3, 0) $, $ N= 6 $, with convex regularization $ ( 1, - 1/2, - 2, - 5/3, - 4/3, - 2/3, 0) $, is given in Fig. a.

Figure: r080940a

To avoid certain pathologies (like $ a _ {i} ^ {( c)} = - \infty $ for all $ i > 0 $), let $ \{ a _ {n} \} _ {n=0} ^ \infty $ be bounded from below. The Newton polygon of this sequence is defined as the limit of the Newton polygons of the finite sequences $ \{ a _ {n} \} _ {n=0} ^ {N} $ as $ N \rightarrow \infty $. It remains true that $ a _ {i} ^ {( c)} $ is determined by the condition that $ ( i, a _ {i} ^ {( c)} ) $ lies on the Newton polygon of $ \{ a _ {n} \} _ {n=0} ^ \infty $.

Let $ K $ be a non-Archimedean valued field with valuation $ v $ (cf. also Norm on a field). Let $ 1+ a _ {1} X + \dots + a _ {N} X ^ {N} = f( X) $ be a polynomial of degree $ N $ over $ K $. The Newton polygon of the polynomial $ f( X) $ is the Newton polygon of the sequence $ ( v( 1), v( a _ {1} ) \dots v ( a _ {N} )) $. It carries immediate information on the valuations of the roots of $ f( X) $ (in a complete algebraic closure of $ K $). Indeed, if $ \lambda $ is the slope of a segment of the Newton polygon of (abscissa) length $ r $, then there are precisely $ r $ roots of valuation $ - \lambda $ (counted with multiplicities); an analogous result holds for roots of power series (this is related to a $ p $-adic Weierstrass preparation theorem, cf. (the editorial comments to) Weierstrass theorem, and [a3]).

The Newton polygon of a sequence $ \{ a _ {n} \} $ can be obtained geometrically as follows. For all $ t, c \in ( - \infty , \infty ) $ one considers the line $ l( t, c) $ in $ \mathbf R ^ {2} $ through $ ( 0, c) $ of slope $ t $; it is given by the equation $ y= tx+ c $. Let $ U( t, c)= \{ {( x, y) } : {x\geq 0, y\geq tx+ c } \} $ be the supergraph of $ l( t, c) $. Let $ A $ be the graph of $ \{ a _ {n} \} $, $ A= \{ {( i, a _ {i} ) } : {i= 0 , 1 ,\dots } \} $. Then the Newton polygon is the lower boundary of the convex set

$$ \bigcap _ {A \subset U( t,c) } U( t, c). $$

As noted, the Newton regularization (convex regularization) of a sequence $ \{ a _ {n} \} $ is determined by its Newton polygon. This construction has been generalized. Let $ \omega ( t) $ be a non-decreasing function of $ t $ with values in $ [ 0, \infty ] $. Let

$$ U ^ \omega ( t, c) = U( t, c) \cup \{ {( x, y) } : {x > \omega ( t) } \} . $$

The lower boundary of

$$ \bigcap _ {A \subset U ^ \omega ( t,c) } U ^ \omega ( t, c) $$

now defines the $ \omega $-regularized sequence $ \{ a _ {n} ^ {( \omega ) } \} $. Newton and exponential regularization correspond to $ \omega ( t) \equiv \infty $ and $ \omega ( t) = \mathop{\rm exp} ( t) $, respectively.

References

[a1] S. Mandelbrojt, "Séries adhérentes, régularisations des suites, applications" , Gauthier-Villars (1952)
[a2] J.A. Siddigi, "On the equivalence of classes of infinitely differentiable functions" Soviet J. Contemp. Math. Anal. Arm. Acad. Sci. , 19 : 1 (1984) pp. 18–29 Izv. Akad. Nauk Arm.SSR Mat. , 19 : 1 (1984) pp. 19–30
[a3] N. Koblitz, "$p$-adic numbers, $p$-adic analysis, and zeta-functions" , Springer (1977) pp. Chapt. IV, §3–4
How to Cite This Entry:
Regularization of sequences. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regularization_of_sequences&oldid=19031