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$  n = 0, 1 \dots $
 
$  n = 0, 1 \dots $
 
be a sequence of real numbers (indexed by the non-negative integers). A regularization of  $  \{ a _ {n} \} $
 
be a sequence of real numbers (indexed by the non-negative integers). A regularization of  $  \{ a _ {n} \} $
is a sequence  $  \{ a _ {n}  ^ {(} r) \} $
+
is a sequence  $  \{ a _ {n}  ^ {( r)} \} $
 
obtained from  $  \{ a _ {n} \} $
 
obtained from  $  \{ a _ {n} \} $
 
by replacing certain  $  a _ {n} $
 
by replacing certain  $  a _ {n} $
Line 20: Line 20:
 
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} \} $
 
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} \} $
 
and  $  \{ L _ {n} \} $
determine the same quasi-analytic class if suitably regularized sequences  $  \{ M _ {n}  ^ {(} r) \} $
+
determine the same quasi-analytic class if suitably regularized sequences  $  \{ M _ {n}  ^ {( r)} \} $
and  $  \{ L _ {n}  ^ {(} r) \} $
+
and  $  \{ L _ {n}  ^ {( r)} \} $
 
are the same, cf. [[#References|[a1]]], [[#References|[a2]]].
 
are the same, cf. [[#References|[a1]]], [[#References|[a2]]].
  
Line 31: Line 31:
 
a _ {i}  \leq   
 
a _ {i}  \leq   
 
\frac{i- r }{s- r }
 
\frac{i- r }{s- r }
  a _ {s} + s-
+
  a _ {s} +
\frac{i}{s-}
+
\frac{s-i}{s-r} a _ {r} ;
r a _ {r} ;
 
 
$$
 
$$
  
 
that is, if the point  $  ( i, a _ {i} ) $
 
that is, if the point  $  ( i, a _ {i} ) $
 
is located below or on the segment in the plane joining  $  ( r, a _ {r} ) $
 
is located below or on the segment in the plane joining  $  ( r, a _ {r} ) $
and  $  ( s, a _ {s} ) $(
+
and  $  ( s, a _ {s} ) $ (cf. [[Convex function (of a real variable)|Convex function (of a real variable)]]).
cf. [[Convex function (of a real variable)|Convex function (of a real variable)]]).
 
  
The convex regularization, or Newton regularization,  $  \{ a _ {n}  ^ {(} c) \} $
+
The convex regularization, or Newton regularization,  $  \{ a _ {n}  ^ {( c)} \} $
 
of  $  \{ a _ {n} \} $
 
of  $  \{ a _ {n} \} $
is the largest convex minorant of  $  \{ a _ {n} \} $(
+
is the largest convex minorant of  $  \{ a _ {n} \} $ (cf. [[Majorant and minorant|Majorant and minorant]], 1)).
cf. [[Majorant and minorant|Majorant and minorant]], 1)).
 
  
 
The log-convex regularization, or convex regularization by means of the logarithm, of a sequence of positive numbers  $  \{ a _ {n} \} $
 
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) \} $
+
is the sequence of positive numbers  $  \{ a _ {n}  ^ {( lc)} \} $
such that  $  \{  \mathop{\rm log}  a _ {n}  ^ {(} lc) \} $
+
such that  $  \{  \mathop{\rm log}  a _ {n}  ^ {( lc)} \} $
 
is the convex regularization of  $  \{  \mathop{\rm log}  a _ {n} \} $.  
 
is the convex regularization of  $  \{  \mathop{\rm log}  a _ {n} \} $.  
 
It is defined by the relations
 
It is defined by the relations
  
 
$$  
 
$$  
T _ {a} ( r)  =  \sup _ { n> } 0  
+
T _ {a} ( r)  =  \sup _ { n>0 }   
 
\frac{r  ^ {n} }{a _ {n} }
 
\frac{r  ^ {n} }{a _ {n} }
 
  ,\ \  
 
  ,\ \  
a _ {n}  ^ {(} lc)  =  \sup _ { r> } 0  
+
a _ {n}  ^ {( lc)} =  \sup _ { r>0 }   
 
\frac{r  ^ {n} }{T _ {a} ( r) }
 
\frac{r  ^ {n} }{T _ {a} ( r) }
 
  .
 
  .
 
$$
 
$$
  
The exponential regularization  $  \{ a _ {n}  ^ {(} e) \} $
+
The exponential regularization  $  \{ a _ {n}  ^ {( e)} \} $
 
of  $  \{ a _ {n} \} $
 
of  $  \{ a _ {n} \} $
 
is defined by the relations
 
is defined by the relations
Line 69: Line 66:
 
\frac{r  ^ {n} }{a _ {n} }
 
\frac{r  ^ {n} }{a _ {n} }
 
  \  ( r \geq  1) ,\ \  
 
  \  ( r \geq  1) ,\ \  
a _ {n}  ^ {(} e)  =  \sup _ {r \geq  n }   
+
a _ {n}  ^ {( e)} =  \sup _ {r \geq  n }   
 
\frac{r  ^ {n}
 
\frac{r  ^ {n}
 
  }{S _ {a} ( r) }
 
  }{S _ {a} ( r) }
Line 76: Line 73:
  
 
The Newton regularization of a sequence  $  \{ a _ {n} \} $
 
The Newton regularization of a sequence  $  \{ a _ {n} \} $
is very much related to the Newton polygon of  $  \{ 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} $,  
+
(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} $
 
its Newton polygon is the highest convex polygonal line in  $  \mathbf R  ^ {2} $
 
joining  $  ( 0, a _ {0} ) $
 
joining  $  ( 0, a _ {0} ) $
 
to  $  ( N, a _ {N} ) $,  
 
to  $  ( N, a _ {N} ) $,  
i.e. it is the polygonal line consisting of the segments joining  $  ( i, a _ {i}  ^ {(} c) ) $
+
i.e. it is the polygonal line consisting of the segments joining  $  ( i, a _ {i}  ^ {( c)} ) $
to  $  ( i+ 1 , a _ {i+} 1 ^ {(} c) ) $,  
+
to  $  ( i+ 1 , a _ {i+ 1}  ^ {( c)} ) $,  
 
$  i= 0 \dots N- 1 $.  
 
$  i= 0 \dots N- 1 $.  
Thus, the number  $  a _ {i}  ^ {(} c) $
+
Thus, the number  $  a _ {i}  ^ {( c)} $
 
is the ordinate of the point of the Newton polygon of  $  \{ a _ {i} \} $
 
is the ordinate of the point of the Newton polygon of  $  \{ a _ {i} \} $
 
with abscissa  $  i $.
 
with abscissa  $  i $.
Line 97: Line 94:
 
Figure: r080940a
 
Figure: r080940a
  
To avoid certain pathologies (like  $  a _ {i}  ^ {(} c) = - \infty $
+
To avoid certain pathologies (like  $  a _ {i}  ^ {( c)} = - \infty $
 
for all  $  i > 0 $),  
 
for all  $  i > 0 $),  
let  $  \{ a _ {n} \} _ {n=} 0 ^  \infty  $
+
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} $
+
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 $.  
 
as  $  N \rightarrow \infty $.  
It remains true that  $  a _ {i}  ^ {(} c) $
+
It remains true that  $  a _ {i}  ^ {( c)} $
is determined by the condition that  $  ( i, a _ {i}  ^ {(} c) ) $
+
is determined by the condition that  $  ( i, a _ {i}  ^ {( c)} ) $
lies on the Newton polygon of  $  \{ a _ {n} \} _ {n=} 0 ^  \infty  $.
+
lies on the Newton polygon of  $  \{ a _ {n} \} _ {n=0}  ^  \infty  $.
  
 
Let  $  K $
 
Let  $  K $
be a non-Archimedean valued field with [[Valuation|valuation]]  $  v $(
+
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) $
+
(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 $
 
be a polynomial of degree  $  N $
 
over  $  K $.  
 
over  $  K $.  
 
The Newton polygon of the polynomial  $  f( X) $
 
The Newton polygon of the polynomial  $  f( X) $
 
is the Newton polygon of the sequence  $  ( v( 1), v( a _ {1} ) \dots v ( a _ {N} )) $.  
 
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) $(
+
It carries immediate information on the valuations of the roots of  $  f( X) $
in a complete algebraic closure of  $  K  $).  
+
(in a complete algebraic closure of  $  K  $).  
 
Indeed, if  $  \lambda $
 
Indeed, if  $  \lambda $
 
is the slope of a segment of the Newton polygon of (abscissa) length  $  r $,  
 
is the slope of a segment of the Newton polygon of (abscissa) length  $  r $,  
 
then there are precisely  $  r $
 
then there are precisely  $  r $
roots of valuation  $  - \lambda $(
+
roots of valuation  $  - \lambda $
counted with multiplicities); an analogous result holds for roots of power series (this is related to a  $  p $-
+
(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]]]).
adic Weierstrass preparation theorem, cf. (the editorial comments to) [[Weierstrass theorem|Weierstrass theorem]], and [[#References|[a3]]]).
 
  
 
The  Newton polygon of a sequence  $  \{ a _ {n} \} $
 
The  Newton polygon of a sequence  $  \{ a _ {n} \} $
Line 137: Line 133:
  
 
$$  
 
$$  
\cap _ {A \subset  U( t,c) } U( t, c).
+
\bigcap _ {A \subset  U( t,c) } U( t, c).
 
$$
 
$$
  
Line 154: Line 150:
  
 
$$  
 
$$  
\cap _ {A \subset  U  ^  \omega  ( t,c) } U  ^  \omega  ( t, c)
+
\bigcap _ {A \subset  U  ^  \omega  ( t,c) } U  ^  \omega  ( t, c)
 
$$
 
$$
  
now defines the  $  \omega $-
+
now defines the  $  \omega $-regularized sequence  $  \{ a _ {n} ^ {( \omega ) } \} $.  
regularized sequence  $  \{ a _ {n} ^ {( \omega ) } \} $.  
 
 
Newton and exponential regularization correspond to  $  \omega ( t) \equiv \infty $
 
Newton and exponential regularization correspond to  $  \omega ( t) \equiv \infty $
 
and  $  \omega ( t) =  \mathop{\rm exp} ( t) $,  
 
and  $  \omega ( t) =  \mathop{\rm exp} ( t) $,  
Line 164: Line 159:
  
 
====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=48491