Regularization of sequences

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