# 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
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Regularization of sequences. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regularization_of_sequences&oldid=51698