Regularization of sequences

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Let , be a sequence of real numbers (indexed by the non-negative integers). A regularization of is a sequence obtained from by replacing certain 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 -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 and determine the same quasi-analytic class if suitably regularized sequences and are the same, cf. [a1], [a2].

Some important regularization procedures are as follows. A sequence of real numbers is called a convex sequence if the function is convex, i.e. if for all ,

that is, if the point is located below or on the segment in the plane joining and (cf. Convex function (of a real variable)).

The convex regularization, or Newton regularization, of is the largest convex minorant of (cf. Majorant and minorant, 1)).

The log-convex regularization, or convex regularization by means of the logarithm, of a sequence of positive numbers is the sequence of positive numbers such that is the convex regularization of . It is defined by the relations

The exponential regularization of is defined by the relations

The Newton regularization of a sequence is very much related to the Newton polygon of (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 , its Newton polygon is the highest convex polygonal line in joining to , i.e. it is the polygonal line consisting of the segments joining to , . Thus, the number is the ordinate of the point of the Newton polygon of with abscissa .

An example of this for the sequence , , with convex regularization , is given in Fig. a.

Figure: r080940a

To avoid certain pathologies (like for all ), let be bounded from below. The Newton polygon of this sequence is defined as the limit of the Newton polygons of the finite sequences as . It remains true that is determined by the condition that lies on the Newton polygon of .

Let be a non-Archimedean valued field with valuation (cf. also Norm on a field). Let be a polynomial of degree over . The Newton polygon of the polynomial is the Newton polygon of the sequence . It carries immediate information on the valuations of the roots of (in a complete algebraic closure of ). Indeed, if is the slope of a segment of the Newton polygon of (abscissa) length , then there are precisely roots of valuation (counted with multiplicities); an analogous result holds for roots of power series (this is related to a -adic Weierstrass preparation theorem, cf. (the editorial comments to) Weierstrass theorem, and [a3]).

The Newton polygon of a sequence can be obtained geometrically as follows. For all one considers the line in through of slope ; it is given by the equation . Let be the supergraph of . Let be the graph of , . Then the Newton polygon is the lower boundary of the convex set

As noted, the Newton regularization (convex regularization) of a sequence is determined by its Newton polygon. This construction has been generalized. Let be a non-decreasing function of with values in . Let

The lower boundary of

now defines the -regularized sequence . Newton and exponential regularization correspond to and , respectively.


[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, "-adic numbers, -adic analysis, and zeta-functions" , Springer (1977) pp. Chapt. IV, §3–4
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