The Role of Asymptotic Mean in the Geometric Theory of Asymptotic Expansions in the Real Domain

We call “asymptotic mean” (at +∞) of a real-valued function [ ) ∈ ∞ loc f L T 1 ,+ the number, supposed to exist, ( ) ∫ x f x T M x f t t 1 + := lim d − → ∞ , and highlight its role in the geometric theory of asymptotic expansions in the real domain of type (*) ( ) ( ) ( ) ( ) ( ) , , → ∞ n n n f x a x a x o x x 1 1 = + + + + φ φ φ  where the comparison functions ( ) ( ) n x x 1 , , φ φ  , forming an asymptotic scale at +∞, belong to one of the three classes having a definite “type of variation” at +∞, slow, regular or rapid. For regularlyvarying comparison functions we can characterize the existence of an asymptotic expansion (*) by the nice property that a certain quantity ( ) F t has an asymptotic mean at +∞. This quantity is defined via a linear differential operator in f and admits of a remarkable geometric interpretation as it measures the ordinate of the point wherein that special curve ( ) ( ) ( ) ( ) n n y a t x a t x 1 1 = + + φ φ  , which has a contact of order n − 1 with the graph of f at the generic point t, intersects a fixed vertical line, say x = T. Sufficient or necessary conditions hold true for the other two classes. In this article we give results for two types of expansions already studied in our current development of a general theory of asymptotic expansions in the real domain, namely polynomial and two-term expansions.

the nice property that a certain quantity ( ) F t has an asymptotic mean at +∞.This quantity is de- fined via a linear differential operator in f and admits of a remarkable geometric interpretation as it measures the ordinate of the point wherein that special curve n n y a t x a t x

Introduction
In our current endeavor to establish a general analytic theory of asymptotic expansions in the real domain [1]- [6], we highlighted that what we called the geometric approach leads in a natural way to a linear differential operator, say  , depending solely on the comparison functions appearing in a possible expansion; certain asymptotic or integral conditions involving the quantity ( ) then characterize an expansion of a given function f either in itself or matched to other expansions obtained by formal differentiation in suitable senses.The theory we are referring to is based on the following ideas.Suppose one wishes to find conditions (sufficient and/or necessary) for the validity of an asymptotic expansion , , where the ordered n-tuple of comparison functions  forms an asymptotic scale at +∞, that is to say: ( ) ( ) , ; 1, , . In this paper we intentionally choose x → +∞ as this is the situation wherein the classical concept of asymptotic mean plays a role.The simplest elementary case is that of an "asymptotic straight line"-( ) ( ) + + → +∞ ,-and it goes back to Newton the "natural" idea of looking at this contingency as the "limit position of the tangent line at the graph of f" as the point of tangency goes to infinity.The German geometer Haupt [7], in 1922, extended this idea to study "nth-order asymptotic parabolas" i.e. "polynomial asymptotic expansions" ( ) ( ) ( ) ( ) 2) looking at them as "limit positions of nth-order osculating parabolas".In [1] we collected various scattered results on such expansions completing them with some missing links and adding a new theory called "factorizational theory".A rich bibliography with historical references is also to be found in [1].For a general expansion (1.1) a rough idea consists in looking at the "generalized polynomial" ( ) as the limit position of a suitable family of "generalized polynomial curves" , n n y a t x a t x as the parameter t → +∞ .Of course a curve (1.3) must have some meaningful link with the graph of f and, from a technical point of view, the simplest choice consists in (1.3) admitting of a contact of order ( )  via a certain set of asymptotic relations for f.At least this is what has been done for the two cases already systematized in the literature: that of polynomial asymptotic expansions in [1] and that of two-term expansions in [4].
In this paper we point out that, whenever the comparison functions admit of an "index of variation at +∞", one can obtain new types of asymptotic results revolving around a classical concept which we label "asymptotic mean".In §2 we first present an overview of the class of functions with an asymptotic mean; then, after introducing classes of slowly-varying, regularly-varying or rapidly-varying functions in a restricted sense, we give new results correlating these last classes, asymptotic means and weighted asymptotic means.In §3 we give characterizations of certain sets of polynomial asymptotic expansions via asymptotic means of the coefficients of nth-order osculating parabolas; in particular we shall study the following Conjecture.An asymptotic expansion (1.2) holds true iff the constant coefficient of the nth-order osculating parabola at the generic point ( ) ( ) , t f t has an asymptotic mean at +∞.This nice statement will be proved true for a class of functions f satisfying a certain differential inequality.In §4 we establish either characterizations or sufficient conditions or necessary conditions for an asymptotic expansion according to the three "types of variation at +∞" of the comparison functions i φ so giving the exact results va- guely mentioned in ( [4]; pp.261-263).Extension of the results to a general asymptotic expansion (1.1), n ≥ 3, is based on information about the asymptotic behavior of Wronskians of regularly-or rapidly-varying functions and this requires a separate nonshort treatment.
Almost all proofs are collected in §5.A recurrent notation is:

General Properties
The following concept is meaningful in itself and often encountered both in classical Analysis (see references throughout this section) and in modern applied mathematics, Sanders and Verhulst [8].
+∞ then its asymptotic mean at +∞ is defined as the number ( ) provided that the limit exists and is finite.(Obviously neither the existence nor the value of f M depend on the particular choice of T.) We shall use the symbol ∞  to denote the class of all functions defined on an interval of the form [ ) , T +∞ and having an asymptotic mean at +∞ ; ∞  is obviously a vector space over  .In order to help the reader grasp the meaning of the quantity f M we shall list various classes of functions contained in ∞  ; at the same time we shall have at our disposal some practical rules for testing the existence and the possible value of d the arithmetic mean of on , .

4)
If f has a bounded antiderivative (i.e.  ( ) ( ) In fact integrating by parts we have where . That the last term on the right is ( ) o x α follows dividing by x α and applying l'Hos- pital's rule. Proposition 2.1 is widely used in asymptotic theory of ordinary differential equations: in a different but equivalent formulation it goes back to Faedo ([10]; lemma, p. 118) and also appears in a paper by Hallam ([11]; lemma 1.1, p. 136).However the nontrivial proofs given by these authors are only valid for one-signed f.The elementary proof given above applies to any f: it essentially goes back to Hukuhara ([12]; Lemma 1, p. 72) and appears again in Ostrowski ([13]; Lemma II).
7) If for some fixed 0 λ > there exists a finite limit ( ) For a proof see Agnew ([14]; Th. 6. does not necessarily imply ( ) f a ′ +∞ = , the converse inference being true; but relation (2.9) is equivalent to f ∞ ′ ∈  and, if this is the case, then The last relation also implies the following version of L'Hospital's rule for functions in This concept is an extension to improper integrals of the concept of arithmetical mean for a sequence, see Hardy ([15]; pp.430-434) and ( [16]; Ch.V and p. 110).It follows from our definition that "f is summable (C, 1) on [ ) : sin log cos log ; sin log , even if f is uniformly continuous on [ ) 1, +∞ .In Blinov [17] there is a more elaborate counterexample of a bounded uniformly-continuous function constructed with the implicit use of almost-periodic functions.
For f bounded, the contingency " f ∞ ∈  " can be characterized via the behavior at the origin of the Lap- lace-transform of f: see either Ditkine and Proudnikov ( [18]; Th. 4, p. 196)  ( ) All the above properties, from 1 to 9, practically are sufficient conditions for f ∞ ∈  , none of them being characteristic.A counterexample for the converse of property in 6 is provided by: where φ stands for some suitable function such that ( ) φ +∞ = +∞ .The number , f L φ is a kind of "weighted asymptotic mean" of f and can be considered, the sign apart, as a "generalized limit of ( ) f x as x → +∞ " for the simple reason that a trivial application of L'Hospital's rule yields ( ( ) ∫ under obvious hypotheses on , f φ .The notion of regular variation gives the key to finding out a large meaningful class of test-functions φ , in- cluding powers, such that (2.17), valid for one fixed φ , is equivalent to f ∞ ∈  .

Preliminaries on Regularly-or Rapidly-Varying Functions
We use the notion of variation, either regular or rapid, in a restricted sense; for the general theory the reader is referred to the monograph by Bingham, Goldie and Teugels [21].We get three different results for the three classes defined in Definition 2.2.Let (I) φ is termed "regularly varying at +∞ (in the strong sense)" if for some constant α ∈  which is called the index of regular variation of φ at +∞.The family of all such functions for a fixed α is denoted by In the case 0 α = the function φ is also termed "slowly varying at +∞ (in the strong sense)".
(II) φ is termed "rapidly varying at +∞ (in the strong sense)" if Accordingly, the index of rapid variation at +∞ is defined to be either +∞ or −∞ and the corresponding families of functions are denoted by (III) φ is said to have an "index of variation at +∞ in the strong sense" if the following limit exists in the ex- tended real line: Remarks 1) Condition " φ ultimately of one strict sign" is essential both in the general and in our restricted definition.The choice 0 , and α , k β 's are any real numbers.
Typical functions in ≠ too has ultimately one strict sign and there are two contingencies for the limit ( ) as inferrred from the identity

∫
For 0 α = all the possible contingencies may occur for this limit as shown by the functions: 1; ( ) ( ) may happen that φ′ has no index of variation at +∞ as shown by the counterexamples: sin , 1 2, oscillatory and unbounded for 2 ; 2 sin log , 0 2e sin e , oscillatory and unbounded ; e e sin e , oscillatory and unbounded .
But if φ′ has an index of variation then there are precise links between the two indexes.
In the case 0 α = and without the stated additional condition on ( )

Relationships between Asymptotic Mean and Weighted Asymptotic Means
We can now give and understand generalizations of the mentioned results by Ostrowski and Agnew.(I) (Regularly-varying functions: extension of a result by Ostrowski, 1976) ) and (2.17) are equivalent to each other.If this is the case then , hence , f L φ does not depend on φ .An equivalent statement is: Under conditions (2.25) the following two asymptotic relations are equivalent to each other: for a constant a which turns out to depend only on f.In one direction we have that the first relation in (2.26), which is trivially true whenever ( ) then for any fixed . (III) (Rapidly-varying functions: extension of a result by Agnew, 1942).
A counterexample for the converse inference in part (II) is provided by: where the last relation can be easily proved by suitably integrating by parts.And a counterexample for the converse inference in part (III) is trivially provided by: ( : e , : sin ; 0; 1 e e sin d sin cos admits of no limit as . 2 Notice that , φ φ′ may be rapidly varying without satisfying (2.28) 3 as shown by the function  : and 0 0.
We end this section by mentioning that the concept of asymptotic mean plays a role also in "Tauberian theorems", Hardy ([16]; Ch. 12), in non-oscillation properties of second-order differential equations, Hartman [22] and ( [23]; pp.365-367), and in the theory of Cauchy-Frullani integrals, Ostrowski [20].In this last paper our Theorem 2.4-(I) appears for the first time in the literature though for the special case ( ) x x φ = and the proof is somewhat involved.In a previous paper Ostrowski ( [13]; Lemma II) had given a quick proof of a lemma correlated to our present context, a proof based on integration by parts; curiously enough he does not apply the same elementary device in proving the result under consideration, which is just the device used by us to prove the general case.Also the original proof by Agnew [14] is indirect; the author is interested in studying the limit ( ) where f M is a real number independent from λ .He first proves the equivalence between (2.8) and (2.35) and then that (2.35) implies f ∞ ∈  .

Polynomial Asymptotic Expansions and Asymptotic Means
t is defined almost everywhere and for each such t let us consider the "nth-order osculating parabola" to the graph of f at the point ( ) ( ) which may be rewritten in the form where ( ) , , lim 0,1, , and running through the set where is defined exist as finite numbers, we say that the parabola ( ) , or equivalently the polynomial , is the "nth-order limit parabola" to [the graph of] f at +∞.A limit parabola of order zero denotes a mere relation F t the "nth-order contact indicatrix" of the curve ( ) with respect to the y-axis as it represents the ordinate of the point of intersection in the x, y-plane between the y-axis and the curve (3.1).
We report here simplified versions of two of the main results in [1].
≥ , the following are equivalent properties: 1) The graph of f has a limit parabola at +∞ of order n i.e., by definition, all the limits .
If this is the case then the following integral representation holds true ( ) ( ) ( ) for a suitable polynomial n Π , the same as above, and a suitable number ,0 n γ , the same as in (3.6).We expressed relations in (3.7) by saying that the asymptotic expansion ( ) ( ) ( ) ; : ; 0 .
2) The single function 3) There exists a polynomial ( ) If this is the case then , , and the following integral representation holds true: .
In the elementary case n = 1 the result is: Then an expansion (3.10) holds true iff If this is the case then, according to Theorem 3.3, the expansion (3.10) is formally differentiable ( ) times in the strong sense.We exhibit an example for the case 1 n = and a counterexample for the case 2 n = ; they seem to be just the same because in both expansions the remainder is exactly the same quantity but a striking difference appears in the behaviors of 1,0 F and 2,0 F .Example for the case 1 ; sin d sin ; 2; sin 1 .
Here 1,0 F is bounded and admits of asymptotic mean ( ) b = but has no limit at +∞; accordingly the expansion ( ) ( ) is not formally differentiable in the strong sense though the differentiated expansions of any order satisfy the remarkable asymptotic estimates in (3.17).
Counterexample for the case 2 n = : Here 2,0 F is unbounded both from below and from above and admits of no asymptotic mean; notwithstanding, an asymptotic expansion ( ) ( ) Hence the equivalence stated in Theorem 3.4 may fail without the restriction in (3.16).According to Theorem 3.3 the expansion of f 2 is not formally differentiable once in the strong sense.
In the elementary case in (3.15) condition ( ) → +∞ is explicitly defined in Giblin ( [24]; p. 279) as the "bounded distance condition" and it is easily checked that it is equivalent to a pair of relations it is the further condition of existence of asymptotic mean that changes the first relation in (3.19) into an asymptotic straight line.

Two-Term Asymptotic Expansions and Asymptotic Means
In this section we give an exhaustive list of results concerning the role of asymptotic mean in the theory of two-term asymptotic expansions involving comparison functions admitting of indexes of variation at +∞.We first report a result from [4].Preliminary notations and formulas ([4]; p. 255).As usual we say that two functions f, g (as well as their graphs) have a first-order contact at a point t 0 if ( ) ( ) ( ) ( ) provided that f, g are defined on a neighborhood of t 0 and the involved derivatives exist as finite numbers.
Let now 1 φ , 2 φ be two real-valued functions differentiable on an interval I such that their Wronskian ( ) ( ) ( ) ( ) φ φ = never vanishes on I and let f be differentiable on I. Then for each 0 t I ∈ there exists a unique function in the family : span , having a first-order contact with f at t 0 .Denoting this function by ( ) where will be called the contact indicatrix of order one of the function f at the point t with respect to the family  and the straight line x T = .
( ) represents the ordinate of the point of intersection between the vertical line x T = and the curve ( ) ( ) ( ) ( ) where t is thought of as fixed.The assumption on ( ) W x implies that 1 φ and 2 φ do not vanish simultaneously hence F * is a nontrivial linear combination of 1 f * , 2 f * .It may happen that, for some choices of T, F * coincides with 1 f * or 2 f * , a constant factor apart, according as ( ) Using (4.2) F * may be represented as , , where we have put For a function , loc f L T x ∈ the following are equivalent properties: 1) It holds true an asymptotic expansion 2) There exists a finite limit 3) There exists a finite limit If this is the case we have the following two representations: The validity of (4.8) may be expressed by the geometric locution: "the graph of f admits of the curve ( ) ( ) as an asymptotic curve in the family ( ) Notice that in the cited reference condition (4.10) is written in the form 0, , for some real number then the following three properties are equivalent: : and suppose that: , ; then an expansion (4.16) implies both conditions (4.17)-(4.18).Under the stated assumptions for the validity of part (I) the equivalence "(4.16) ⇔ (4.18)" admits of the following geometric reformulation: "The graph of f admits of an asymptotic curve in the family ( ) , as x → +∞ , iff the contact in- dicatrix of order one of the function f with respect to  has an asymptotic mean at +∞".
Notice that this result for two-term expansions requires no restrictions on the signs of 2 f * , F * .

Proofs
Proof of Lemma 2.3.By hypothesis the following two limits exist in  : ( ) ( ) ( ) ( ) lim , lim .We now evaluate α by L'Hospital's rule first noticing that: 0 α < ≤ +∞ implies ( ) φ +∞ = and the first limit in (5.1) implies ( ) In both cases the rule may be applied and It remains the case α = −∞ which implies ( ) 0 φ +∞ = and this condition leads to excluding the following contingencies for the indicated reasons: 1) and this would imply, by L'Hospital's rule as in (5.2): ( ) ( ) The case 1 α ′ = − must be treated in a different way.A basic property of our class of functions, directly inferred from the limits in (5.1), claims the validity of the following asymptotic estimates: Now in our present proof we have α = −∞ and 1 α ′ = − , hence ( )  and there are two a-priori contingencies about the integral φ +∞ ′ ∫ .Its divergence would imply ( ) φ +∞ = +∞ which cannot be; in the other case we would have which contradicts the second relation in (5.3).Notice that the procedure used to prove this last case works for any α ′ ∈  as well.

.1).
A. Granata and estimate the behavior of , I x I x as x → +∞ .From (2.1) and 5.8) we get: As concerns 2 I we have: ( , from whence and (2.1) we get: As ( ) , we obtain the convergence of ( ) . By (5.16) and (5.18) we may integrate by parts as follows: which is (2.17 and from (5.17): ( ) ( ) ( ) Now we retrace all steps in the second part of the proof of part (I) checking the validity of the corresponding formulas for 0 α = .Instead of the first relation in (5.16) we have: and, instead of (5.18): The convergence of A f +∞ ⋅ ∫ follows as above.And using the same integration by parts as in (5.19) we get the same final relation.
(III) Let us first show that the three conditions in (2.28) imply that both , φ φ′ are rapidly-varying at +∞.
Conditions in (2.28) 1,2 are equivalent to ( ) ( ) Now we retrace all steps in the first part of the proof of part (I) and again use decomposition (5.10); instead of (5.11) we get: and instead of (5.12) we get, using (5.24): whence our claim follows dividing both sides by x.  Proof of Theorem 3.3.Let us assume (3.12) and start from the integral representation ( [1]; formula (6.3), p. 185): and the last relation, when replaced into (5.29),yields: But the first relation in (5.31) implies that the iterated improper integral ( ) having used one of the following elementary identities (to be used again): To prove the formal differentiabilty we put: Calling ( ) n P x the last sum on the right in (5.34), which differ by a constant from the sum on the right in (5.33), and applying Leibniz's rule to (5.33) we get: The expressions of ( ) x and its derivative: (5.39)So far we have proved that (3.12) implies relations in (3.13) for ( ) , 1 1 k f k n ≤ ≤ − , without any information on ( ) n f , and, for the time being, n P is a non-better specified polynomial of degree n ≤ .To prove (3.11) we estimate the behavior, as x → +∞ , of ( ) , n k F x for 1 k n ≤ ≤ using its known expression in terms of f, ([1]; formula (2.6), p. 168): as the first sum is nothing but the expression of the coefficient of the power k x in the polynomial n P has asymptotic mean "zero" and the same is true for a term ( ) 1 o ; so the sum of the last three terms above represents a function with asymptotic mean equalling k a .We have proved that "2) ⇒ 1) ∧ 3)".It remains to show "3) ⇒ 2)".First step.Let us first evaluate x Now we start as in (5.40) from the expression of ,0 n F : which implies the convergence of the improper integral ( )  = .Now we do calculations just like those from (5.41) to (5.43) but starting from representation (5.45) and paying attention to the signs, so getting:  It follows that any result on formal differentiability of a polynomial asymptotic expansion involving g admits of a literal transposition to a polynomial asymptotic expansion involving f.Our assumption are now: expansion (3.10) and one-signedness of ,0 n F , and the proof (which we make explicit here) is a word-for-word repetition of that in ( [1]; Proof of Th. 4.2, pp. 193-195) with a slight modification at the conclusive passage.From representation (5.29) we infer ( ) ( ) ( ) ( )       where the coefficients k a are those in (3.10).From (5.58) we infer that ( ) ( ) its role in the geometric theory of asymptotic expansions in the real domain of type (*) asymptotic scale at +∞, belong to one of the three classes having a definite "type of variation" at +∞, slow, regular or rapid.For regularlyvarying comparison functions we can characterize the existence of an asymptotic expansion (*) by , i.e.

4 )
This requires suitable assumptions: the regularity of the i φ 's and f and a special structure of the n-tuple

M
= .This is the condition appearing in the classical Dirichlet test for convergence of improper integrals of type This follows from the previous case when p = 1 and from Hölder's inequality, when 1 p > : not know if part (III) remains true when replacing the three conditions (2.28) by the weaker conditions: following isolated result, needed in the sequel, without placing it in a general context.Proposition 2.6.If f ∞ ∈  then: the example in(2.13),no information on the growth-order of ( ) n f may be obtained in the context of Theorem 3.3, generally speaking.For 2 n ≥ a characterization similar to that in (3.15) holds true under a restriction on the sign of ,0 n F and we have the following analogue of Proposition 3.2.-sided boundedness condition:

(
the two polynomials n P and the sum appearing in (5.45) have the same leading coefficient: n n a c Proof of Theorem 3.4.The only thing to be proved is that an expansion (3.10) plus condition (3.16) imply ,0 n F ∞ ∈  .We first show that it is enough to prove our claim with (3.16) replaced by the condition of one-signedness: fact it is known, ([1]; Lemma 2.2, p. 169), that: p be a polynomial of type (5.50) and define: Hence representation (5.29) can be rewritten in the form

F
this last limit exists in the extended real line, hence it must be a finite number.This means the convergence of The last relation implies that n c coincides with the n a in (3.10) and we get: Th. 3.1, p. 173), which we shall not presently use.by monotonicity, condition (3.6) holds true).If this is the case then we also have the properties in Proposition 3.1, hence the expansion (3.10) automatically implies its formal differentiability n times in the strong sense.
[1]formally differentiable n times in the "strong sense" because in the same paper we characterized another weaker set of differentiated expansions, ([1]; ( ) , By the above argument involving L'Hospital's rule we arrive at the convergence of the iterated integral