_{1}

^{*}

We call “asymptotic mean” (at +∞) of a real-valued function the number, supposed to exist, , and highlight its role in the geometric theory of asymptotic expansions in the real domain of type (*) where the comparison functions , forming an asymptotic scale at +∞, belong to one of the three classes having a definite “type of variation” at +∞, slow, regular or rapid. For regularly varying 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 , 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.

In our current endeavor to establish a general analytic theory of asymptotic expansions in the real domain [

where the ordered n-tuple of comparison functions

looking at them as “limit positions of nth-order osculating parabolas”. In [

as the parameter

This requires suitable assumptions: the regularity of the

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 [

Conjecture. An asymptotic expansion (1.2) holds true iff the constant coefficient of the nth-order osculating parabola at the generic point

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

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 non- short treatment.

Almost all proofs are collected in §5. A recurrent notation is:

・

・

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 [

Definition 2.1. If

provided that the limit exists and is finite. (Obviously neither the existence nor the value of

We shall use the symbol

1) If

2) If f is periodic on

A direct elementary proof may be found in Corduneanu ([

3) If f is almost periodic on

4) If f has a bounded antiderivative (i.e.

in the classical Dirichlet test for convergence of improper integrals of type

5) If

and from Hölder’s inequality, when

6) If the improper integral

which follows from the hypothesis and the next

Proposition 2.1. If

In fact integrating by parts we have

where

Proposition 2.1 is widely used in asymptotic theory of ordinary differential equations: in a different but equiv- alent formulation it goes back to Faedo ([

7) If for some fixed

then

8) If there exists a finite limit

then

9) If

does not necessarily imply

The last relation also implies the following version of L’Hospital’s rule for functions in

Proposition 2.2. If

For the proof just write

10) The space

This concept is an extension to improper integrals of the concept of arithmetical mean for a sequence, see Hardy ([

11) Two negative properties concerning functions in

a) Not any bounded function belongs to

even if f is uniformly continuous on

For f bounded, the contingency “

b) In general no information on the order of growth of a function in

we have

but

All the above properties, from 1 to 9, practically are sufficient conditions for

12) However in Ostrowski ([

The number

and, if this is the case,

This result, used by Ostrowski, e.g., in the study of Frullani’s integral, may also yield the nice geometric characterization of a rectilinear asymptote, see (3.15) below. But in other asymptotic investigations a more general form of condition (2.16) is encountered, namely

where

under obvious hypotheses on

The notion of regular variation gives the key to finding out a large meaningful class of test-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 [

Definition 2.2. Let

(I)

for some constant

(II)

Accordingly, the index of rapid variation at +∞ is defined to be either +∞ or −∞ and the corresponding families of functions are denoted by

(III)

Remarks 1) Condition “

2) Typical functions in

Typical functions in

index of variation is:

3) For

as inferrred from the identity

For

with

4) If

But if

Lemma 2.3. If

In the case

but it cannot be

We can now give and understand generalizations of the mentioned results by Ostrowski and Agnew.

Theorem 2.4. Let

(I) (Regularly-varying functions: extension of a result by Ostrowski, 1976). If

then for any fixed

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

(II) (Slowly-varying functions). If

then for any fixed

(III) (Rapidly-varying functions: extension of a result by Agnew, 1942). If

(which imply that both

Corollary 2.5. Special cases reformulated:

For

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:

Notice that _{3} as shown by the function

We add the following isolated result, needed in the sequel, without placing it in a general context.

Proposition 2.6. If

We end this section by mentioning that the concept of asymptotic mean plays a role also in “Tauberian theorems”, Hardy ([

where

If

which may be rewritten in the form

where

exist as finite numbers, we say that the parabola

or equivalently the polynomial

We shall call the function

We report here simplified versions of two of the main results in [

Proposition 3.1. For

1) The graph of f has a limit parabola at

2) The single limit

3) There exists a polynomial

If this is the case then the following integral representation holds true

for a suitable polynomial

We expressed relations in (3.7) by saying that the asymptotic expansion

is formally differentiable n times in the “strong sense” because in the same paper we characterized another weaker set of differentiated expansions, ([

Proposition 3.2. If

iff its nth-order contact indicatrix

Now we give analogues of the two foregoing propositions with condition (3.6) replaced by the weaker condition

Theorem 3.3. For

1) All the functions

2) The single function

3) There exists a polynomial

If this is the case then

In the elementary case n = 1 the result is:

Notice that the representation of

For

Theorem 3.4. Let

Then an expansion (3.10) holds true iff

We exhibit an example for the case

Example for the case

Here

Counterexample for the case

Here _{2} is not formally differentiable once in the strong sense.

In the elementary case in (3.15) condition

it is the further condition of existence of asymptotic mean that changes the first relation in (3.19) into an asymptotic straight line.

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 [

Preliminary notations and formulas ([_{0} if _{0} and the involved derivatives exist as finite numbers.

Let now_{0}. Denoting this function by

where

If _{0}. The function

will be called the contact indicatrix of order one of the function f at the point t with respect to the family

Using (4.2)

where we have put

Proposition 4.1. (Characterization of a two-term asymptotic expansion: [

For a function

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

Notice that in the cited reference condition (4.10) is written in the form

however (4.5) implies

and (4.10) follows.

The two limits in (4.9), (4.10) are of the type studied in §2 and a direct application of Theorem 2.4 gives the following results.

Theorem 4.2. In assumptions (4.6)-(4.7) let it be:

(I) (Regularly-varying comparison functions). If

then the following three properties are equivalent:

(II) (Slowly-varying comparison functions). If

then each condition (4.17) or (4.18) implies an expansion (4.16).

(III) (Rapidly-varying comparison functions). Put

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) Û (4.18)” admits of the following geometric reformulation:

“The graph of f admits of an asymptotic curve in the family

Notice that this result for two-term expansions requires no restrictions on the signs of

Proof of Lemma 2.3. By hypothesis the following two limits exist in

We now evaluate

It remains the case

1)

2)

which is a positive real number; hence

3)

as in (5.2):

4) The case

Now in our present proof we have

and there are two a-priori contingencies about the integral

which contradicts the second relation in (5.3). Notice that the procedure used to prove this last case works for any

The last assertion in the statement of Lemma 2.3, namely “it cannot be

Proof of Theorem 2.4. (I) We make explicit the assumptions writing:

which in turn imply the following relations to be used in the sequel:

First part: (2.17) Þ (2.1). If we put

then, by (2.17), we may write

From (5.9) and (2.17):

Using (5.11) and (5.13) in the left side of (5.10) we get

Second part: (2.1) Þ (2.17). First step: convergence of

and estimate the behavior of

As concerns

from whence and (2.1) we get:

As

Second step: asymptotic behavior of

which is (2.17) with

(II) From the first assumption in (2.27) we infer:

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

and, instead of (5.18):

The convergence of

(III) Let us first show that the three conditions in (2.28) imply that both _{1,2} are equivalent to_{3} is equivalent to

which implies, by (2.28)_{1},

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

From (5.25), (5.26), (5.27) we get (2.1) with

Proof of Proposition 2.6. Integration by parts gives:

whence our claim follows dividing both sides by x. W

Proof of Theorem 3.3. Let us assume (3.12) and start from the integral representation ([

which for

From (5.30) the elementary equivalence in (3.14) easily follows, hence we suppose

and the last relation, when replaced into (5.29), yields:

But the first relation in (5.31) implies that the iterated improper integral

together with the expansion:

having used one of the following elementary identities (to be used again):

To prove the formal differentiabilty we put:

and from (5.31) we infer relations:

Calling

The expressions of

So far we have proved that (3.12) implies relations in (3.13) for

as the first sum is nothing but the expression of the coefficient of the power

By (2.34) the function

we get:

Now we start as in (5.40) from the expression of

whence we get

which implies the convergence of the improper integral

Comparing (5.45) and the assumed relation

having used the identity

which, by (2.29), implies

Proof of Theorem 3.4. The only thing to be proved is that an expansion (3.10) plus condition (3.16) imply

In fact it is known, ([

Let now g be any function,

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

and, by (3.10), the following limit:

For

and (3.10) implies that “

For

By the one-signedness of

The last relation implies that

By the above argument involving L’Hospital’s rule we arrive at the convergence of the iterated integral

which implies representation

where the coefficients

and applications of L’Hospital’s rule

which, by (2.29), is equivalent to

In passing notice that the last calculations and (5.34) prove that:

For a given function