_{1}

^{*}

Part II-B of our work continues the factorizational theory of asymptotic expansions of type (*)
,
,
where the asymptotic scale
,
, is assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x
_{0}. The main result states that to each scale of this type it remains as-sociated an important class of functions (namely that of generalized convex functions) enjoying the property that the expansion (*), if valid, is automatically formally differentiable n ? 1 times in the two special senses characterized in Part II-A. A second result shows that formal applications of ordinary derivatives to an asymptotic expansion are rarely admissible and that they may also yield skew results even for scales of powers.

This is a continuation of a previous paper [

・ §8 contains the main result in the paper: to each Chebyshev asymptotic scale

・ In §9, discussing formal application of standard derivatives to an asymptotic expansion, we characterize the existence of certain polynomial expansions at an endpoint where derivatives may fail to exist and such that the growth-order estimates of the remainders of the differentiated expansions follow unexpected algebraic rules.

・ §10 contains the proofs and §11 contains a few remarks about our theory.

Whereas the results in Part II-A show that “formal differentiation of asymptotic expansions” is usually admissible only if suitable operators linked to the given scale are used, the results in this Part II-B shed further light on this classical problem by exhibiting a meaningful and not too special case wherein suitable formal differentiations are automatically admissible and by showing that standard derivatives are admissible in very special cases only and that they may yield formulas algebraically skew from a classical viewpoint.

We continue the numbering of sections and formulas in [

The operator

is the unique linear ordinary differential operator of type (2.1)_{1,2}, acting on the space

and we are supposing

The theory developed in Part II-A becomes particularly simple when the involved improper integrals are absolutely convergent and still more expressive for a function f satisfying the nth-order differential inequality

Under the assumptions (7.1) and (7.3) this is a subclass of the so-called “generalized convex functions with respect to the system

Theorem 8.1 (Complete asymptotic expansions). If

1) There exist

2) There exist n real numbers

3) The following set of asymptotic expansions holds true:

4) The following set of asymptotic expansions holds true:

5) The following integral condition is satisfied:

6) The following integral condition is satisfied:

To this list we may obviously add the other properties in Theorem 5.1 and if this is the case the remainder

whence it follows that

The above equivalence “1) Û 2)” simply means that, under condition (8.1), a relation “

In addition to the equivalence “3) Û 4)” there is another remarkable circumstance wherein the two types of formal differentiations are simultaneously admissible namely when the convergence of the pertinent improper integrals is absolute.

Theorem 8.2. For

Hence each of these three conditions implies both sets of asymptotic expansions (4.31) and (5.5)-(5.6) (here the signs of the Wronskians are immaterial).

An indirect brief proof of the equivalence “(8.11) Û (8.12)” can be based on Theorem 8.1, but it also follows from the following remarkable relation valid for any signs of the Wronskians in (7.3), (7.4):

Using Theorems 4.4 and 5.2, we can also get the analogues of Theorems 8.1-8.2 for incomplete asymptotic expansions and here is a concise statement, all asymptotic relations referring to

Theorem 8.3 (Incomplete asymptotic expansions). Let

(which last relations are written in (4.28) in an expanded form);

To the foregoing list we may obviously add property 2) or property 5) in Theorem 4.4 and properties 2)-3) in Theorem 5.2. For

Notice that relation (8.13) and the definition of

hence (8.18) does not in general imply the convergence, as

Moreover each of the “O”-estimates in (8.17) is meaningful whenever the involved integral diverges as

one-signedness, it is possible to infer from condition (8.20) sharper estimates not depending on

Theorem 8.4 (Sharper estimates for

Then the estimates in (8.17) for

In the present context the above estimates are by no means obvious or “natural”: they have been obtained by adapting the standard calculations in the proof of the Abel-Dirichlet’s test for convergence of weighted improper integrals (Lemma 10.1 below). As a simple check of their validity we reobtain classical estimates for the derivatives of nth-order convex functions, and to be consistent with the meaning of n in the present series of papers, namely “n = dimension of the Chebyshev system

Corollary 8.5 (Rates of increase of derivatives of higher-order convex functions). Assume that:

Then the following asymptotic relations hold true as

Here the asymptotic scale is:

The case

The estimates in Corollary 8.5 also follow from old results by Landau, Hardy and Littlewood about differentiation of asymptotic relations involving real powers, under assumptions of monotonicity on the derivatives, results that were discussed in [

An important remark. In Theorem 8.1 the two types of formal differentibility

lent facts whereas it is not so for a generic f such that

borhood of

Open problem. For

We shall not dwell on this marginal aspect of the theory though it leaves unsolved whether or not we may use representation formula (14.38), in alternative to (15.12)-(15.13), under condition (15.10).

Before investigating cases wherein standard derivatives

Definition 9.1 (Asymptotically-admissible operators). Let

without any further regularity assumptions.

(I) (A definition valid in special cases but highlighting the concept).

if its formal application to both sides of (9.2) yields a new asymptotic expansion

This implicitly implies that

Put in these terms the definition is well-posed if none of the functions

(II) (A general definition). First, if

and say that

after suppression of all the zero terms. An alternative locution for an asymptotically-admissible

A first group of examples clarifies the necessity of specifying “after suppression of all the zero terms”. In each of the following three examples the standard operator of differentiation

That the standard operator of differentiation does not preserve asymptotic hierarchies is quite elementary but a second group of examples shows that it may not preserve asymptotic hierarchies even when acting on an n-tuple forming a Chebyshev asymptotic scale (signs apart) on a neighborhooh of

1) Elementary examples showing that if

2) Examples of Chebyshev asymptotic scales

3) Example of a Chebyshev asymptotic scale

The above examples are variations on the examples in Bourbaki ([

4) Example of a Chebyshev asymptotic scale

and there exists a function g such that

Just take

with different values of

Let us ask the question: What are the natural scales granting the asymptotic admissibility of the standard operators

disproved even for the familiar scale at

for some function

For

Here the operators

Lemma 9.1. (I) If

(II) If

The reader must not think that we are now filling a few pages with trivialities about Taylor’s formula; as a matter of fact if we apply our theory to the operator

Theorem 9.2. Let:

whereas the one associated to factorization (9.20) is the standard derivative

Consider now a generic polynomial of order

(I) (The continuity property of Taylor’s formula). The following are equivalent properties for a fixed

1) The set of asymptotic expansions as

As concerns the bound

2) The improper integral (involving n−i iterated integrations)

which for

On account of the hypothesis

which is equivalent to condition

Hence relations (9.29) are nothing but Taylor’s formula of order i of

(II) (A polynomial expansion at an endpoint where derivatives may fail to exist). The following are equivalent properties for a fixed

3) The set of aymptotic expansions as

4) The set of aymptotic expansions as

where the two estimates coincide for

The nondifferentiated expansion is written differently in (9.34) than in (9.35) to correctly apply the operators

5) The iterated improper integral (involving i+1 integrations)

We must comment on the above claims. Part (I) is a classical elementary property which may be traced back to Walter and Ford ([

then the estimate for

Examples. The following simple examples involving oscillatory functions will reassure the doubtful readers (and the author himself was in the number). For

an example being provided by

which is differentiable at

For

It is obvious that in both cases f can be extended so as to be of class

for which

A final comment. The discussion in this section shows that formal applications of ordinary derivatives to an asymptotic expansion is not admissible generally speaking, and even for the very special asymptotic scale (9.22) the first (but not the second) factorizational approach can give seemingly-unnatural results. It is in principle true that each of the two sets of expansions characterized in Part II-A, §§4,5, can provide asymptotic information (not always meaningful and not necessarily expansions) for the ordinary derivatives; however this is easily achieved for the first-order derivative but is practically unmanageable for higher-order derivatives and yields no theoretical result. It is also true that for expansions in arbitrary real powers as

Proof of Theorem 8.1. The only thing to be proved is the inference “1) Þ 5) Ù 6)”, the other properties being included in Theorems 4.5 and 5.1. We use a procedure already used in ([

By the assumption (8.2) the left-hand side has a finite limit as

after applying L’Hospital’s rule

with suitable constants

Here again the left-hand side has a finite limit as

after applying L’Hospital’s rule

with suitable constants

with a suitable constant

and (10.6) in turn implies:

By the positivity of the integrand this last relation implies (8.6) and the first representation in (8.8) for

which implies

and (5.1) can be rewritten as

with suitable constants

Evaluating the limit of the right-hand side by L’Hospital’s rule and using formula in (2.31),

and this last limit, which exists in

and (10.11) can be rewritten as

with suitable constants

for some

exists in

which, by the positivity of the integrand, exists in

with some constant

and if we try to evaluate the limit of the ratio on the left applying L’Hospital’s rule

lows the second representation in (8.8) for the remainder in (8.3). The proof is over.

Proof of Theorem 8.2. The equivalence between (8.10) and (8.11) easily follows from Fubini’s theorem by interchanging the order of integrations in (8.10) whereas the equivalence between (8.11) and (8.12) is by no means an obvious fact. A concise proof based on Theorem 8.1 is as follows; putting

we have

hence F satisfies

and (8.12). Now we prove relation (8.13) recalling that all the involved functions and Wronskians are strictly one-signed on the interval. The symbol of asymptotic equivalence is referred to

valid for any ordered

where in the last but one passage we have applied formula (10.22) to the ordered triplet

Now, for a fixed

As in (10.24) we can prove that (10.26) is true for

that is (10.26). Using this relation for

that is (8.13).

Proof of Theorem 8.3. The only thing to prove is the O-estimates in (8.17). From representation (5.2)

for some constant c, whence the estimate for

Proof of Theorem 8.4 is contained in the following lemma only valid under the stated one-signedness restrictions.

Lemma 10.1 (Growth-order estimates for iterated improper integrals with nonnegative integrands). Assumptions:

Thesis. The following estimates hold true:

Proof. All functions

which implies the first relation in (10.33). To estimate

Now the integral on the left is convergent by hypothesis and the first integral on the right is convergent by (10.34), hence the second integral on the right converges as well and we get the equality

whence, again by (10.34),

The nondecreasingness of

which yields the estimate for

we want to prove

whence

And now the nondecreasingness of

Relations (10.39)-(10.40) allow iteration of the procedure to estimate

Proof of Theorem 9.2. Part (I) is nothing but Theorems 5.1 and 5.2 applied to the present case wherein f is replaced by

Lemma 10.2. Let f be of class

For a fixed

is equivalent to the set of asymptotic relations, as

where the two estimates coincide for

For

Proof. It is easily proved by induction that the expanded expressions of the

with suitable constants

whence

Second step. From the second equality in (10.45) we infer

having used (10.47), and from these last estimates we get

Suppose now that the relations in (10.43) for the derivatives have been proved true for

whence

Now for

For the remaining values of i we must use the second estimate in (10.42) for

Now inside the sum each

In the first case the restriction

Finally from (10.51) we get the sought-for estimates for

The proof of “(10.42) Þ (10.43)” is over. The converse implication is checked at once replacing the estimates in (10.43) into the sum (10.45) expressing

We show by two examples that the use of non-C.F.’s is unreliable to construct a general theory. Let us refer, e.g., to the characterizations of an asymptotic expansion for a generalized convex function.

First example. For

and that the inference “Þ” can be easily proved when a C.F. of the operator

and the related representation

Assuming the validity of the expansion in (11.1), we try to find a necessary integral condition involving

and by reasons of constant sign, we infer

a much weaker condition than

Second example. Let us consider the operator

We have

obtained from the results in §8 based on the use of C.F.’s. However, in this case the equivalence in (11.8) can be also obtained using the non C.F. in (11.7) if one starts from the corresponding integral representation for f and reapplies the same procedure used in the proof of Theorem 8.1.

We start noticing that the convergence of an iterated integral

where

Now an integral representation linked to a C.F. of type (I) and more general than (4.15) in Part II-A is

wherein the

If there exists an n-tuple

then, recalling that

with an appropriate representation of the remainder.

But, by the initial remark, such an integral condition is almost useless for general results as well as for practical applications if the

Moral. Working with the C.F. of type (I) at

The situation is technically different when working with a C.F. of type (II). Referring to an integral representation of type

more general than (5.1) in Part II-A, we see that as soon as we may choose

does not “whimsically” depend on