_{1}

^{*}

After studying finite asymptotic expansions in real powers, we have developed a general theory for expansions of type (*) ,x → x_{0} where the ordered n-tuple forms an asymptotic scale at x_{0} , i.e. as x → x_{0}, 1 ≤ i ≤ n – 1, and is practically assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x _{o}. As in previous papers by the author concerning polynomial, real-power and two-term theory, the locution “factorizational theory” refers to the special approach based on various types of factorizations of a differential operator associated to . Moreover, the guiding thread of our theory is the property of formal differentiation and we aim at characterizing some n-tuples of asymptotic expansions formed by (*) and n -1 expansions obtained by formal applications of suitable linear differential operators of orders 1,2,…,n-1. Some considerations lead to restrict the attention to two sets of operators naturally associated to “canonical factorizations”. This gives rise to conjectures whose proofs build an analytic theory of finite asymptotic expansions in the real domain which, though not elementary, parallels the familiar results about Taylor’s formula. One of the results states that to each scale of the type under consideration it remains associated 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 two special senses.

In this paper, we give the main results concerning a general analytic theory of asymptotic expansions of type

where

and the Hardy notation

the

(i) It yields applicable analytic characterizations of an expansion (1.1) matched to other asymptotic relations obtained by formal differentiations in suitable senses.

(ii) For each Chebyshev asymptotic scale there are at least two well-defined

(iii) A special family of functions is associated to each Chebyshev asymptotic scale namely that of generalized convex functions, for which the validity of the sole relation (1.1) automatically implies its formal differentiability

The introductions in [

Notations

・

・

・ For

・

・ If no ambiguity arises we use the following shorthand notations or similar ones:

wherein each integral

・ The symbol

・ The acronyms we systematically use: T.A.S. := “ Chebyshev asymptotic scale” as in Def. 2.1;

C.F. := “canonical factorization” defined in Proposition 2.1-(iv) and (v).

Our theory is built upon appropriate integral representations stemming from a special structure of the asymptotic scale

where

Proposition 2.1 (Disconjugacy on an open interval via factorizations). For an operator

(i)

(ii)

or equivalently

(iii)

where the

(iv)

and a similar “C.F. of type (I) at the endpoint b”, i.e. with the

(v) For each

and

Remarks. Conditions (2.5) or (2.6) are required to hold for the index

and this is a special contingency characterized in [

which are C.F.’s of type (II) at both the endpoints “0” and “

C.F.’s are naturally linked to bases of ker

Proposition 2.2 (Wronskians of asymptotic scales and their hierarchies). Let

(i) Its kernel has some basis

(ii) For each such basis:

noticing the reversed order of the

(iii) For any strictly decreasing set of indexes

we have:

and in particular we have the inequalities:

(iv) For each

Notice the ordering of the

To visualize (2.12), we list a few asymptotic scales at

It is quite important to note the order of the

Definition (Chebyshev asymptotic scales). The ordered n-tuple of real-valued functions

Whenever the

they remain associated to the operator:

which is the unique linear ordinary differential operator of type

Remarks. 1. Condition (2.15) is the usual regularity assumption in approximation theory (Chebyshev systems and the like), whereas in matters involving differential equations/inequalities it is natural to assume (2.19). Choosing an half-open interval is a matter of convenience: the point

2. In the definition we have merely supposed the non-vanishingness of various functions instead of specifying their signs as in Proposition 2.2; this avoids restrictions that are immaterial in asymptotic investigations. If the

3. As concrete examples of such asymptotic scales on

When comparing our notations with other authors’ results the reader must carefully notice the numbering of the

The concept of Chebyshev asymptotic scale, even under the weak regularity (2.15), admits of useful characterizations which generalize a classical result, ([

Proposition 2.3 For

for any set of indexes satisfying (2.9) and we also have the hierarchies between the Wronskians stated in Propo- sition 2.2-(iv) and referred to

Notice that the converse of the inference “(2.18)

on the interval

In the next proposition we collect all the facts essential to develop our theory of asymptotic expansions.

Proposition 2.4 (Formulas concerning T.A.S.’s linked to differential operators). Let the ordered

(i) Define the following

Then the

Their reciprocals, left apart

on the interval

Our operator admits of the following factorization on

which is a global C.F. of type (II) at both endpoints

(ii) Our T.A.S. (apart from the signs) admits of the following integral representation in terms of the

hence the

In the special case where all the Wronskians in (2.18) are strictly positive, i.e. when

(iii) Analogously we define the following

They satisfy the same regularity conditions on the half-open interval

on the interval

hence, apart from constant factors, the associated factorization

is “the” global C.F. of

(iv) The special fundamental system of solutions to

satisfies the asymptotic relations:

Relations (2.38) uniquely determine the fundamental system

The construction of the two above factorizations starting from the given expressions of the coefficients

In the elementary case of Taylor’s formula, the simple condition

is not a mere sufficient condition for the validity of the asymptotic expansion

it in fact characterizes the set of the

which is formed by (3.2) together with the relations obtained by formal differentiation

If we strenghten condition (3.1) by assuming

we also have the representation

which, besides implying the validity of (3.3) for

in two quite different senses and under suitable integrability conditions. But in the analogous theory for expansions in arbitrary real powers

developed in [

are likely to be formally applicable to an expansion (1.1) because they preserve the hierarchy (2.17) after suppressing the identically-zero terms, which means that they transform an asymptotic expansion with a zero remainder

into a similar expansion, namely:

For instance, we have the identity:

wherein

for each fixed

Now in the Wronskians (3.9) a permutation of

where “nested” refers to the inclusions of their kernels and the problem consists in finding sufficient, and possibly necessary, conditions for the validity of the set of asymptotic relations

with proper choices of the

which we label as “weighted derivatives of orders 0, 1, 2 etc. with respect to the weights

Referring to the factorization of type (I) in (2.36), with the

which satisfy the recursive formula

And referring to the factorization of type (II) in (2.29), with the

which satisfy the recursive formula

Now representations (2.30) and (2.36) imply that:

hence, there exists never-vanishing functions

It follows that

for each fixed

both equivalent to (2.17). Hence, applying each

Conjecture. For each chosen C.F. of

there exists a linear subspace

(ii) each

The problem consists in finding out analytic conditions characterizing the elements of

There is another kind of considerations suggesting a special role of C.F.’s of type (II). If we wish to investigate the possible expressions of the coefficients of an asymptotic expansion alternatively to the recurrent formulas (1.3), so generalizing (3.4), it is clear from the study of polynomial expansions in [

Proposition 3.1 (The coefficients of an asymptotic expansion with zero remainder). Referring to the T.A.S. in Proposition 2.4 and to the special factorization (2.29) the following facts hold true for the differential operators

(I) The

(II) For a fixed

iff

If (3.31)-(3.32) hold true on a left neighborhood of

where, for

(III) In the special case where all the Wronskians in (2.18) are strictly positive then the constants in (3.28)-(3.29) have the values:

We stress that the equivalence “

Conjecture. If all the limits in (3.33) exist as finite numbers for some function

holds true matched to other expansions obtained by formal applications of the operators

Our study gives complete answers to the above Conjectures and the main results are reported in the next sections.

We start from the “unique” C.F. of our operator

with suitable non-zero constants

Moreover any function

with suitable constants

Here is one of the main results obtainable by this approach.

Theorem 4.1 (Asymptotic expansions formally differentiable according to the C.F. of type (I)). For

(i) The set of asymptotic expansions as

where the last term in each expansion is lost in the successive expansion.

(ii) The iterated improper integral

(iii) There exist

If this is the case

The phenomenon appearing in (4.5) is intrinsic in the theory; it occurs even in the seemingly elementary case of real-power expansions, [

Starting from an “incomplete asymptotic expansion”

our study would characterize a set of more involved expansions not reported here.

Now, we face our problem starting from a C.F. of type (II) at

In this new context, a representation of the following type is appropriate for any function

with suitable constants

To simplify formulas and to leave no ambiguity about the signs of the involved quantities we assume in this section that the Wronskians in (2.18) are strictly positive.

Hence, by (3.34)

Theorem 5.1 (Asymptotic expansions formally differentiable according to a C.F. of type (II)). Let our T.A.S. be such that all the Wronskians in (2.18) are strictly positive and let

(I) The following are equivalent properties:

(i) There exist

where the first term in each expansion is lost in the successive expansion as in Taylor’s formula. (The relation that would be obtained in (5.5) for

(ii) All the following limits exist as finite numbers:

where the

(iii) The single last limit in (5.6) exists as a finite number, i.e.

and (5.7) is nothing but the relation in (5.5) for

(iv) The improper integral

and automatically also the iterated improper integral

(v) There exist

where we remind that, by (2.25),

(II) Whenever properties in part (I) hold true we have integral representation formulas for the remainders

namely:

for

Under the stronger hypothesis of absolute convergence for the improper integral we get:

Similar estimates can be obtained for the

Remarks. 1. As noticed in [

2. Condition (5.8) involves the sole coefficient

For

3. In Theorem 4.1, generally speaking, no such estimates as in (5.15)-(5.16) can be obtained due to the divergence of all the improper integrals in (4.6) if the innermost integral is factored out.

4. It has been proved in [

The foregoing theory becomes particularly simple when the involved improper integrals are absolutely convergent and still more expressive for a function

If

Theorem 6.1. If all the Wronskians in (2.18) are strictly positive and if

(i) There exist

(ii) There exist

(iii) The following set of asymptotic expansions holds true:

(iv) The following set of asymptotic expansions holds true:

(v) The following integral condition is satisfied:

(vi) The following integral condition is satisfied:

To this list, we may obviously add the other properties in Theorem 5.1.

If this is the case, the remainder

whence it follows that

In addition to the equivalence (iii)

Theorem 6.2. For

Hence, each of these three conditions implies both sets of asymptotic expansions (4.4) and (5.5). (Here the signs of the Wronskians are immaterial.)

The equivalence between (6.9) and (6.10) easily follows from Fubini’s theorem by interchanging the order of integrations in (6.9) whereas the equivalence between (6.10) and (6.11) is by no means an obvious fact. A proof may be obtained by showing a stronger result, namely the following asymptotic relation:

The foregoing results are well illustrated by the special class of scales of the form:

where

We also assume:

Now, using a proper device it can be given an elementary proof of the formula:

where

Proposition 7.1. Under the above assumptions and notations:

(I) The “unique”

which also gives the differential operators

(II) A special

which also gives the differential operators

Identities (7.5)-(7.6) can be proved either using (7.4) and formulas in Proposition 2.4 or writing out the n- tuple (2.37) and checking that its span coincides with ker

Proposition 7.2. (I) Referring to Theorem 4.1 we have the equivalence of the following three properties:

(i) The set of asymptotic expansions as

(ii) The improper integral

(iii) For suitable constants

(II) Referring to Theorem 5.1, we have the equivalence of the following three properties:

(iv) The set of asymptotic expansions as

(v) The improper integral

(vi) For suitable constants

We visualize Proposition 7.2 for five remarkable choices of

Corollary 7.3. (I)

(II)

see the theory developed in [

(III)

(IV)

(V)

The author thanks the referees for their helpful suggestions.