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This paper, divided into three parts (Part II-A, Part II-B and Part II-C), contains the detailed 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 . It follows two pre-viously published papers: the first, labelled as Part I, contains the complete (elementary but non-trivial) theory for ; the second is a survey highlighting only the main results without proofs. All the material appearing in §2 of the survey is here reproduced in an expanded form, as it contains all the preliminary formulas necessary to understand and prove the results. The remaining part of the survey—especially the heuristical considerations and consequent conjectures in §3—may serve as a good introduction to the complete theory.

Following the line of thought in [

where

and the Hardy notation

the

Our theory parallels: 1) the classical Taylor’s formula at a point

・ In §2, we collect all the preliminary material concerning factorizations of a disconjugate operator and the nonvanishingness of various Wronskians involving certain bases of its kernel. The scale of comparison functions

・ In §3, we exhibit the operators used in formal differentiation together with some of their elementary properties.

・ In §4, we state characterizations of a set of asymptotic expansions obtained from (1.1) by formal applications of the differential operators implicitly defined by the “unique” canonical factorization, termed by us of type (I), which chronologically is the first to be introduced, studied and applied.

・ In $5, we do the same job for the differential operators implicitly defined by a special canonical factorization termed by us of type (II) and constructed using the given asymptotic scale.

・ In $6, all proofs are collected.

・ In Part II-B, we specialize the results to the important class of functions satisfying a differential inequality linked to the scale, so obtaining many nice characterizations.

・ In Part II-C, we exhibit two algorithms for constructing the canonical factorizations upon which our theory is built. These algorithms are simple to describe, admit of “natural” asymptotic interpretations (also showing the appropriateness of the used differential operators) and are of considerable help in building examples and counterexamples concerning formal differentiation of asymptotic expansions.

Occasionally, an asymptotic expansion

will be called “incomplete”―with respect to the given scale

We stress that the guiding thread of our work is “formal differentiation of asymptotic expansions”, a theme going back to the early 20th century (Landau, Hardy, Boas and others), but usually referring to the use of standard derivatives. The factorizational approach clearly shows that formal differentiation is admissible only if one uses suitable differential operators strictly linked to the involved scale together with related factorizations. The case of standard derivatives is very special and is highlighted in Part II-B and Part II-C.

Notations

・

・

・ For

・

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

wherein each integral

・ The symbol

・ Two acronyms systematically used are T.A.S. = “Chebyshev asymptotic scale” as in Def. 2.1, and C.F. = “canonical factorization” defined in Proposition 2.1-(iv) and (v);

・ Propositions are numbered consecutively in each section irrespective of their labelling as lemma, theorem and so on.

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

In this section,

where

Proposition 2.1 (Disconjugacy on an open interval via factorizations). For an operator _{1,2},

1)

2)

or equivalently

3)

where the

4)

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

5) For each c,

and

Remarks. 1) In the definition of a C.F. conditions (2.5) or (2.6) are required to hold for the index i running from 1 to

2) A global C.F. of type (I) at a specified endpoint does always exist for a disconjugate operator on an open interval and is “essentially” unique in the sense that the functions

which 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).

(I) (Results involving a differential operator). Let _{1,2} disconjugate on an open interval

1) Its kernel has some basis

2) For each such basis

noticing the reversed order of the

3) For any strictly decreasing set of indexes

we have

and in particular we have the inequalities

4) For each k,

Notice the ordering of the

(II) (Results involving scales with less regularity). Let

and let there exist an integer

where the symbol

together with the above-stated properties in 3) and 4). Notice that in (2.17)-(2.18) the signs of the Wronskians are well defined even if they remain undefined in the assumptions (2.15)-(2.16).

To visualize (2.14), we list a few asymptotic scales at

It is quite important to note the order of the

The above results substantiate the following definition of special asymptotic scales wherein we merely fix the neighborhood of b left undefined in Proposition 2.2 whose part (I) grants the existence of such scales whereas part (II) implies a lot of useful properties even for scales with less regularity. From now on the interval will be denoted as in the two-term theory [

Definition 2.1 (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 (2.1)_{1,2}, acting on the space

Remarks. 1) Condition (2.21) 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 (2.25).

2) Choosing an half-open interval in this definition is a matter of convenience: the point

3) In the above definition we have merely supposed the nonvanishingness of various functions instead of specifying their signs as in Proposition 2.2; this avoids restrictions that are immaterial in asymptotic investigations. If the

4) 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

though the converse generally fails as it may be easily checked for the scale

which satisfies (2.27) on

Proposition 2.3 (Several characterizations and additional properties of T.A.S.’s). Let the ordered n-tuple of real-valued functions

(I) The following are equivalent properties:

1)

2) Both sets of inequalities (2.24) and (2.27) hold true.

3) The ordered n-tuple

4) The n-tuple

with suitable functions

If this is the case the

Conversely we have the following formulas for the Wronskians of the

(II) For

for any set of indexes satisfying (2.11) and we also have the hierarchies between the Wronskians stated in Proposition 2.2-4) and referred to

Part (I) of Proposition 2.3 generalizes a classical result, ([

Under condition (2.25) formulas in Proposition 2.3-2) are related to C.F.’s of type (II) at

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

1) 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 T and

2) 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 (2.24) are strictly positive, i.e. when

3) Analogously we define the following

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

on the interval

hence the associated factorization

is (up to constant factors) “the” global C.F. of type (I) at

4) The special fundamental system of solutions to

satisfies the asymptotic relations:

Relations (2.49) uniquely determine the fundamental system

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

A quick proof of the existence of C.F.’s. The global existence of C.F.’s of type (I) was for the first time proved by Trench [

In this section, we collect some facts concerning those special operators associated to canonical factorizations: properties and formulas which our theory is constructed upon. We do not report the heuristic considerations which justify our approach and show how “natural” the obtained results are; we refer the reader to ([

Referring to the factorization of type (I) in (2.47), with the

which satisfy the recursive formula

And referring to the factorization of type (II) in (2.39), with the

which satisfy the recursive formula

We call L_{k} [respectively M_{k}] “the weighted derivative of order k with respect to the weights

hence there exist never-vanishing functions

It follows that L_{k} and M_{k} preserve the hierarchy (2.23), namely we have the following asymptotic scales:

for each fixed k,

both equivalent to (2.23). Hence if we apply any n-tuple of operators L_{k} and M_{k},

we get again an asymptotic expansion with a zero remainder and in this sense we may say that “the asymptotic expansion (3.11) is formally differentiable _{k} have a remarkable asymptotic link with the coefficients in (3.11) as claimed in the following

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 (2.39) the following facts hold true for the differential operators M_{k} in (3.3):

(I) The

(II) For a fixed k,

if and only if

where, for h = k, (3.18) is the identity (3.16).

(III) In the special case wherein all the Wronskians in (2.24) are strictly positive then the constants in (3.13)- (3.14) have the values:

We stress that the equivalence “(3.16) Û (3.17)” is an algebraic fact based on (3.12)-(3.13) whereas the inference “(3.16)-(3.17) Þ (3.18)” is an asymptotic property whose validity requires that

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

with suitable nonzero constants

and viceversa

with suitable constants

Lemma 4.1. The following relations are checked at once:

Hence, we have the following chains of asymptotic relations:

The first chain in (4.8) coincides with the second chain in (2.49) apart from the ordering and the factor

Lemma 4.2. If a solution

for some

Moreover,

with the

Last, with the

Lemma 4.3. Any function

with suitable constants

By (4.13) the linear combination

We shall now characterize various situations wherein relations (4.16)-(4.17) become asymptotic expansions. In the following two theorems we state separately three cases of a single claim lest a unified statement be obscure. The reader is referred to the first remark after Theorem 4.4 to grasp the meaning of the differentiated asymptotic expansions which exhibit a special non-common phenomenon.

Theorem 4.4 (Asymptotic expansions formally differentiable according to the C.F. of type (I)). Let

(I) The following are equivalent properties for a suitable constant

1) The set of asymptotic relations

2) The single asymptotic relation

which is the explicit form of the relation in (4.18) for

3) The improper integral

Under condition (4.20) we have the representation formula

(II) For a fixed

4) The set of asymptotic expansions as

5) The second group of asymptotic expansions in (4.22), i.e.

where we point out that the last meaningful term in the right-hand side is a constant.

6) The following improper integral, involving “i” iterated integrations,

Under condition (4.24), we have the representation formula

for

Remarks. 1) Relations in (4.22) may be read as follows. The first relation, involving

and the relations involving

where

The loss of the last meaningful term, where it occurs, is caused by formula (4.12) for

Notice that in the second group of expansions in (4.28) i.e. those with remainder “

2) It is shown in §6, after the proof of Theorem 4.4, that the set (4.23) is not equivalent in general to the single relation

as in part (I) of the theorem (case

3) Suitable weighted differentiations of (4.25) yield integral representations of the remainders in the differentiated expansions of orders greater than

For

Theorem 4.5 (The case

1) The set of asymptotic expansions as

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

2) The improper integral

3) There exist n real numbers

If this is the case

The phenomenon described in (4.28) and (4.31) is intrinsic in the theory; it occurs even in the seemingly elementary case of real-power expansions, ([

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

with suitable constants

Warning! To simplify formulas and to leave no ambiguity about the signs of the involved quantities we assume throughout this section that the Wronskians in (2.24) are strictly positive. Hence, by (3.19)

By (3.8), the ordered linear combination in (5.2),

is an asymptotic expansion at

Theorem 5.1 (Complete expansions formally differentiable according to a C.F. of type (II) ). Let our T.A.S. be such that all the Wronskians in (2.24) are strictly positive and let

(I) The following are equivalent properties:

1) There exist n real numbers

where the first term in each expansion is lost in the successive expansion, just the same phenomenon as in Taylor’s formula. Notice that the relation that would be obtained in (5.6) for

2) All the following limits exist as finite numbers:

where the

3) The single last limit in (5.7) exists as a finite number, i.e.

and (5.8) is nothing but the relation in (5.6) for

4) The improper integral

and automatically also the iterated improper integral

5) There exist n real numbers

where, by (2.35),

(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 (5.9) involves the sole coefficient

hence (5.9) can be rewritten as

For

3) In Theorem 4.5, generally speaking, no such estimates as in (5.16)-(5.17) can be obtained due to the divergence of all the improper integrals in (4.33) if the innermost integral is factored out.

4) Theorem 5.1 changes the perspective of the elementary characterizations in (1.3) of the coefficients

In the following result about incomplete expansions formal differentiation is in general legitimate a number of times less than the “length” of the expansion (see Remark 2 after the statement).

Theorem 5.2 (A result on incomplete asymptotic expansions). Let our T.A.S. be such that all the Wronskians in (2.24) are strictly positive and let

(I) For a fixed

1) There exist i real numbers

2) All the following limits exist as finite numbers:

where the

3) The single last limit in (5.22) exists as a finite number, i.e.

and (5.23) coincides with the relation in (5.21) for

4) The improper integral (involving n−i+1 iterated integrations)

and automatically also the iterated improper integral

(II) For

holds true for some real number

Remarks. 1) We shall see in the proof of Theorem 5.2, formula (7.44), that the representations of the quantities

2) As concerns estimates of the quantities

for some constant c. If, as

Theorem 5.3 (The analogue of Theorems 5.1-5.2 with “O”-estimates). Let our T.A.S. be such that all the Wronskians in (2.24) are strictly positive, let

1) There exist

2) All the following relations hold true:

where the

3) It holds true the single last relation in (5.31) i.e.

4) We have the following estimate instead of condition (5.24):

For

and representation (5.11)-(5.12) must be replaced by

For

holds true iff

Proof of Proposition 2.1. For the equivalence of the two properties in 1), see Coppel ([

Proof of Proposition 2.2. Part (I) is contained in ([

Proof of Proposition 2.3. 1) Þ 2). Let

In particular, we have

and we may apply part (II) of Proposition 2.2 (regardless of the signs) because the second condition in (6.2) coincides with the condition in (2.16) for

which imply (2.27). Proposition 2.2 also implies all the claims in part (II).

1) Û 3). We refer to the standard definition of the concept of “extended complete Chebyshev system on a generic interval J”, based on the maximum number of zeros for their linear combinations, see, e.g., ([

2) Þ 4). Here, we are retracing the steps of the proof in ([

This implies three facts: 1) the convergence of the improper integrals

2) the representations for

Before using induction, we prove the representation of

whence

which implies the representation of

we immediately infer from (1.5) and from (2.14) referred to

(The n-tuple

We may now apply our inductive hypothesis inferring that

where the

and (6.13) becomes

which, by (2.23), gives the sought-for formula for

we have as in ([

hence,

and this shows the converse inference “4) Þ 2)”.

Proof of Proposition 2.4. 1)-2). Properties in (2.36) follow directly from the assumptions, and relations in (2.37) are a standard fact as remarked in the preceding proof. As concerns (2.38), the continuity of the

Factorization (2.39) is then the classical factorization arising from (2.35) and discovered for the first time by Pólya [

3) The very same reasonings prove the properties of the

and in general, by (2.10), these calculations prove the existence of a C.F. of type (I) at

Proof of Proposition 3.2. Relations (3.12) to (3.14) are directly checked using representations (2.40). Relation (3.15) follows from the second relation in (3.6) replacing u by

(3.18) for

For

where the remainder “

Proof of Lemma 4.2. From the chain

for suitable constants

now (4.10) follows from (4.7), and (4.11) follows from (4.4). If in (6.22) we replace

Proof of Theorem 4.4. Part (I). From (4.12) and (4.17), with

denoting by L our operator

If (4.19) holds true we have (4.21), and representations in (4.16) can be rewritten as

Now we have

by (4.1) and (4.10)

where

is independent of k. To show that c coincides with the

Part (II). Case

First, we prove “(6.31) Û (6.32)”. If (6.32) holds true then, by part (I) of our theorem, we have all relations in (4.18) and in particular the second relation in (6.31). Moreover we can rewrite representation in (4.16) for

where

By comparison with the first relation in (6.31) we get (6.32) because

as

Here, the constant

Proof of Theorem 5.1. 1) Þ 2). Relation (5.5) implies the existence of

same value of k because of (3.13)-(3.14). 2) Þ 3) is obvious. 3) Û 4). It follows from (5.3) that the limit in (5.8) exists in

where, as above,

4) Þ 1). We have already proved (6.36) which is (5.6) for

whence, by (6.36) and (2.38), we get

for a suitable constant

which is the relation in (5.6) for

and integrate (6.38) after dividing by

for a suitable constant

with a suitable constant

Proof of Theorem 5.2. 1) Þ 2) follows from (3.13)-(3.14). 2) Þ 3) is obvious. 3) Û 4): by (3.13)-(3.14) the representation in (5.2) for

whence our equivalence follows at once. If this is the case (5.2) can be rewritten as

where

4) Þ 1). As in the corresponding inference in Theorem 5.1 we integrate (6.44) starting from

whence, by (2.38), (3.4) and (3.14), we get

where the constant

Proof of Theorem 5.3. This is almost a word-for word repetition of the proofs of Theorems 5.1-5.2. 1) Þ 2). For

By iteration, we get all relations in (5.28)-(5.29).

The author thanks the referees for their valuable suggestions.

In the above reference [

・ On p. 255, first line under title of §4: “gaphs” reads “graphs”.

・ On p. 260: the reader may notice that (4.29)_{2} is just a reformulation of (4.27).

・ On p. 261, first line from above: delete “t” in locution “t limit position”.

・ On p. 266: there is a redundant sign of absolute value “|” inside the integrals in (5.27) and (5.28).

・ On p. 284: in the right-hand side of formula (8.26) the quantity

・ On p. 286: in reference [

In the above reference [

・ On p. 3: in the right-hand side of formula (2.3) the symbol u is missing in the innermost position so that the formula correctly reads:

・ On p. 19: in reference [