This is a continuation of a previous paper [1] , about the factorizational theory of asymptotic expansions in the real domain.

・ §8 contains the main result in the paper: to each Chebyshev asymptotic scale it remains associated an important class of functions enjoying the property that an asymptotic expansion according to this scale, if valid, is automatically formally differentiable times in the two special senses characterized in §§4,5 in Part II-A. Under the regularity assumptions of the factorizational theory, this class is characterized by an nth-order differential inequality whereas in the yet-to-be-developed geometric theory it will be the class of generalized convex functions as in the special case of polynomial expansions ([2] , §4).

・ 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 [1] , but we adopt an independent numbering of the references in the bibliography. In order to agree with some classical terminology about the matter in this paper, it is convenient to specify the signs of certain Wronskians, so we list the fundamental properties of the scale we shall use taken from ([1] , Def. 2.1 and Prop. 2.3):

The operator

is the unique linear ordinary differential operator of type (2.1)_1,2, acting on the space and such that Expansions we are studying are of type

and we are supposing as the two-term theory has been thoroughly studied in [3] . Operators and are defined in formulas (3.1) to (3.4) in Part II-A; properties of the’s are reported in the first few lemmas in §4 and properties of the’s are to be found in Proposition 3.1 with the signs specified by (3.19), due to our present assumption (7.3). We recall the acronym C.F. for “canonical factorization” ([1] , Prop. 2.1).

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”. The nice result stated in the next theorem claims that: if such a function admits of an asymptotic expansion (7.7) then this expansion is automatically differentiable times in the senses of both relations (4.31) and (5.6).

Theorem 8.1 (Complete asymptotic expansions). If satisfies (8.1) then the following are equivalent properties:

1) There exist real numbers such that

2) There exist n real numbers such that

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 of the expansion in (8.3) admits of both representations:

whence it follows that

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

,” implies the existence of a finite “”.

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 the following three integral conditions are equivalent:

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 of course.

Theorem 8.3 (Incomplete asymptotic expansions). Let satisfy (8.1) and let be fixed. Then the following are equivalent properties:

(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 relation (8.14) reads and the first group of expansions in (8.17) reduces to relation in (5.26).

Notice that relation (8.13) and the definition of in (8.11) imply that,

hence (8.18) does not in general imply the convergence, as, of any of the inner integrals appearing in (8.20); it is the stronger condition (8.10) which implies the convergence af all the integrals in (8.20) (remember that, by (2.38) and (2.45), and are not subjected to any integrability constraint).

Moreover each of the “O”-estimates in (8.17) is meaningful whenever the involved integral diverges as i.e. whenever the asymptotic expansion in (8.15) cannot be improved by adding more meaningful terms of the form. As soon as one of these integrals converges to a real number as then we may apply the theorem with a greater value of i. And in the case of divergence, under the present assumption of

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

Theorem 8.4 (Sharper estimates for,). Under the assumptions in Theorem 8.3 let, and suppose that all the integrals appearing in the “O”-estimates in (8.17) diverge as, i.e.

Then the estimates in (8.17) for can be replaced by:

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”, we state the result for convex functions of order according to a standard terminology.

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

Then the following asymptotic relations hold true as:

Here the asymptotic scale is:;; and. The special case asserts that if an th-order convex function f on is bounded at then as:

The case shows that the estimates in Theorem 8.4 are the best possible, generally speaking.

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 [4] and then extended in [5] to asymptotic expansions in real powers. The special case has also been obtained independently by Popoviciu ([6] , p. 28). Specialization of Theorems 8.3-8.4 to the scale, yields analogous estimates at ([2] , Th. 4.2 and Remark 1, p. 181), if use is made of a technical result ([2] , Prop. 5.2, p. 183).

An important remark. In Theorem 8.1 the two types of formal differentibility times are equiva-

lent facts whereas it is not so for a generic f such that changes sign on each deleted left neigh-

borhood of. This has been proved for polynomial expansions [2] and for real-power expansions [5] in an indirect way by expressing the two sets of differentiated expansions as suitable sets of expansions involving the standard operators; the new sets of expansions made evident that what we called “weak formal differentiability”, linked to the C.F. of type (I), is indeed a weaker property than what we called “strong formal differentiability”, linked to a C.F. of type (II). This will be also proved true in Part II-C, §15, for a special class of expansions including the real-power case. The same circumstance occurs for a general two-term expansion ([3] ; Remarks, p. 261) but is not a self-evident fact. In each of these three cases direct proofs could be also provided working on the corresponding integral conditions. Hence in these cases the locutions of “weak or strong formal differentiation” are legitimate. But in the general theory for we face a nontrivial situation and state

Open problem. For consider the two types of formal differentiability characterized in Theorems 4.5 and 5.1. Investigate whether or not the property in Theorem 5.1 always implies the one in Theorem 4.5, the two properties being equivalent in the case of absolute convergence described in Theorem 8.2.

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).

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 ([2] , p. 193) and in ([5] , p. 213). From representation in (4.15) we get (using the simplified notation):

By the assumption (8.2) the left-hand side has a finite limit as, and for the right-hand side we have:

after applying L’Hospital’s rule times (which is legitimate as all the denominators diverge to). By the positivity of the integrand this last limit exists in and coincides with the limit of the left-hand side in (10.1) hence it must be a real number and (4.15) can take the form:

with suitable constants. From this we get:

Here again the left-hand side has a finite limit as whereas the limit of the right-hand side, by (4.1), equals:

after applying L’Hospital’s rule times. Hence this last limit, which exists in, must be a real number and (10.3) can be rewritten as:

with suitable constants. It is now clear how this procedure works and by induction one can prove the validity of representation:

with a suitable constant. As a last step we observe that (8.2) implies:

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. To prove (8.7) we apply the same ideas starting from representation (5.1) and dividing by; recalling that we get

which implies

and (5.1) can be rewritten as

with suitable constants. From this we get

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

and this last limit, which exists in, must be a real number. This means that

and (10.11) can be rewritten as

with suitable constants. For the clarity’s sake we make explicit the steps in this second part of our proof. Assume by induction that the following two conditions hold true:

for some, and suitable constants. Dividing both sides of (10.16) by and taking account of (8.2) we infer that the limit of the quantity

exists in. Applying L’Hospital’s rule i times to evaluate this limit we get the new limit

which, by the positivity of the integrand, exists in hence it must be a real number. We infer that condition (10.15) holds true with i replaced by and this implies representation (10.16) with i replaced by and suitable constants. By this inductive procedure we arrive at representation:

with some constant. Dividing by and using (2.2) we may now conclude that

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

, which exists in and must be a finite number. This is condition (8.7) which al-

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 a.e. on and Theorem 8.1 implies the equivalence between (8.11)

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 of course, and, whenever used, is granted by the hierarchies of the Wronskians in (2.14) and the divergence of the involved integrals. We report a classical differentiation formula used in Proposition 2.4:

valid for any ordered -tuple of functions at any point where the required derivatives exist. Now for we have:

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

Now, for a fixed, we use the procedure in (10.24) to prove by induction on i that

As in (10.24) we can prove that (10.26) is true for; assuming it to be true with i replaced by we have:

that is (10.26). Using this relation for and the expression in (2.43) for we, finally, get:

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 follows. And so on for the other estimates.

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 are nonnegative and nondecreasing and satisfy. For we have the simple estimate

which implies the first relation in (10.33). To estimate we integrate by parts as follows

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 implies

which yields the estimate for. Relations (10.36)-(10.37) give the key to proceed by induction. Putting

we want to prove. For the sake of simplicity, we write out the calculations for using (10.36), which reads, and integrating by parts in the integral appearing in (10.36):

whence

And now the nondecreasingness of implies

Relations (10.39)-(10.40) allow iteration of the procedure to estimate and so on.

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 and. The equivalence between (9.32) and (9.33) is an elementary fact under the tacit assumption for. In part (II) relations (9.34) are those given in Theorems 4.4-4.5 specialized to our case and the only thing to be proved is the equivalence “3) Û 4)” contained in the next lemma where we simplify all formulas by assuming and.

Lemma 10.2. Let f be of class on some deleted neighborhood of and let

For a fixed the following set of asymptotic relations as:

is equivalent to the set of asymptotic relations, as, involving only standard derivatives:

where the two estimates coincide for, namely

For in (10.42) we have for all k,; and in (10.43) we have , for all k,.

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

with suitable constants whose explicit values are not needed in our calculations. Let us now prove “(10.42) Þ (10.43)”. First step. From the first equality in (10.45) we get

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, , and let us prove the corresponding relations for. From (10.45) we get

whence

Now for we use the first estimate in (10.42) for and the first estimates in (10.43) for the derivatives so getting from (10.51)

For the remaining values of i we must use the second estimate in (10.42) for and the suitable estimates in (10.43) for so getting from (10.51)

Now inside the sum each must be replaced by one of the two estimates in (10.43), assumed to be true, and we have two possible contingencies

In the first case the restriction implies, and in the second case the restriction implies as well. In each case the whole sum in (10.53) is as and (10.53) gives

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: