_{1}

^{*}

This part II-C of our work completes the factorizational theory of asymptotic expansions in the real domain. Here we present two algorithms for constructing canonical factorizations of a disconjugate operator starting from a basis of its kernel which forms a Chebyshev asymptotic scale at an endpoint. These algorithms arise quite naturally in our asymptotic context and prove very simple in special cases and/or for scales with a small numbers of terms. All the results in the three Parts of this work are well illustrated by a class of asymptotic scales featuring interesting properties. Examples and counterexamples complete the exposition.

We continue the numbering of sections used in the preceding two parts of this work: Part II-A [

where the asymptotic scale

is subject to certain Wronskian restrictions, let us try to find out expressions for the coefficients

where the existence (as finite numbers) of the involved limits characterize the coefficients

where by (2.35) M_{1}, M_{2} are the operators defined in (3.3), apart from the signs. By iteration we may conjecture that this heuristic procedure leads to the formulas

provided the involved limits exist in

・ In §13 we show that different organizations of the above calculations give rise to two algorithms for constructing canonical factorizations starting from a given asymptotic scale; the procedures seem quite natural in the context of formal differentiation of an asymptotic expansion and shed a light of “easiness”, so to say, on the formulas of asymptotic differentiability obtained so far and seemingly complicated in themselves.

・ In $14 an example illustrating the two algorithms is given.

・ In §15 a useful class of scales is studied highlighting some pecularities concerning various types of formal differentiabilty and here the idea underlying the two algorithms plays a role even if manipulated differently.

・ The last §16 contains additional remarks on the algorithms.

The original procedure used by Trench [

Let us consider a generic element

which we interpret as an asymptotic expansion at

Theorem 13.1 (The algorithm for a special C.F. of type (II)). Let

al C.F. of _{0} in (2.39) [

viding by the first meaningful term on the right, concide with the expansions obtained by applying to (13.1) the operators

(A) Verbal description of the algorithm.

1^{st} step. Divide both sides of (13.1) by the first term on the right, which is the term with the largest growth- order at

Suppressing the derivative the left-hand side in (13.2) is the operator M_{0}u.

2^{nd} step. Divide both sides of (13.2) by the first term on the right and take derivatives so obtaining

Suppressing the outermost derivative the left-hand side in (13.3) is the operator M_{1}u.

3^{rd} step. Repeat the procedure on (13.3) dividing by the first term on the right and then taking derivatives so getting

Iterating the procedure each of the obtained relation is an identity on _{0}, hence at each step we are dividing by the term on the right with the largest growth-order at x_{0}. Notice that at each step the asymptotic expansion loses its first meaningful term and this is the same phenomenon occurring in differentiation of Taylor’s formula. After n steps we arrive at an identity

where the

(B) Schematic description of the algorithm.

Step “1”:

Step “2”:

Step “3”:

Step “4”:

and so on, where the symbol “d & d” stands for the two operations “divide” both sides by the underbraced term on the right and then “differentiate” both sides (the equation in each step being the result of the preceding step).

Theorem 13.2 (The algorithm for the C.F. of type (I)). Let _{0} and

(A) Verbal description of the algorithm.

1^{st} step. Divide both sides of (13.1) by the last term on the right, which is the term with the smallest growth- order at

2^{nd} step. Divide both sides of (13.6) by the last term on the right and then take derivatives so obtaining

3^{rd} step. Repeat the procedure on (13.7) dividing by the last term on the right and then taking derivatives so getting

Iterating the procedure each of the obtained relation is an identity on _{0}, hence at each step we are dividing by the term on the right with the smallest growth-order at x_{0}. Also notice that at each step the asymptotic expansion loses its last meaningful term and this is a phenomenon different from that occurring in differentiation of Taylor’s formula (see the foregoing proposition). After n steps we arrive at an identity

where the

(B) Schematic description of the algorithm.

Step “1”:

Step “2”:

Step “3”:

Step “4”:

and so on with the symbol “d & d” reminding of the two operations “divide” both sides by the underbraced term on the right and then “differentiate” both sides (the equation in each step being the result of the preceding step).

Remarks. 1) In order to obtain any C.F. by the above procedures one may simply choose

2) If some operator

then the algorithm in Theorem 13.2 yields the C.F. of type (I) at x_{0}, valid on the whole given interval _{0} valid on some left neighborhood of x_{0}, the largest of them being characterized by the nonvanishingness of all the Wronskians

3) In practical applications of the algorithms there is a fatal pitfall to avoid, namely the temptation at each step of suppressing brackets, cancelling possible opposite terms and rearranging in an aestetically-nicer asymptotic scale. This in general gives rise to a factorization of an operator quite different from

4) The algorithms are of course applicable to obtain C.F.’s at a left endpoint valid on each neighborhood whereon the Wronskians never vanish.

Proof of Theorem 13.1, that of Theorem 13.2 being exactly the same after replacing _{0}, q_{1}, q_{2} coincide with Pólya’s coefficients in (2.35) [

hence it is enough to show that Pólya’s expression for

It is also clear that the various identities obtained are nothing but those obtained by applying to (13.1) the operators

Comparing the expressions in (2.37) [

We shall show the validity of this identity even if the outermost derivatives are suppressed. Using the elementary formula

by (13.13) with

We have also proved that at the k-th step the linear combination on the right, before dividing for the next step, coincides with the expression

which, by (3.8) in Part II-A, is an asymptotic expansion at

Consider the fourth-order operator L of type (2.1)_{1,2}, [

acting on

the algorithm in Theorem 13.2 yields in sequence:

Hence

and this is “the” global C.F. of L of type (I) at

(The underbraced terms on the right are those by which one must divide and then differentiate.) Hence:

where

Hence (14.4) is a C.F. of L of type (II) at

which is easily seen to be the interval

In the various steps of the above procedures one must carefully avoid the temptation of rearranging the terms in the right-hand side in (supposedly) nicer asymptotic scales. For instance the first procedure involves quite simple terms and only at the last-but-one step we may split the remaining term on the right by writing

and taking

This gives a fifth-order operator

distinct from the given fourth-order operator. On the contrary the second procedure offers a great number of temptations! For instance if one rewrites the result of the first step as

and then goes on applying the second algorithm to (14.8) as if the right-hand side would be an asymptotic expansion with three meaningful terms, one gets:

the only difference between the two expressions on the right being the term-grouping. From the upper relation in (14.9), considered as an asymptotic expansion at

and then

whose left-hand side is a fourth-order operator distinct from our operator. If, instead, one starts from the lower relation in (14.9), considered as an expansion at

and so forth in an endless process leading nowhere!!

We specialize the results of our theory for the special class of Chebyshev asymptotic scales

where

Our assumptions will be:

This class has been cursorily presented in ([

To apply our theory we observe that by a proper device it can be given an elementary proof of the formula (which is anyway a classical result):

where

of the differential operator

various constant factors appearing in the expressions of the Wronskians. Hence, apart from immaterial constant factors, the functions

Proposition 15.1 (C.F.’s for the asymptotic scale (15.1)). Let

(I) The “unique” C.F. of type (I) at

and we denote by

(II) A special C.F. of type (II) at

and we denote by

The above factorizations may quite simply be obtained via our algorithms as well; for instance starting from

the procedure of the second algorithm yields in sequence:

and so on (15.7) follows. If, as a first step, both sides of (15.9) are divided by

and no matter how many times we iterate the procedure the right sides is a non-identically zero expression except for special sequences of the exponents such as

seem natural contingencies quite likely to be encountered. We shall show that the two main sets af expansions characterized in our theory actually are equivalent to expansions involving iterates of the simple operator

noticing the identities

The second identity in (15.13) is the main operational property of

We report only on “complete” asymptotic expansions; for “incomplete” expansions it might be complicated to list all the circumstances concerning estimates of the remainders: see Theorem 9.2-(II) in Part II-B.

Proposition 15.2. (Characterizations of “weak” formal differentiability for the scale (15.1)).

(I) Referring to Theorem 4.5 in [

1) The set of asymptotic expansions as

where

2) The set of asymptotic expansions as

3) The improper integral

4) For suitable constants

(II) The linear combinations in the two types of differentiated expansions are explicitly given by

wherein the second sum actually denotes the expression of

Proposition 15.3. (Characterizations of “strong” formal differentiability for the scale (15.1)).

(I) Referring to Theorem 5.1 in [

1) The set of asymptotic expansions as

2) The set of asymptotic expansions as

3) The improper integral

4) For suitable constants

(II) The linear combinations in the two types of differentiated expansions are explicitly given by

wherein the second sum, generally speaking, contains all the terms in the expression of

Comparing (15.16) and (15.21) we see that each remainder in (15.16) has a growth-order greater than the corresponding one in (15.21) hence we may say that (15.16) and (15.21) are obtained from the first expansion in (15.15) by formal differentiation respectively in a “weak” and in a “strong” sense: see Remark and Open Problem at the end of §8 in [

Propositions 15.2 and 15.3 generalize the technical lemmas used in ([

Proposition 15.4. (A case of equidistant exponents). (I) If

be

we have the expansion for f together with the equivalent sets of differentiated expansions

with the coefficients

we have the expansion for f together with the equivalent sets of differentiated expansions

with the coefficients

(II) If

quence of exponents as

we have the expansion for f together with the equivalent sets of differentiated expansions

with the coefficients

we have the expansion for f together with the equivalent sets of differentiated expansions

with the coefficients

Writing down the first two differentiated expansions for each group may help the reader grasp the different circumstances. In the situation of Proposition 15.4-(I) and in a concise notation let us start from the expansion

Then the three expansions in (15.27) and (15.29) respectively run as follows:

For a quick check of (15.37) the reader is suggested to use the algorithm in Proposition 13.1.

Remark. The above results give a glimpse of the great variety of differentiated expansions that may be encounterd in applications. In the general case of equidistant exponents

The proofs of Propositions 15.2, 15.3 rely on formulas linking the various involved operators.

Lemma 15.5. (I) For weak differentiability the formula linking

with suitable coefficients

(II) For strong differentiability the formula linking

with suitable coefficients

Proof. Let us prove (15.38). For

and suppose (15.38) hold true for a certain k; then for

which is (15.38) with k replaced by

Due to the linearity of our operators it is enough to prove our claims for the case

Lemma 15.6. With shortened notations we have:

Proof. The equivalence in (15.41) is easily checked for

We assume this equivalence to hold true for a certain range

Viceversa suppose that

whence

which is the sought-for estimate. The equivalence in (15.42) is similarly proved using (15.39).

We specialize the foregoing results for particular choices of

where

Examples for weak differentiability. (I)

And under the restrictions

they are equivalent to:

But it must be observed that, though the comparison functions appearing in each summation in (15.47) form an asymptotic scale due to the easily-checked relation

the asymptotic expansions in (15.47) are, so to say, impure in so far that neither all the terms in the explicit expressions of

Under the restrictions _{2} implies the following weaker set of expansions than in (15.47)

but not viceversa. The claims about (15.47) follow from the formula

which implies

under conditions (15.46). For generic values of the

(II)

(III)

(IV)

The equivalence between (15.54)_{2} and (15.54)_{3} follows from the formula

which implies

(V)

Examples for strong differentiability. (I)

Quite surprisingly we don’t have any characterization in terms of _{2} implies (but is not implied by) the expansions

wherein all the terms into the summation symbol must be taken into consideration but not all the terms in the explicit expressions of

(II)

(III)

(IV)

The equivalence between (15.62)_{2} and (15.63)_{3} follows easily from (15.55).

(V)

To illustrate Theorems 8.3-8.4 in Part II-B we exhibit the estimates of the remainders in a case of incomplete expansions for a generalized convex function with respect to the scale (15.1)-(15.3).

Proposition 15.7. Let

then the following asymptotic relations hold true as

consistently with Corollary 8.5 in Part II-B.

Proof. Using the expressions of the

with suitable non-zero constants

and by an easy induction the remaining estimates are proved.

What about applying the above algorithms to (13.1) with a random choice of the term to be factored out at each step? If one carefully checks that at each step one is dividing by a nowhere-vanishing function one may well obtain, after n steps, a factorization valid on a certain subinterval of the given interval but, in general for

and is such that all the possible Wronskians constructed with these three functions do not vanish on

And here are the 6 so-obtained factorizations arranged in the same order:

all valid on

A few remarks about the factorizations of

Let

Then by Proposition 2.2 in Part II-A all the possible Wronskians

coinciding

where the functions on the right can be arranged in an asymptotic scale at

Examples of formal differentiability according to canonical or non-canonical factorizations. The function

admits of the expansion

with the remainder explicitly given by

“inferred from the left side of

and it has property I-B if

“inferred from the right side of

Properties II-A, II-B, III-A, III-B are similarly defined looking at

Separating various cases for

For

which, in our case, happens iff