Motivated by a general theory of finite asymptotic expansions in the real domain for functions f of one real variable, a theory developed in a previous series of papers, we present a detailed survey on the classes of higher-order asymptotically-varying functions where “asymptotically” stands for one of the adverbs “regularly, smoothly, rapidly, exponentially”. For order 1 the theory of regularly-varying functions (with a minimum of regularity such as measurability) is well established and well developed whereas for higher orders involving differentiable functions we encounter different approaches in the literature not linked together, and the cases of rapid or exponential variation, even of order 1, are not systrematically treated. In this semi-expository paper we systematize much scattered matter concerning the pertinent theory of such classes of functions hopefully being of help to those who need these results for various applications. The present Part I contains the higher-order theory for regular, smooth and rapid variation.
In a previously-published series of papers ( [
by two special procedures of formal differentiation are expressed via improper integrals involving both the quantity
and for practical applications it is quite useful to have some information on the asymptotic behavior of the ratio of Wronskians. For
and then recalling that in various but related contexts the following remarkable asymptotic relations linking the ratios
or
for different classes of functions. Bourbaki ( [
Balkema, Geluk and de Haan, ( [
with the property that the function
such a
All these approaches for infinitely-differentiable functions have in common the existence of the following limit
wherein the three contingencies are special cases of the more general classes of functions traditionally labelled as “slowly, regularly or rapidly varying at
- §2 contains a detailed and integrated exposition of basic properties (algebraic, differential and asymptotic) concerning regular and rapid variation in the strong sense. Much, but not all, the material concerning regular variation is standard and the most elementary proofs have been reported. Some facts concerning the index of variation of the first derivative in §2.3 are essential both to give a correct definition of higher-order regular variation and to understand possible restrictions on the indexes.
- In §3 we give an integrated exposition of higher-order regular variation (a concept indirectly encountered in the context, e.g., of Hardy fields) and smooth variation (a concept explicitly present in the literature concerning some applications of regular variation), both traditionally (but not in our approach) referred to
- In §4 an analogous exposition for higher-order rapid variation is given with several characterizations. To be useful for applications a restriction must be added to the “spontaneous” concept of higher order for this class of functions.
- In §5 there is a discussion about various useful asymptotic functional equations satisfied by the functions in the previously-studied classes.
In part II we exhaustively describe results about algebraic operations on higher-order asymptotically-varying functions and treat concepts related to exponential variation and some of their basic applications.
General notations
-
-
-
-
- For
- If
- The logarithmic derivative
- Hardy’s notations:
“
“
- The relation “
- The relation of asymptotic equivalence:
- When describing properties related to exponential variation it is convenient to use the following nonstandard notation:
and a similar definition for notation
We shall formally use these notations like the familiar “
- Factorial powers:
where
Propositions are numbered consecutively in each section irrespective of their labelling as lemma, theorem and so on.
Notations for iterated natural logarithms and exponentials
The special definitions for
The general theory of finite asymptotic expansions we constructed in the cited papers essentially deals with functions of the regularity class
Unlike the traditional concept of “order of growth” which involves one specified comparison function we use the generic locution of “type of growth”, or better “type of asymptotic variation”, to denote one of the classes of functions which are either regularly or smoothly or rapidly or exponentially varying; and these are classes which in our exposition are defined via “asymptotic differential equations” whereas for order 1 they may be included in larger classes defined through “asymptotic functional equations”.
Definition 2.1. Let
(I)
for some constant
(II)
Accordingly, the index of rapid variation at
(III) f is said to have an “index of variation at
with the tacit agreement that the limit is taken for x such that
We sometimes omit the specification “in strong sense” as this is the only meaning we are using for this concept.
Remarks. 1. Condition “f ultimately of one strict sign” is essential both in the general and in our restricted definition. The choice
2. The locution “in strong sense” is a reminder of the fact that our class of functions is a proper subset of the class of regularly- or rapidly-varying functions in more general senses. The first larger class is that of those real-valued functions f defined on a neighborhood of
according to Definition 5 in Bourbaki ( [
In the monograph [
3. Typical (indeed the most usual and useful) functions in
- Typical functions in
whose index of variation is: “
- Typical monotonic functions in
and their products
Separating the cases
each of these may be viewed as an “asymptotic (ordinary) differential equation of first order” and it is easily shown (Proposition 2.1 below) that the solutions of the first one of them share the asymptotic properties of the solutions of the ordinary differential equation
we get the characterization: An absolutely continuous function f belongs to the class
And an analogous statement holds true for a rapidly-varying function with
As a first rough asymptotic information:
Notice that either representation “
which is in the class
The general classes of regularly- or rapidly-varying functions enjoy many useful algebraic and analytic properties but it is not self-evident that the same is true for our restricted classes, in particular that they are closed with respect to various operations. In the next subsection we give a list of the main properties omitting those proofs which are quite elementary based on the property of the logarithmic derivative:
Proposition 2.1. (Algebraic and asymptotic properties of regularly-varying functions). The following properties hold true:
(i) Factorization:
(ii) Growth-order estimates:
But for
The third and the fourth of these functions are not ultimately monotonic: The third with bounded oscillations and the fourth, call it f, with unbounded oscillations:
(iii) Algebraic operations. If
For
(iv) Composition. If
In particular, if
(v) The particular case
The examples in (2.22) show that a pair
(vi) Inversion. If
(vii) Asymptotic comparison. If
whereas no inference can be drawn if
Proof. (i) is trivial and (ii) follows from (2.13) as
(iii) For the linear combination in (2.27) in the case
In the case
To prove (iv) write
and notice that
If
To prove (vi) evaluate the following limit by the change of variable
Last: (2.32) follows from (2.25) and (2.21); relations in (2.31) follow from (2.32).,
Remarks. 1. The properties in (iii), (iv) and (vi) are the same as those valid for the standard powers. The first inference in (2.31) can be interpreted by saying that the class
2. In (2.27) it is essential that all the involved quantities (functions and constants) have one and the same sign for
3. The “Zygmund property” cited after (2.6) and concerning the ultimate strict monotonicity of
4. A less direct proof of (2.27) uses the decomposition:
similar to a device which reveals efficient in the case of slow variation in the general weaker sense: see Seneta ( [
Examples. 1. Referring to the third function in (2.22) we mention that it can be proved that the function “
2. If
3. The function
Proposition 2.2. (Algebraic and asymptotic properties of rapidly-varying functions). The following properties hold true:
(i) Growth-order estimates:
(ii) Algebraic operations:
with no inference about the quotient
(iii) Compositions:
with no inference about
In particular
(iv) Inversions. Roughly speaking the inverse of a rapidly-varying function is slowly varying and viceversa. To be precise, if
then
Proof. For the first three groups of relations we write down the proof only for
Both claims in (iv) are proved as in (2.38).,
Examples about rapid variation in weak or strong sense. Comparing with the examples preceding Proposition 2.2 the function “
Quite differently, if “
also shows that “
In the next proposition we collect various results about linear combinations, results particularly useful in asymptotic contexts.
Proposition 2.3. (I) (Positive linear combinations of different types of asymptotic variations).
If
without any further restriction on
provided that:
These inferences, together with (2.27), are summarized in
whatever the positive constants
(II) (Arbitrary linear combinations of asymptotic scales). Let the functions
and let one of the following conditions be satisfied, either
or
with
so that: if
(III) If
then
(IV) Besides (2.56) let
and put
Then
with
Proof. We may include the constants
respectively for the first and the second inference and the claims follow. For the third inference in (2.52) we have “
A different elementary proof is achieved writing:
hence
because we shall prove in a moment that “
In part (II) the result involving (2.57) trivially follows from factoring out
Claims in parts (III), (IV) are corollaries of the result in part (II) involving (2.58).,
Remarks. 1. Using the decomposition in (2.39) the first two inferences in (2.52) may be proved with the restriction “
2. Conditions in (2.57) and (2.58) are independent. Any pair f, g where f is any function of type in (2.7) and g is any function of type in (2.8) with
Counterexamples concerning suppression of conditions (2.57)-(2.58). In the following we use three pairs of functions in ( [
In the last example the function
3. As concerns an additional restriction in the two inferences in (2.53) we have already remarked that the assumptions imply “
But for the time being we do not know any such function: see Proposition 2.5-(III).
For integrals of functions in our classes we report the classical results (with elementary proofs) to highlight a difference between the two cases.
Proposition 2.4. (I) (Integrals of regularly-varying functions). Let
The inferences in (2.74)-(2.75) are respectively equivalent to the following asymptotic relations expressing the behavior of the integral
In the two cases (2.76)-(2.77) we may only assert, generally speaking, that
(II) (Integrals of rapidly-varying functions). We have the rough estimates:
But under the stronger assumption
we have the exact principal parts:
Remarks. Notice that the formal rule in (2.84)-(2.85) does not coincide with that in (2.78)-(2.79) in accordance with relations in (2.104) below. If
Proof. The convergence or divergence of the integral in case
and this, in turn, together with condition
see Pólya-Szegö ( [
and analogously for (2.82). To prove (2.84) just write
with a suitable constant c. Using the third condition in (2.83) we get
The divergence of
Remarks. 1. There is a difference between the two cases: though the character of regular or rapid variation of an antiderivative is elementarily checked, a useful result about the asymptotic behaviors in the rapid-variation case requires a restrictive assumption. Condition
Here:
2. The proof based on the device in (2.87) could be adapted to the case of a regularly-varying function observing that, for
and imposing the extra-assumption of formal differentiation of this last relation, i.e.
One would re obtain the inferences in (2.74)-(2.75) but under the unnecessary restrictions: (2.91) and
The following properties of the first derivative are essential to develop the theory of higher-order variation.
Proposition 2.5. (Elementary asymptotic properties of the first derivative). The following hold true with all asymptotic properties referring to
(I) (Regular variation). The estimates in (2.19) imply that:
For
Moreover, for each
(II) (Slow variation).
and
(III) (Rapid variation).
For
We do not know if relations in (2.102), or even the simple relation
Proof of part (III). The estimates for
where both factors tend to
Notice that it is easy to give an example of a function
but in this case the limit “
As far as the possible index of variation of the first derivative is concerned notice that if
But if
Proposition 2.6. (Index of variation of the first derivative). (I) If
In the case
but it cannot be
(II) If
If
for some
(III) If either “
and we do not know whether the partial converse holds true i.e. if both conditions “
Proof. Proof of part (I) is taken from ( [
We now evaluate
The same argument is valid for
(i)
(ii)
which is a positive real number; hence “
(iii)
(iv) The case
and there are two a-priori contingencies concerning the integral
contradicting the first relation in (2.114). Notice that the procedure used to prove this last case works for any
The last assertion in the statement, namely “it cannot be
For part (II) the assumptions for (2.108) are:
whence (2.108) follows. Viceversa assume
which means that
Last, relation (2.110) follows from the decomposition
as the factors on the right diverge either both to
By the foregoing proposition we can define unambiguously some concepts of “higher-order asymptotic variation” separating the cases of regular variation (in this section) and rapid variation (in the next section).
Definition 3.1. (Regular variation of higher order). A function
Whenever needed we denote the indexes of the derivatives as follows:
Remarks. 1. It is essential to consider the absolute values in order to not impose a-priori restrictions on the signs of the derivatives. Saying that “
2. A nonzero constant belongs to the class
3. If (3.1) holds true then, by Proposition 2.6-(I):
By (2.106) the inference in (3.3) may well hold true without the stated restriction whenever “
Proposition 3.1. (Principal parts of higher derivatives in case of regular variation).
(I) If
hold true whichever
(II) (Partial converse). If
with suitable constants
If this is the case then:
Proof. (I) Both claims for
Replacing the relation for
and iterating the procedure yields
which by (2.106) coincides with (3.5) under the assumptions in (3.3). Under the assumptions in (3.4) we get relations in (3.5) for
In any case (3.5) hold true for
For part (II) we must prove that relations (3.7)-(3.8) imply
The claim for
the inductive hypothesis implies
and we must prove the relation in (3.7) with k replaced by
and replace this expression into the relation in (3.14) for
For
and the proof is over.,
As noticed in the proof, (3.5) holds true for any
wherein
Second counterexample:
though the relations in (3.7) hold true for each
Third example:
and relations in (3.7) hold true with
The second counterexample above shows that the set of relations in (3.5) in themselves do not grant that all the involved derivatives be regularly varying: it may well occur an abrupt transition from regular variation to rapid variation at a certain order of derivation. This is the main motivation for our Definition 3.1. But the asymptotic relations for
Proposition 3.2. (Several characterizations). For an
The reader will notice in the proof that the differential expressions “
Proof. We use notation
from which, putting
which implies the relation in (3.23) for
the sum in square brackets being
The equivalence (3.22) Û (3.24) is contained in the following:
First, it is elementary to prove by induction the formula
wherein
having used the relation in (3.29) for
wherein we have used the expression of
and we shall prove the following representation:
where
If now (3.34) is assumed true for a certain k then, differentiating both sides and using (3.35) in the left-hand side, we get
whence
where we have put
the right-hand side being a polynomial in
Balkema, Geluk and de Haan ( [
Definition 3.2. (Smooth variation of higher order). A function
Notice that in our definition
the reason of the strict inclusion being that some derivatives of a smoothly-varying function may vanish or change sign infinitely often. Examples:
In the third example all derivatives
If as in (3.24) we associate to
In fact we have
implying
The anomalies in these examples make the definition of smooth variation a bit unsatisfying from a theoretical viewpoint unlike the definition of higher-order regular variation; they also show that the possible more complete locution “smooth regular variation” would not be appropriate; however it turns out that relations in (3.21)-(3.24), regardless of
Proposition 3.3. (Derivatives and integrals of smoothly varying functions). (I)
(II) If
The very same inferences in the case of regular variation, i.e. with
Proof. We report the more elementary arguments used in ( [
To prove the first inference in (3.45) put
A similar proof in case of divergence.,
Notice that, with
using “
To end this section let us ask ourselves what can be said about relations in (3.21) holding true with some unknown coefficients
Proposition 3.4. Let a function
for some unspecified constants
Proof. The claim amounts to state that the
using (3.51) and then the secod relation in (3.53)
hence
,
Before giving the proper definition of higher-order rapid variation it is good to add some remarks about the additional condition
appearing in Proposition 2.4-(II). The counterexample in (2.89) shows that this supplementary condition is almost necessary to obtain a meaningful general result about the asymptotic behavior of the antiderivatives of a rapidly-varying function. Now if in (2.84)-(2.85) we put
and, changing again notation, we have one of the following two equivalent relations:
For some applications conditions like those in (4.3) are necessary for meaningful general results as, e.g., in determining asymptotic expansions of antiderivatives and in another class of expansions studied in Part II, §11, of this work: this justifies the following restricted concept of rapid variation.
Definition 4.1. (Rapid variation of higher order).
(I) (First order). A function
(II) (Higher order). A function
If
Remarks. 1. According to our definitions when we speak of a function
2. Conditions in (4.7) obviously imply those in (4.6) whereas, viceversa, complicated calculations in the attempt of proving (4.7) in addition to (4.6) may be usually saved using the classical result (already mentioned in the proof of Proposition 2.5) that: “
As concerns the analogue of Proposition 3.1 it happens that relations in (3.5) have no analogues for rapidly-varying functions of higher order whereas those in (3.6) have so yielding a useful characterization of this class of functions.
Proposition 4.1. (Principal parts of higher derivatives in case of rapid variation). Let
It follows that even
whereas a different way of writing relations in (4.6) would give the weaker scale:
Proof. Relations in (4.9) and in (4.11) simply are different ways of rewriting relations respectively in (4.8) and in (4.10). Now inspecting (4.7) we have for
which is (4.9) and (4.10) for
hence (4.7) are equivalent to (4.9). It remains to prove the equivalence between (4.9) and (4.10) for
If
If
Using (4.18) and the relation in (4.8) involving the ratio
which is the relation in (4.9) for
An instructive counterexample concerning Definition 4.1 and the associated function. Let us consider the following function
wherein ultimately “
whence we infer that
and condition (4.1) is not satisfied as
and
though for
For
As concerns the associated function in (3.24),
Hence for any
A remark about the ratios
A remark about
an integration yields
Moreover the identity in (4.14) gives
i.e.
then
and
i.e.
Corollary 4.2. (Summing up the behaviors of the higher derivatives). Let
Examples.
Using representations in (2.12) it is easy to prove certain useful asymptotic relations satisfied by regularly-varying functions and in particular (2.6) which has been assumed by Karamata as the definition of a general concept of regular variation. The standpoint in this section is that of highlighting how a given function acts upon various asymptotic relations and we give these properties the collective name of “asymptotic functional equations”.
Proposition 5.1. (Slow and regular variation). (I) A function
which states that a slowly-varying function transforms the relation of “asymptotic similarity between functions diverging to
which, by the presence of the parameter
(II) A function
They mean that a regularly-varying function preserves the relations of “asymptotic similarity” and “asymptotic equivalence” between functions diverging to
in particular:
The asymptotic functional equation (2.6) is a special case of (5.6) and it expresses the “power-like” type of growth of
referring to §11 in Part II for expansions with more terms. Two special cases of (5.7) are:
Another useful consequence of (5.6) is:
and in particular:
Proof. Rewrite (2.12) as
To prove (5.1) and (5.3) let
then
By (5.13):
and
so that
For
which are the precise meaning of the thesis in (5.3). And for
which is (5.4), equivalent to (5.5). The asymptotic expansion in (5.7) is similarly proved:
Last, (5.10) follows from the mean-value theorem of integral calculus:
as “
and then using (2.78) or (2.79).,
A comment on uniform convergence. Refining the calculations in (5.19) the last expression in (5.19) may be replaced by
where the symbols “
and it is easily seen that the quantity on the right tends to zero, as
In the next proposition special cases of the asymptotic relation in (5.7) are commented upon.
Proposition 5.2. (I) (Asymptotic sublinearity). If
a property enjoyed by all functions
Such a property may be interpreted as a kind of “asymptotic sublinearity”.
(II) (Asymptotic linearity). For a function
a property enjoyed by all functions
The property in (5.27) may be interpreted as a kind of “asymptotic linearity”. For
an equation satisfied by all functions
“A function
(III) (A subclass of slowly-varying functions). The strong asymptotic functional equation
states that
but neither by
For rapid variation, which formally refers to the limit cases
Proposition 5.3. (Rapid variation). (I) If
In particular
(II) If
In particular
The asymptotic functional relations (5.32) and (5.34) where assumed by de Haan as definitions of the general classes of (measurable) rapidly-varying functions of index
Proof. (I) From relation (2.2) we get that
whence
The limits in (5.31) follow by applying the exponential as in (5.19) and those in (5.33) follow by applying the just-proved result to the function
Granata, A. (2016) The Theory of Higher-Order Types of Asym- ptotic Variation for Differentiable Functions. Part I: Higher-Order Regular, Smooth and Rapid Variation. Advances in Pure Ma- thematics, 6, 776-816. http://dx.doi.org/10.4236/apm.2016.612063