 Research Article
 Open Access
 Published:
New Limit Formulas for the Convolution of a Function with a Measure and Their Applications
Journal of Inequalities and Applications volume 2008, Article number: 748929 (2008)
Abstract
Asymptotic behavior of a convolution of a function with a measure is investigated. Our results give conditions which ensure that the exact rate of the convolution function can be determined using a positive weight function related to the given function and measure. Many earlier related results are included and generalized. Our new limit formulas are applicable to subexponential functions, to tail equivalent distributions, and to polynomialtype convolutions, among others.
1. Introduction
This paper investigates the existence of the limit of the ratio of a convolution and a positive valued weight function. The limit is given by an explicit formula in terms of the elements in the convolution and of the weight function. Our results are formulated for the convolution of a function with a measure and also for the convolution of two functions.
Our work was inspired by two different applications. One of them is the asymptotic stability theory of differential and integral equations, where an important question is to determine the exact convergence rate to the steady state. The second one is related to the asymptotic representation of the distribution of the sum of independent random variables. In the above and several similar problems, the weighted limits of convolutions play important role with different types of weights.
Let be a given measure on the Borel sets of and let be a measurable function. The convolution is defined by
for all for which the integral exists.
The convolution of two locally Lebesgue integrable functions is defined by
for for which the integral exists.
The motivation of our work came from the following three known results.
The first wellknown result has been used frequently in the asymptotic theory of the solutions of differential and integral equations (see, e.g., [1]).
Theorem 1.1.
Let be locally integrable and assume that
Then
The next wellknown simple result plays a central role, for instance, in the asymptotic theory of fractional differential and integral equations (see, e.g., [2–5])
Theorem 1.2.
Let be given. Then
where is the wellknown Beta function.
The third known result is formulated for continuous subexponential weight functions. A continuous function is subexponential if
The terminology is suggested by the fact that (1.7) implies that for every .
The next result has been proved in [6] and it plays a central role to get exact rates of subexponential decay of solutions of Volterra integral and integrodifferential equations (see, e.g., [6–8]).
Theorem 1.3.
Let the weight function be continuous and subexponential. If are continuous functions such that
are finite, then
Based on the above three known results, we conclude the next observations.
(i) All of the above theorems give different limit formulas for the ratio at . In fact in Theorem 1.1 and , in Theorem 1.2.
(ii) The weight functions in Theorems 1.1 and 1.2 satisfy condition (1.7), but they do not satisfy condition (1.6).
(iii)The condition for in (1.8) is not necessarily true in Theorem 1.1. Instead of that holds, where .
(iv) in Theorem 1.2 and at the same time is not zero.
Our first goal is to prove results which unify the abovementioned theorems. Second, we want to extend the limit formulas for the convolution of a function with a measure. This makes possible the applications of our theorems to not only density but also distribution functions.
In fact we prove limit formulas which contain three terms, and the weight function does not satisfy condition (1.6). The major idea in the proofs of the main results is borrowed from the theory of subexponential functions. Namely, for large enough , in fact the convolution can be split into three terms:
Under suitable assumptions and some timetricky and technical treatments of the above three terms, we get the limit formula
where the following limits are finite:
In the limit formula (1.11), the terms and are interpreted as zero whenever and , respectively. So the values of and need not be finite in the applications.
The limit formula (1.11) can be reformulated for the convolution of two functions and . Formally, it can be done if the measure is such that for every Borel set .
In that case and , where
These indicate that our remarks (i)–(iv) are taking into account and the known Theorems 1.1, 1.2, and 1.3 are unified in our results.
The organization of the paper is as follows. Section 2 contains notations and definitions. Section 3 lists and discusses the main results both for the convolution of a function with a measure and for the convolution of two functions. In Section 4 we present the corollaries of our main results for subexponential and longtailed distributions. In Section 5 we show that our results can be easily reformulated to an extended set of weight functions. Section 6 gives some corollaries of our main results for the case when the weight function is of polynomial type. These results have possible applications in the asymptotic theory of fractional differential and integral equations. The proofs of the main results are given in Section 8 based on some preliminary statements stated and proved in Section 7.
2. The Basic Notations and Definitions
First we introduce some notations. The set of real numbers is denoted by , and denotes the set of nonnegative numbers.
In our investigations we will make use of different sets of measures and functions given in the next definitions.
Definition 2.1.
Let be the algebra of the Borel sets of . denotes the set of measures defined on such that the measure of any compact subset of is a nonnegative number.
Note that the classical Lebesgue measure defined on , denoted by , is an element of .
Let . In this paper we will write for the integral of on the closed interval . The integral of on the interval is written as When , instead of we also write
Definition 2.2.
denotes the set of functions which are Lebesgue integrable on any compact subset of . As usual,
Definition 2.3.
is the set of the Borel measurable functions which are bounded on any compact subset of
Definition 2.4.
A measure from belongs to the set if it is absolutely continuous with respect to . In this case means that is a nonnegative function from such that for every ( is the RadonNikodym derivative of with respect to ).
It is not difficult to show that for any and the convolution
of and is well defined on . It is known (see, e.g., [9]) that for any the convolution
of and is well defined for almost every (shortly a.e.) . It follows that for any and the convolution is well defined for a.e. and
where .
In this paper our major goal is to give conditions—possibly sharp—which guarantee the existence of the finite limit of the ratio
as . The weight function will belong to some special classes of the functions given in the following definitions.
Definition 2.5.
Let be the set of the functions such that
The set of the functions for which the above convergence is uniform on any compact interval is denoted by .
It is clear that if , then
holds for and hence it holds for any . Therefore for all we have
that is the function is so called regularly varying at infinity. Thus applying the Karamata uniform convergence theorem (see, e.g., [10]) it follows that the convergence in (2.8) is uniform in on any compact set of assuming that is Lebesgue measurable. From this we get that the convergence in (2.6) is uniform on any compact set of assuming that is Lebesgue measurable. Thus contains the Lebesgue measurable members of On the other hand from [10] we know that there exists a nonmeasurable function such that and hence is a proper subset of .
To give an explicit formula for the weighted limit of the convolution at we should assume some limit relations between and and between and .
Definition 2.6.
Let .
(a) denotes the set of functions such that the limit
is finite.
(b) denotes the set of measures such that for any fixed the limit
is finite.

(c)
Let
(2.11)
Definition 2.7.
Let and . denotes the set of functions for which
holds for any fixed
Remark 2.8.
A measure belongs to if and only if for any fixed the limit
is finite, where . Moreover for any It can be shown (see Proposition 7.3) that if and then
We close this section with the following definition.
Definition 2.9.
A function is said to be oscillatory on if there exist two sequences , such that and as moreover .
3. Main Results
In this section we state our main results. Their proofs are relegated to Section 8.
We use the following hypothesis.
(H), and the improper integral
is finite whenever is oscillatory.
Note that if then (H) is satisfied for any
In the next result we give an explicit limit formula for the weighted limit of the convolution of and at +
Theorem 3.1.
Assume (H). Then the following results hold.

(a)
The following three statements are equivalent.
(a_{1}) The limit
is finite.
(a_{2}) For some the limit
is finite.
(a_{3}) The values
are finite, moreover

(b)
Assume that one of the statements is true. Then the limit (3.3) is finite for any and
(3.6)
where
are finite.
Remark 3.2.
Our theorem is applicable for the case when and also when Namely, if , then (3.7) yields that is zero independently on the value of Similarly if , then from (3.9) it follows that is independently zero on . We will see that this character of our theorem is important for getting limit formulas for polynomialtype convolutions (see Corollary 6.2 in Section 6).
Now consider the case that is, . In this case we can apply Theorem 3.1 by using the hypothesis.
, the function is nonnegative such that
is finite for every , and the improper integral is finite whenever is oscillatory
Theorem 3.3.
Assume . Then the following results hold.

(a)
The following three statements are equivalent.
(a_{1}) The limit
is finite.
(a_{2}) For some the limit
is finite.
(a_{3}) The values
are finite, moreover

(b)
Assume that one of the statements is true. Then the limit (3.12) is finite for any , and
(3.15)
where
are finite.
When , then (see Proposition 7.3), and we get the following.
Theorem 3.4.
Let , and assume that
(i) the improper integral
is finite, whenever is oscillatory and is not oscillatory;
(ii)the improper integral
is finite, whenever is not oscillatory and is oscillatory.
Then the statements of Theorem 3.3 are valid and (3.15) can be written in the form
Remark 3.5.

(a)
If , and is nonnegative such that defined in (3.10) is finite for every , then the conditions of Theorem 3.3 hold.

(b)
If , and , then Theorem 3.4 is applicable.
Remark 3.6.
The wellknown result Theorem 1.1 (see, e.g., [1]) is a straightforward consequence of Theorem 3.3.
4. Applications of The Main Results to Subexponential Functions
In this section we concentrate on the socalled subexponential functions which are strongly related to the subexponential distributions. Such distributions play an important role, for instance, in modeling heavytailed data. Such appears in the situations where some extremely large values occur in a sample compared to the mean size of data (see, e.g., [11] and the references therein).
First we consider the "densitytype" subexponential functions.
Definition 4.1.
Assume that a function is called subexponential if and
Remark 4.2.
Let such that is finite. Then is measurable and hence . Thus
which shows that
Therefore and the normalized function is a subexponential density function. This gives the meaning of the "densitytype" subexponentiality.
From Theorems 3.4 and 6.1, we get the following.
Theorem 4.3.
If is a subexponential function and , then
It is worth to note that formula (4.4) has been obtained by Appleby et al. [7] in the case when the functions and are continuous on . These types of limit formulas were used effectively for studying the subexponential rate of decay of solutions of integral and integrodifferential equations (see, e.g., [6, 12]).
Now we apply our main results to subexponential and longtailedtype distribution functions.
Definition 4.4.
Let be a distribution function on such that and for all . Then
(a) is called subexponential if
or equivalently
where denotes the tail of , that is,
(b) is called longtailed if
The definition of the subexponential distribution was introduced by Chistyakov [13] in 1964 and there are a large number of papers in the literature dealing with them. For the major properties and also for applications, we refer to the nice introduction and review paper by Goldie and Klüppelberg [11] and the references in it.
Now we show the consequences of our main results for the abovedefined class of distribution functions. The proofs will be explained in Section 8.
It is noted in [14, 15] (see also [11]) that the set of the subexponential distributions is a proper subset of the set of the longtailed distributions.
In the first theorem, we give equivalent statements for subexponential distributions; and in the second one, we give a limit formula for the more general longtailed distributions.
Theorem 4.5.
Let be a distribution function such that and . Then the following statements are equivalent.
(a) is subexponential.
(b) is longtailed and there is a such that the limit
is finite and
(c) is longtailed and there is a such that
is finite and
Theorem 4.6.
Let be distribution functions, , and is longtailed. If
are finite, and
then
The above theorem can be easily applied for tailequivalent distributions defined as follows (see [11]).
Definition 4.7 (tailequivalence).
Two distributions with the conditions and are called tailequivalent if is a positive number.
Corollary 4.8.
Let be distribution functions, , and is longtailed. If the conditions of Theorem 4.6 are satisfied, then and are tail equivalent if and only if , that is, at least one of the distribution functions and is tail equivalent to .
5. Further Corollaries for an Extended Set of Weight Functions
First we consider the extension of the set
Definition 5.1.
Let . By one denotes the set of the functions such that
for all . By one denotes the set of the functions for which the convergence in (5.1) is uniform on , for any .
Remark 5.2.
It is clear that , and if and only if , where is defined by .
Let , and . Then
where , and for .
Thus our earlier results are applicable if and But from Remark 5.2 we have that if and only if Moreover, if and only if the limit
is finite for any
Remark 5.3.
if and only if . Namely, let . Thus , and hence Now let . Then for any such that . Therefore almost every and hence . From this it follows that is absolute continuous with respect to , that is, .
It can be seen that if and only if .
Remark 5.4.
if and only if . Here denotes the set of functions for which
for any
The above remarks show that our main results, Theorems 3.1–3.4, can be easily reformulated for the class , assuming that we replace the hypotheses (H) and by and , respectively. In fact we use the following modified hypotheses.
are such that defined in (5.3) is finite for any and the improper integral
is finite, whenever is oscillatory.
is nonnegative such that
is finite for every , and the improper integral is finite, whenever is oscillatory.
The extended form of Theorem 3.1 is as follows.
Theorem 5.5.
Assume . Then the following results hold.

(a)
The following three statements are equivalent.
The limit
is finite.
For some the limit
is finite.
The values
are finite, moreover

(b)
Assume that one of the statements is true. Then the limit (5.8) is finite for any and
(5.11)
where
and , defined in (3.8), are finite.
The extensions of Theorems 3.3 and 3.4 are similar and are left to the reader.
6. PowerType Weight Function and The Role of The Middle Term
The introduction of our middle term was motivated by two independent papers [2, 4]. In both papers powertype estimations have been proved for the solutions of functional differential equations and of the wave equations with boundary condition, respectively. The joint idea was to transform the original problems into a convolutiontype form. By treating the convolution form, powertype estimations were given without investigating any limit formula.
As a consequence of Theorem 3.4, we prove the next result, and as a corollary of it we give a powertype limit formula.
Theorem 6.1.
Let , and let be positive such that the limit is finite. If and , then the limit is finite and
where
and , defined in (3.16), are finite.
The following corollary is a generalization of Theorem 1.2 and shows the importance of our middle term when is a powertype function.
Corollary 6.2.
Let and assume that the limits
are finite, where are given constants. Then
where is the Beta function, that is, ( is the wellknown Gamma function).
In the above limit formula, and the middle term whenever .
7. Preliminary Results
In this section we state and prove preliminary and auxiliary results. They will be used in the proofs of our main results in the next section. denotes the set of the positive integers.
Proposition 7.1.
Let and such that is finite for any with a fixed Then .
Proof.
Let and such that . Then
and this yields .
Proposition 7.2.
Let and . Then the following hold.
(a) for any .
(b) where is the only element of the set .
Proof.

(a)
First we show that is additive. In fact for we have
(7.2)
for . This yields . Therefore can be extended in a unique way to such that it is additive. Now (a) follows since is nonnegative on .

(b)
For any we have
(7.3)
therefore
Since is arbitrarily chosen, statement (b) is proved.
Proposition 7.3.
Let and assume that , that is, . If then
Proof.
is a positive function, therefore Thus for any there exists a such that
From this it follows that
On the other hand there is a such that
where we used that .
Thus
therefore
From this it follows
for any fixed . This completes the proof as .
Definition 7.4.
For any and , let be defined by
It is clear that for any fixed is a measure on (the unit mass at ), and .
Proposition 7.5.
Let be a given sequence in such that
and let be a sequence of nonnegative numbers. Suppose .

(a)
If the measure belongs to then and

(b)
If and then
Proof.
It is clear that

(a)
Let be fixed. Since for every we have
(7.13)
and hence This and statement (a) of Proposition 7.2 imply that Therefore
But and hence statement (a) is proved.

(b)
Let and be fixed. Then
(7.15)
for Thus
for But and therefore
The proof is complete.
Definition 7.6.
Let be fixed.

(a)
denotes the algebra of the Borel sets of

(b)
denotes the set of the finite measures on .

(c)
A topology defined on is said to be the weak topology on if it is the weakest one which makes the mapping
(7.18)
continuous for all continuous .
Definition 7.7.
For a fixed and , define the shift operator
Let . For
Proposition 7.8.
Let , and be fixed. Then
where the convergence is in the weak topology of .
Proof.
We should prove that for any fixed continuous function , we have
For any the function denotes the characteristic function of
Let
where and
Then from the statement (b) of Proposition 7.2, it follows
It is known that for a fixed continuous function , there exists a sequence of step functions such that it converges to uniformly on Thus for arbitrarily fixed there is an index such that
In that case
for all large enough. Here we used the conclusion of the first part of our proof and statement (b) of Proposition 7.2. Since is fixed but arbitrary, the proof is complete.
Corollary 7.9.
Let and .
(a)If is Borel measurable and Riemann integrable on any interval , then .
(b) If and , then .
(c)Let . If and , then .
Proof.
From Proposition 7.8, it follows (see, e.g., [12]) that if is a.e. continuous, then From this we get statements (a) and (b).

(c)
Let and be fixed. Since , there is a such that
(7.27)
and hence
Since there is a such that
Thus
From the general transformation theorem for integrals (see, e.g., [12]) and from the translation invariance of the Lebesgue measure , we get: for any and
But Proposition 7.3 shows that and So
Since is arbitrary, this completes the proof.
Proposition 7.10.
Let , and assume that is not oscillatory on . Then the following mappings have limits in as :
Proof.
Let be nonnegative on , where is large enough. Then for and , we get
Thus the abovedefined mappings are decreasing, and hence their limits exist in as . When is eventually nonpositive, then the above procedure can be applied for The proof is complete.
In the next two results, we give explicit formulas for the limit inferior and limit superior of the weighted convolution of and at
Theorem 7.11.
Assume (H). Then the following results hold.

(a)
The following two statements are equivalent.
The limit inferior
is finite.
For some , the limit inferior
is finite.

(b)
If the limit inferior (7.36) is finite for a fixed , then it is finite for any and
(7.37)
where
and (they are defined in (3.7) and (3.9), resp.) are finite.
Proof.
Let be fixed. Then for any and we get
First we show that
In fact for and we have
But therefore for there exists such that
Thus (7.41) yields
and hence
which implies (7.40).
Since we have
Assume that holds. Then (7.39), (7.40), and (7.45) imply . On the other hand, from (7.39), (7.40), and (7.45) we get that yields , and hence statement (a) is proved. This also verifies the first part of statement (b).
Now we prove the second part of statement (b). Assume that (7.36) is finite for any . Then (7.39) yields
Now assume that is not oscillatory. Then there exists such that either for every or for every . We consider the case when for , the other case can be handled similarly.
All the three terms on the righthand side of (7.46) have limit as in In fact
and by Proposition 7.10, we get
moreover
Now the second part of (b) is proved, since the lefthand side of (7.46) is finite and independent on .
Now assume that is oscillatory on and as we assumed is finite. In that case and hence
Thus by using similar arguments to those we used above, statement (b) is proved again.
Theorem 7.12.
Assume (H). Then the following results hold.

(a)
The following two statements are equivalent.
The limit superior
is finite.
For some the limit superior
is finite

(b)
If the limit superior (7.52) is finite for a fixed then it is finite for any and
(7.53)
where
and (they are defined in (3.7) and (3.9), resp.) are finite.
Proof.
Its proof is similar to the proof of Theorem 7.11, therefore it is omitted.
Theorem 7.13.
Let , and assume that

(i)
the improper integral
(7.55)
is finite, whenever is oscillatory and is not oscillatory,

(ii)
the improper integral
(7.56)
is finite, whenever is not oscillatory and is oscillatory.
Then the following results hold.

(a)
The following two statements are equivalent.
The limit inferior
is finite.
For some the limit inferior
is finite.

(b)
If the limit inferior (7.58) is finite for a fixed , then it is finite for any and
(7.59)
where
and (they are defined in Theorem 3.3) are finite.
Proof.
Let be fixed. Then for each and , we have
The proof of (7.40) can easily be modified to show that
Since
it follows from (7.62) that
By using (7.61), (7.62), and (7.64) instead of (7.39), (7.40), and (7.45), the argument employed in the proof of Theorem 7.11(a) and the first part of (b) extends to give (a) and the first part of (b).
Consider now the proof of (7.59). Suppose that (7.58) is finite for every . By (7.61),
for each . We separate the proof into four steps.

(h)
Suppose first that and are oscillatory. Then , hence (7.65) implies that
(7.66)
It now follows that is finite and

(j)
Suppose next that exactly one of the functions and is oscillatory. Without loss of generality, we can assume that is oscillatory and is not oscillatory. Then , hence (7.65) shows that
(7.68)
By (i), is finite and

(k)
Suppose that there is such that and for every . Then it follows from and
(7.70)
that
A similar argument gives that
Since
we have
Using (7.65), we deduce from (7.71), (7.72), and (7.74) that the limits in (7.71) and (7.72) are finite, and therefore exists and is finite. This gives (7.59). If and for every , then a similar proof can be applied.

(l)
Suppose finally that there is such that and for every , or and for every . This case follows by an argument entirely similar to that for the case (k). Here the limits (7.71) and (7.72) are in , and (7.74) is nonpositive.
Theorem 7.14.
Under the hypotheses of Theorem 7.13 the following results hold.

(a)
The following two statements are equivalent.
The limit superior
is finite.
For some , the limit superior
is finite.

(b)
If the limit superior (7.76) is finite for a fixed , then it is finite for any and
(7.77)
where
and (they are defined in Theorem 3.3) are finite.
Proof.
The proof is similar to the proof of the previous theorem, therefore it is omitted.
8. The Proofs of The Main Results
In this section, we give the proofs of the results stated in Sections 3–6.
Proof of Theorem 3.1.
A similar argument employed in the proof of Theorem 7.11 gives the equivalence of and , and part (b). It is clear from and (b) that implies . If holds, then by Theorems 7.11 and 7.12, the values of and are finite. Since , it follows from (7.37) and (7.53) that . This shows that yields .
Proof of Theorem 3.3.
This is an immediate consequence of Theorem 3.1.
Proof of Theorem 3.4.
The argument of Theorem 7.13 can easily be generalized to prove the equivalence of and , and part (b). and (b) imply . If holds, then we can apply Theorems 7.13 and 7.14. It now follows from (7.59) and (7.77) that , and therefore is satisfied.
Proof of Theorem 1.1.
First we suppose that is nonnegative. It follows from that
for every . We can see that the hypothesis is satisfied with , and . This shows that Theorem 3.3 can be applied. Consider now the proof that holds, and . There exists such that
This implies that
hence
and therefore, by ,
Since
it follows from (8.5) that is true, and . Now (3.15) gives the result.
In the general case, the preceding can be applied to both and .
Proof of Theorem 4.3.
is a subexponential function, and therefore and
Theorem 3.4 may now be applied with , and is obtained. Thus the result follows from Theorem 6.1 with .
Proof of Theorem 4.5.
Suppose that is subexponential. Then is longtailed as is well known (details can be found in [11]), and thus . The distribution function generates a distribution . Since
we can see that for every , and therefore . It follows that the hypothesis (H) is satisfied with and . This shows that Theorem 3.1 can be applied. Since is subexponential, the equivalence of and implies (4.9), and (3.6) gives (4.10). We have proved that (b) comes from (a). On the other hand, (c) obviously follows from (b).
By the equivalence of and in Theorem 3.1, (c) implies (a).
Proof of Theorem 4.6.
is longtailed, hence . The distribution function characterizes a distribution . Then
giving
and therefore and .
It follows that the condition (H) is satisfied with and .
According to Theorem 3.1 and (3.6), we now have
Applying this and taking into account Proposition 7.2(b), the result follows, since
Proof of Theorem 5.5.
By the correspondence between hypotheses (H) and (H()), Theorem 3.1 implies the result.
Proof of Theorem 6.1.
By Theorem 3.4, we deduce that
Let . Since and , we can find such that
and therefore also
This implies that
Equation (8.13) shows that is finite, hence the definition of and the previous inequality give
and therefore
The result now follows from Theorem 3.4 (see Theorem 3.3) by applying
and .
Proof of Corollary 6.2.
If
then . Let be defined by
Then , and therefore . By (6.3) and Theorem 6.1, it is enough to prove
which comes from the definition of the Beta function.
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This work is supported by Hungarian National Foundation for Scientific Research Grant no. K73274.
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Győri, I., Horváth, L. New Limit Formulas for the Convolution of a Function with a Measure and Their Applications. J Inequal Appl 2008, 748929 (2008). https://doi.org/10.1155/2008/748929
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Keywords
 Weight Function
 Asymptotic Theory
 Beta Function
 Middle Term
 Subexponential Distribution