Convolutions with the Continuous Primitive Integral
If is a continuous function on the real line and is its distributional derivative, then the continuous primitive integral of distribution is . This integral contains the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals. Under the Alexiewicz norm, the space of integrable distributions is a Banach space. We define the convolution for an integrable distribution and a function of bounded variation or an function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. For of bounded variation, is uniformly continuous and we have the estimate , where is the Alexiewicz norm. This supremum is taken over all intervals . When , the estimate is . There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.
1. Introduction and Notation
The convolution of two functions and on the real line is . Convolutions play an important role in pure and applied mathematics in Fourier analysis, approximation theory, differential equations, integral equations, and many other areas. In this paper, we consider convolutions for the continuous primitive integral. This integral extends the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals on the real line and has a very simple definition in terms of distributional derivatives.
Some of the main results for Lebesgue integral convolutions are that the convolution defines a Banach algebra on and such that . The convolution is commutative, associative, and commutes with translations. If and then and . Convolutions also have the approximation property that if () and then as , where and . When is bounded and continuous, there is a similar result for . For these results see, for example, ; see  for related results with the Henstock-Kurzweil integral. Using the Alexiewicz norm, all of these results have generalizations to continuous primitive integrals that are proven in what follows.
We now define the continuous primitive integral. For this, we need some notation for distributions. The space of test functions is . The support of function is the closure of the set on which does not vanish and is denoted . Under usual pointwise operations, is a linear space over field . In we have a notion of convergence. If then as if there is a compact set such that for each , , and for each we have uniformly on as . The distributions are denoted and are the continuous linear functionals on . For and we write . For and we have . Moreover, if in then in . Linear operations are defined in by for ; and . If then defines a distribution . The integral exists as a Lebesgue integral. All distributions have derivatives of all orders that are themselves distributions. For and the distributional derivative of is where . This is also called the weak derivative. If is a function that is differentiable in the pointwise sense at then we write its derivative as . If is a bijection such that for any then the composition with distribution is defined by for all . Translations are a special case. For , define the translation on distribution by for test function where . All of the results on distributions we use can be found in .
The following Banach space will be of importance: . We use the notation and . The extended real line is denoted by . The space then consists of functions continuous on with a limit of at . We denote the functions that are continuous on that have real limits at by . Hence, is properly contained in , which is itself properly contained in the space of uniformly continuous functions on . The space is a Banach space under the uniform norm; for . The continuous primitive integral is defined by taking as the space of primitives. The space of integrable distributions is . If then for . The distributional differential equation has only constant solutions so the primitive satisfying is unique. Integrable distributions are then tempered and of order one. This integral, including a discussion of extensions to , is described in . A more general integral is obtained by taking the primitives to be regulated functions, that is, functions with a left and right limit at each point, see .
Examples of distributions in are for functions that have a finite Lebesgue, Henstock-Kurzweil, or wide Denjoy integral. We identify function with the distribution . Pointwise function values can be recovered from at points of continuity of by evaluating the limit for a delta sequence converging to . This is a sequence of test functions such that for each , , , and the support of tends to as . Note that if is an increasing function with for almost all then the Lebesgue integral but and . For another example of a distribution in , let be continuous and nowhere differentiable in the pointwise sense. Then and for all .
The space is a Banach space under the Alexiewicz norm; where the supremum is taken over all intervals . An equivalent norm is . The continuous primitive integral contains the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals since their primitives are continuous functions. These three spaces of functions are not complete under the Alexiewicz norm and in fact is their completion. The lack of a Banach space has hampered application of the Henstock-Kurzweil integral to problems outside of real analysis. As we will see in what follows, the Banach space is a suitable setting for applications of nonabsolute integration.
We will also need to use functions of bounded variation. Let . The variation of is where the supremum is taken over all disjoint intervals . The functions of bounded variation are denoted . This is a Banach space under the norm . Equivalent norms are and for each . Functions of bounded variation have a left and right limit at each point in and limits at , so, as above, we will define .
If then the essential variation of is where the supremum is taken over all with . Then . This is a Banach space under the norm . Let . For define . For left continuity, and for right continuity . The functions of normalized bounded variation are . If then such that almost everywhere. For each there is exactly one function such that almost everywhere. In this case, . Changing on a set of measure zero does not affect its essential variation. Each function of essential bounded variation has a distributional derivative that is a signed Radon measure. This will be denoted where for all .
We will see that and that . Similarly for . Convolutions for and will be defined using sequences in that converge to in the norm. It will be shown that and that .
Convolutions can be defined for distributions in several different ways.
Definition 1.1. Let and . Define : (i) , (ii) for each , let ; (iii) .
In (i), . This definition also applies to other spaces of test functions and their duals, such as the Schwartz space of rapidly decreasing functions or the compactly supported distributions. In (ii), . In , it is shown that definitions (i) and (ii) are equivalent. In (iii), . However, this definition requires restrictions on the supports of and . It suffices that one of these distributions has compact support. Other conditions on the supports can be imposed (see [3, 6]). This definition is an instance of the tensor product, where now .
Under (i), is in . It satisfies , , and . Under (iii), with appropriate support restrictions, is in . It is commutative and associative, commutes with translations, and satisfies . It is weakly continuous in , that is, if in then in see [1, 3, 6, 7] for additional properties of convolutions of distributions.
Although elements of are distributions, we show in this paper that their behavior as convolutions is more like that of integrable functions.
An appendix contains the proof of a type of Fubini theorem.
2. Convolution in
In this section, we prove basic results for the convolution when and . Under these conditions, is commutative, continuous on and commutes with translations. It can be estimated in the uniform norm in terms of the Alexiewicz and norms. There is also an associative property. We first need the result that forms the space of multipliers for , that is, if then for all . The integral is defined using the integration by parts formula in the appendix. The Hölder inequality (A.5) shows that is the dual space of .
We define the convolution of and as where . We write this as .
Theorem 2.1. Let and let . Then (a) exists on (b) Let (c) Let (d) Assume , (e) If then . (f) Let , then . (g) For each define by . Then is a bounded linear operator and . There exists a nonzero distribution such that . For each define by . Then is a bounded linear operator and . There exists a nonzero function such that . (h) .
Proof. (a) Existence is given via the integration by parts formula (A.1) in the appendix. (b) See [4, Theorem 11] for a change of variables theorem that can be used with . (c) This inequality follows from the Hölder inequality (A.5). (d) Let . From (c), we have The last line follows from continuity in the Alexiewicz norm [4, Theorem 22]. Hence, is uniformly continuous on . Also, it follows that The limit can be taken under the integral sign since is of uniform bounded variation, that is, . Theorem 22 in  then applies. Similarly, as . (e) First show . Let be disjoint intervals in . Then Hence, . The interchange of sum and integral follows from the Fubini-Tonelli theorem. Now (d) shows . Write We can interchange orders of integration using Proposition A.3. For (ii) in Proposition A.3, the function is in for each fixed . Since is of bounded variation, it is bounded so and condition (iii) is satisfied. (f) This follows from a linear change of variables as in (a). (g) From (c), we have . Let be in . If then and so . To prove , note that . Let . Then . (h) Suppose . Note that we can write in terms of a Henstock-Stieltjes integral, see  for details. This integral is approximated by Riemann sums where , and there is a gauge function mapping to the open intervals in such that . If then since is open, there is an open interval . We can take such that for all . Also, is constant on each interval in . Therefore, and only tags can contribute to the Riemann sum. However, for all we have so . It follows that .
Similar results are proven for in [1, Section 8.2].
If we use the equivalent norm then . Also, integration by parts gives . Now, given let . Then , and . Hence, and . We can have strict inequality in . For example, let , then but integration by parts shows for each .
Proposition 2.3. The three definitions of convolution for distributions in Definition 1.1 are compatible with for and .
Proof. Let , and . Definition 1.1(i) gives Since and , Proposition A.3 justifies the interchange of integrals. Definition 1.1(ii) gives Definition 1.1(iii) gives The interchange of integrals is accomplished using Proposition A.3 since and .
The locally integrable distributions are defined as . Let and let with support in the compact interval . By the Hake theorem [4, Theorem 25], exists if and only if the limits of exist as and . This gives
There are analogues of the results in Theorem 2.1. For example, . There are also versions where the supports are taken to be semi-infinite intervals.
We can also define the distributions with bounded primitive as . Let and let be its unique primitive. If such that then
It follows that .
It is possible to formulate other existence criteria. For example, if and for some then and are not in , or for any but exists on because and if then .
The following example shows that needs not to be of bounded variation and hence not absolutely continuous. Let . For we have where is the primitive of . However, needs not to be of bounded variation or even of local bounded variation. For example, let and let be its primitive in . Finally, although is continuous, it needs not to be integrable over . For example, let then and only exists if .
3. Convolution in
We now extend the convolution to and . Since there are functions in that are not of bounded variation, there are distributions and functions such that the integral does not exist. The convolution is then defined as the limit in of a sequence for such that in the norm. This is possible since is dense in . We also give an equivalent definition using the fact that is dense in . Take a sequence such that . Then is the limit in of . In this more general setting of convolution defined in we now have an Alexiewicz norm estimate for in terms of estimates of in the Alexiewicz norm and in the norm. There is associativity with functions and commutativity with translations.
Definition 3.1. Let and let . Let such that . Define as the unique element in such that .
To see that the definition makes sense, first note that is dense in since step functions are dense in . Hence, the required sequence exists. Let be a compact interval. Let be the primitive of . Then
The interchange of orders of integration in (3.1) is accomplished with Proposition A.3 using . Integration by parts gives (3.2) since . As is continuous and the function is absolutely continuous, we get (3.3). Taking the supremum over gives
We now have
and is a Cauchy sequence in . Since is complete, this sequence has a limit in which we denote . The definition does not depend on the choice of sequence , thus if such that then as . The previous calculation also shows that if then the integral definition and the limit definition agree.
Definition 3.2. Let and let . Let such that . Define as the unique element in such that .
To show this definition makes sense, first show is dense in .
Proposition 3.3. is dense in .
Proof. Let be the functions that are absolutely continuous on each compact interval and which are of bounded variation on the real line. Then, if and only if there exists such that for almost all . Let be given. Let be its primitive. For , take such that for and for . Due to the Weierstrass approximation theorem, there is a continuous function such that for , for , for and is a polynomial on . Hence, and .
In Definition 3.2, the required sequence exists. Let be a compact interval. Then, by the usual Fubini-Tonelli theorem in ,
Take the supremum over and use the Hölder inequality to get
It now follows that is a Cauchy sequence. It then converges to an element of . However, (3.7) also shows that this limit is independent of the choice of . To see that Definitions 3.1 and 3.2 agree, take with and with . Then
Theorem 3.4. Let and . Define as in Definition 3.1. Then (a). (b) Let . Then . (c) For each , . (d) For each define by . Then is a bounded linear operator and . There exists a nonzero distribution such that . For each define by . Then is a bounded linear operator and . There exists a nonzero function such that . (e) Define for . We have . Then as . (f) Let .
Proof. Let be as in Definition 3.1. (a) Since , (3.4) shows . (b) Let such that . Then such that . Since there is such that . Then such that . Now,
Finally, use Theorem 2.1(e) and (3.4) to write
The Alexiewicz norm is invariant under translation [4, Theorem 28] so . Use Theorem 2.1(f) to write . Translation invariance of the norm completes the proof. (d) From (a), we have . We get equality by considering and to be positive functions in . To prove , note that . We get equality by considering and to be positive functions in . (e) First consider . We have
By dominated convergence, we can take the limit inside the integral (3.14). Continuity of in the Alexiewicz norm then shows as .
Now take a sequence such that Define and . We have
By the inequality in (a), . Whereas, and . Given fix large enough so that . Now let in (3.15).
The interchange of order of integration in (3.13) is justified as follows. A change of variables and Proposition A.3 give
Note that by Corollary A.4. (f) This follows from the equivalence of Definitions 1.1 and 3.1, proved in Proposition 3.5, see [6, Theorems 5.4-2 and 5.3-1].
The fact that convolution is linear in both arguments, together with (b), shows that is an -module over the convolution algebra, see  for the definition. It does not appear that is a Banach algebra under convolution.
We now show that Definition 1.1(iii) and the aforementioned definitions agree.
Proof. Let . Then and . Also, so Dominated convergence then shows . Integration by parts now gives (3.18).
Let such that . Since convergence in implies convergence in , we have
Proposition A.3 allows interchange of the iterated integrals. Define . Then, for large enough . Hence, is of uniform bounded variation. Theorem 22 in , then gives . The last step follows since .
If then needs not to be continuous or bounded. For example, take and let . Then, and . We have for . For we have . Hence, is not continuous at . If then is unbounded at .
As another example, consider and . Then and for each we have . And,
Therefore, by Theorem 2.1(d), and but for each we have .
4. Differentiation and Integration
If is sufficiently smooth, then the pointwise derivative is . Recall the definition of primitives of functions given in the proof of Proposition 3.3. In the following theorem, we require pointwise derivatives of to exist at each point in .
Theorem 4.1. Let , and for each . Then and for each .
Proof. First consider . Let . Then To take the limit inside the integral we can show that the bracketed term in the integrand is of uniform bounded variation for . Let . Since it follows that the variation is given by the Lebesgue integrals Since we have . The second integral on the right of (4.2) gives the derivative of in the limit see [1, page 246]. Hence, in (4.1), we can use [4, Theorem 22] to take the limit under the integral sign. This then gives . Theorem 2.1(d) now shows . Induction on completes the proof.
For similar results when , see [1, Proposition 8.10].
Note that does not imply . For example, . The conditions for imply those in Theorem 4.1. To see this, it suffices to consider . If then exists at each point and is bounded. Hence, the Lebesgue integral exists for each and is absolutely continuous. Since we then have . Similarly, for . The example shows that the condition in the theorem is weaker than the aforementioned condition since but does not exist so .
We found that when then . We can compute the distributional derivative where is a primitive of .
Proposition 4.2. Let and write . Let . Then and .
Proof. Let . Then by the usual Hölder inequality,
Hence, is continuous on Dominated convergence shows that . Therefore, .
Let . Then
Integrate by parts and use the change of variables to get
This gives an alternate definition of for and .
Theorem 4.3. Let , let be the primitive of and let . Define as in Definition 3.1. Then .
Proof. Let . Let such that . By Proposition 4.2, we have