Abstract and Applied Analysis

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, [1]; see [2] 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 [3].

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 [4]. 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 [5].

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 [1], 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 [4] 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 [4] 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 .

Remark 2.2. If and one can use Definition A.2 to define where almost everywhere and . All of the results in Theorem 2.1 and the rest of this paper have analogues. Note that .

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 ,