Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 356724, 11 pages

http://dx.doi.org/10.1155/2013/356724

Review Article

Compatibility Conditions and the Convolution of Functions and Generalized Functions

Andrzej Kamiński and Svetlana Mincheva-Kaminska

Faculty of Mathematics and Natural Sciences, University of Rzeszów, Profesora Pigonia 1, 35-310 Rzeszów, Poland

Received 23 July 2013; Accepted 6 September 2013

Academic Editor: Henryk Hudzik

Copyright © 2013 Andrzej Kamiński and Svetlana Mincheva-Kaminska. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The paper is a review of certain existence theorems concerning the convolution of functions, distributions, and ultradistributions of Beurling type with supports satisfying suitable compatibility conditions. The fact that some conditions are essential for the existence of the convolution in the discussed spaces follows from earlier results and the proofs given at the end of this paper.

In memory of Professor Jan Mikusiński on the 100th anniversary of his birthday

1. Introduction

The convolution and its various generalizations play a very important role in the classical and abstract analysis as well as in other fields of mathematics, in particular in the theory of distributions (see [1–7]), ultradistributions (see [3, 4, 8–18]), hyperfunctions (see [19–21]), and other generalized functions considered for various spaces, subspaces, and approaches. The notion of convolution is a starting point in algebraic approaches to certain generalized functions: the convolution algebra of continuous functions on extends, due to the Titchmarsh theorem, to the field of Mikusiński operators (see [22–25]) which are generalized functions of another type than Schwartz distributions, while Boehmians stand for a common generalization of regular Mikusiński operators of Boehme (see [26]), Schwartz distributions, and other classes of generalized functions on the real line (see [27–31]). An important part of investigations connected with the convolution is the study of convolution operators and convolution semigroups for various spaces of functions and generalized functions (see, e.g., [32–37]). Therefore the problems concerning the existence of the convolution in various spaces of functions and generalized functions are crucial.

The theory developed by Colombeau (see [38]; see also [39]) and his followers (see, e.g., [40–43]) has led to constructions of algebras of new generalized functions related to the distributions and other classical generalized functions due to certain quotient procedures; consequently the algebras of new generalized functions are closed with respect to multiplication as well as other nonlinear operations. However the problem of existence of the product and the convolution of distributions and other generalized functions in the standard sense, without using Colombeau’s approach, remains important. We will analyse in this paper the existence of the convolution of distributions and tempered distributions on meant in the classical sense of general equivalent definitions introduced independently by several authors (see [2, 44–50]). Also the existence of the convolution of ultradistributions and tempered ultradistributions of Beurling type introduced in [18, 51] (see also [52]), corresponding to the above mentioned general definitions of the convolution of distributions and tempered distributions, will be discussed. The convolution of ultradistributions of Roumieu type will be not considered in this paper; for recently obtained results concerning the convolution of Roumieu ultradistributions we refer to the paper [53].

There exist various sufficient conditions guaranteeing existence of the convolution in various spaces of functions and generalized functions. Some of the conditions are given in the form of suitable assumptions concerning their growth at infinity, but there are conditions of another type expressed in terms of supports of given (generalized) functions. We will call them after Mikusiński (see [54], and [7, p. 124–127]) compatibility conditions. We mean the support of a given function (generalized function) in the standard way as the smallest closed set in outside which the function (generalized function) vanishes almost everywhere (everywhere). For modifications of the notion of support in case of distributions and tempered distributions see [7, p. 241] and [55].

We recall in this paper various versions of compatibility conditions imposed on supports of functions, distributions, and ultradistributions of Beurling type which guarantee that the convolution in the respective spaces exists (see [5, 7, 54, 56–59]; see also [60–63]). In Section 7, we prove some inverse theorems which mean that the considered compatibility conditions are optimal for the existence of the convolution in some spaces in terms of supports; that is, the conditions cannot be relaxed. We also mention new cases of compatibility of supports in which the convolution exists in the considered spaces. In particular, we show examples of the so-called spiral sets in with the property that the convolutions of functions, distributions, ultradistributions having supports contained in such sets always exist in the respective spaces, though the sets are unbounded in each direction of .

2. Preliminaries

We will use the standard multidimensional notation concerning and , using traditional symbols even in case they are formally inconsistent, because the proper meaning is easily seen from the context; for example, denotes the Euclidean norm of in and

but we write also for , though .

The standard notation will be also used for the known spaces of (complex-valued) functions on : for (with the norm denoted by ), , , , , , , , and the known (sub) spaces of distributions on : , , and (cf. [1, 7]).

For a set we denote . It will be convenient to use the following notation for and a function on :

Clearly, for any , the set is bounded (compact) in if and only if . Therefore if , then , but only in case . Moreover, the symbol will be used in the following sense:

The words “measure” and “measurable” concerning the sets in are considered in the sense of Lebesgue; the Lebesgue measure of a given Lebesgue measurable set will be denoted by . If is a measurable set, we use the following notation:

That is, in case and in case , where with . Clearly,

We call a measurable set positively bounded in whenever . An element of a measurable set is called a massive point of the set if for each open neighbourhood of in .

A measurable function on is called slowly increasing (resp., rapidly decreasing) if (resp., ) for a certain polynomial on (resp., for every polynomial and some positive constant and for almost all . The sets of all measurable slowly increasing and rapidly decreasing functions on will be denoted by and , respectively.

In our considerations concerning the convolution in the spaces of ultradistributions and of tempered ultradistributions of Beurling type (the respective definitions and notation are introduced in Section 4) we always assume that a given sequence of positive numbers satisfies the following three conditions (see [10–12, 18, 51, 52, 59]):

(M.1); (M.2); (M.3),

where and are some constants. We extend the sequence to its multidimensional version as follows:

By the associated function for the sequence we mean the function on given by

where and .

3. Convolution of Functions and Distributions

Definition 1. Let and be Lebesgue measurable functions on . For a given one defines

saying that exists, whenever the function under the integral sign in (8) is Lebesgue integrable as a function of for the fixed . If exists or, equivalently, exists, for all (almost all) , then one says that the convolution exists everywhere (almost everywhere) on .

Definition 2. Let . One says that the convolution exists in if exists almost everywhere on and .

Let us list some classical results concerning the existence in of the convolution of functions of certain classes:

(a)if , then exists in and ;(b)if and , where such that , then exists in and , where (Young);(c)if and , then exists in and .

However, in general, the existence of the convolution of functions in does not imply any restriction of growth of the convolution at infinity. For instance, the convolution of two measurable slowly increasing functions may exist in , but their convolution may be a function of arbitrarily fast increase at infinity.

The following result, much stronger than one may expect, was proved in [56, 57] (see also [58, 63]).

Theorem 3. Let be an arbitrary continuous (complex-valued) function on . There exists a nonnegative function on such that

the convolution exists everywhere in , and the following inequality is satisfied:

Moreover, the function can be constructed in such a way that its support is compatible with itself (see Definition 16).

Let us underline that inequality (10) in the assertion of Theorem 3 is satisfied for each ; the convergence in (9) can be slow: the faster the increase of the function (i.e., the required increase of the convolution ) is at infinity the slower the approach of to is as tends to (for details of the construction see [56, 57, 63]).

Theorem 3 has a general value. From the theorem one may easily see, for various spaces of functions or generalized functions and their various subspaces, that the convolution in the sense of the considered space leads out of the examined subspace. This concerns, in particular, the convolution in and in (see Definitions 5 and 12) of elements of the subspaces and of the spaces and of Schwartz distributions and Beurling ultradistributions, respectively (see the definitions of the spaces, given in this and in the next section, and the comments at the end of this section).

Let us shortly recall (see [1]; see also, e.g., [5–7]) that distributions are elements of the strong dual of the basic space of test functions, that is, linear continuous functionals on the space defined as the inductive limit:

where the symbol means here that are compact sets growing up to . Recall that , for a fixed , means the space of all functions on whose support is contained in , with the topology given by the family of the norms defined as follows:

Locally integrable functions on may be treated as distributions (so-called regular distributions), because the space can be naturally embedded into . The known structural theorem describes every distribution as a derivative (in the distributional sense) locally, that is, on each relatively compact set in , of finite order of a continuous function (see [1, 5–7]).

Recall that tempered distributions are elements of the dual of the complete metric space of rapidly decreasing functions, that is, of all such that

for all , where is defined by (1), with the metric topology induced by the family of the norms defined in (13). Since is dense in , the space of tempered distributions can be naturally embedded into . The known structural theorem describes each tempered distribution as a derivative (in the distributional sense) of finite order of a continuous slowly increasing function on (see [1–7]).

There are several general definitions of the convolution in of Schwartz distributions given consecutively by Chevalley [2], Schwartz [44], Shiraishi [45], and Vladimirov [46] (see also [47–50, 63–72]).

Most of the mentioned definitions are equivalent (for details see, e.g., [50]). We will recall only one of them, the sequential definition of Vladimirov [46] (see also [6]), based on the notion of strong approximate unit.

Definition 4. A sequence of elements of is said to be a strong approximate unit on if for every there exists an such that for and (hence in as ) and, in addition,

One denotes the set of all strong approximate units on by .

Definition 5. For given the convolution in is defined by

whenever the above limit exists for every strong approximate unit and . One says then that the convolution exists in .

By the Lebesgue theorem, if and the convolution exists in , then it exists in and represents a regular distribution in (see [6, p. 63]).

Analogously to and independently of Definition 5, one may define the convolution in of tempered distributions in various ways (see, e.g., [45, 49, 50] and other references given earlier). Again, we present below only one of several equivalent definitions of the convolution in , namely, the respective counterpart of the aforementioned sequential definition of Vladimirov (cf. [6, 46]).

Definition 6. For given one defines the convolution in by

whenever the above limit exists for every strong approximate unit and . One says then that the convolution exists in .

In [45], Shiraishi posed the problem of whether the assumption that the convolution of two tempered distributions exists in implies the existence of the convolution in (in particular, whether ).

The negative answer to Shiraishi’s problem follows directly from Theorem 3, proved by Kamiński in [56, 57] (see also [58, 63]) and presented in Professor Jan Mikusiński’s seminar in Katowice in 1971 and during the international conference on generalized functions in Rostock in 1972. This also follows from a result obtained independently by Dierolf and Voigt and published in [49]. Dierolf and Voigt constructed in [49] two tempered measures and , concentrated on a countable set in , such that the convolution exists in , but .

Theorem 3 is much stronger than the result of Dierolf and Voigt (it supplies counterexamples concerning the growth of the convolution in various spaces of functions and generalized functions).

To get the negative answer to Shiraishi’s problem it is enough to take in Theorem 3 for the function defined by

(or any continuous function of even faster increase). Clearly, represents a distribution which is not tempered, but from Theorem 3 it follows that there exists a bounded (even vanishing at infinity) function on representing a tempered distribution for which the convolution exists everywhere and exceeds everywhere on the function . Consequently, represents a distribution but not a tempered distribution.

4. Convolution of Beurling Ultradistributions

We will recall the definitions of the space of of Beurling ultradistributions (see [9–12, 18, 51]) and the space of Beurling tempered ultradistributions (see [18, 52, 59, 73]) as well as the corresponding structural theorems characterizing elements of these spaces for a fixed numerical sequence satisfying conditions (see Section 2). Ultradistributions of Roumieu type (see [8]) are not discussed in this paper.

We start by defining Beurling spaces of ultradifferentiable functions. For a given and a regular compact subset of (see [10–12]), we define the space , consisting of all functions from with support contained in such that

with the topology induced by the above norm. Then the basic space of test functions is defined by means of the projective and inductive limits as follows:

where the symbol means that are regular compact sets growing up to .

In addition, for a fixed , we denote by the space of all functions such that

where the symbol is defined in (1), equipped with the topology induced by the above norm . Then we define

The strong dual of , denoted by , is called the space of Beurling ultradistributions.

The following structural theorem (see [10], p. 76) says that Beurling ultradistributions are locally infinite derivatives of measures (continuous functions).

Theorem 7. Let . Then, for each open, relatively compact set in , there are measures for and positive constants and such that

The space of all Beurling tempered ultradistributions is meant as the strong dual of the space defined previously; it was introduced by Pilipovi in [73] (see also [18, 52, 59]). Since is dense in and the inclusion mapping is continuous, we have .

For more details concerning the definitions and properties of the aforementioned spaces of test functions and the spaces of Beurling ultradistributions and of Beurling tempered ultradistributions we refer to [18, 51, 52, 59, 60, 73–75].

The following result is a representation theorem for Beurling tempered ultradistributions (see [59]).

Theorem 8. Suppose that . Then there are functions in for and positive constants , , and such that

where for .

There are various general definitions of the convolution in of Beurling ultradistributions (see [51]) and of the convolution in of Beurling tempered ultradistributions (see [52]). They are suitable counterparts of the known general definitions of the convolution in and the convolution in mentioned in Section 3. The fact that the mentioned definitions of the convolution in of Beurling ultradistributions are equivalent and that the corresponding definitions of the convolution in of Beurling tempered ultradistributions are equivalent was proved in [51, 52], respectively (see also [18]).

We will recall here only these definitions of the convolution in and in which correspond to Vladimirov’s definition of the convolution in and in , respectively. The definitions are based on the notions of strong -approximate unit and strong -approximate unit.

Definition 9. A sequence of elements of is said to be a strong -approximate unit on if for every there exists an such that for and (hence in as ) and, in addition, if there exists a positive constant such that

One denotes the set of all strong -approximate units on by .

Definition 10. If in the previous definition the assumption that for is replaced by for and the remaining assumptions are preserved, then the sequence is called a strong -approximate unit. One denotes the set of all strong -approximate units on by .

Remark 11. According to the known Denjoy-Carleman-Mandelbrojt theorem, the previously defined class of strong -approximate units as well as the class of strong -approximate units contains sufficiently many sequences, for example, all sequences of the form

where is a function of class or , respectively, such that in some neighbourhood of .

Vladimirov’s version of the definition of the convolution in of Beurling ultradistributions has the following form.

Definition 12. For given Beurling ultradistributions the convolution in is defined by

whenever the limit in (26) exists for every strong approximate unit and .

Analogously, the convolution in of Beurling tempered ultradistributions can be defined as follows.

Definition 13. For given two Beurling tempered ultradistributions we define the convolution in by

whenever the limit in (27) exists for every strong approximate unit and .

5. Compatibility Conditions

Let us present various forms of the condition imposed on sets in the context of the convolution of distributions (see [5, p. 383]; see also [6, 7]). They can be expressed in a shorter way by means of the notation introduced in (3).

Let us begin by recalling known equivalent forms of the condition in case the considered sets and are closed in (see, e.g., [5, p. 383]).

Proposition 14. Let be arbitrary closed sets. The following conditions are equivalent:

() for every ;() for every ;() for every .

The meaning of the conditions for can be seen in Figure 1 (where and mean ).

Figure 1: Compatible sets.

As proved in [5, pp. 383–384], conditions are equivalent if the sets and are closed in . In general, we have the following equivalence (see [76]).

Proposition 15. Let be arbitrary sets. The following conditions are equivalent:

() is bounded in for every bounded set in ;() is bounded in for every bounded set in ;() is bounded in for every bounded set in ;()if and for , then implies .

If the sets are closed, then each of the above conditions is equivalent to each of conditions .

Definition 16. Two sets , are called compatible if one of equivalent conditions is satisfied.

Remark 17. The condition in the form and the name “compatible sets” were introduced in [54] (see also [7, p. 124–127]).

There are two well known particular cases of compatible sets and in :

at least one of the sets and is bounded;both and are bounded from the left or both from the right.

Case extends clearly to for and case can be described in in the following form: are (or are contained in) suitable cones with vertices at such that is an open convex cone and , where means the cone dual to (see [18, pp. 4–6]; [7, pp. 129-130]; [6, pp. 63-64]). For , case is illustrated in Figure 2 (where and mean ).

Figure 2: Cones.

Another, less obvious, case of compatible sets in is the following:

both sets and in are unbounded from both sides, that is, unbounded both from below and from above in .

In [63] (see also [56–58]) various pairs of compatible sets and in , both unbounded from both sides, are considered; for in particular. They were constructed as countable sums of intervals (of length ) situated in in a specific way. A general idea lies in constructing a sequence of intervals on the positive half-line (with suitable distances between them) and in shifting in a proper way their symmetric counterparts on the negative half-line. Let be a sequence of positive integers. First denote the sets of indices for . Consider the (doubly indexed) intervals and in and , respectively, defined in the following way:

where

for any and (the first sum on the right in (29) and (30) is meant to be equal to for ). Now define the set by

It is proved in [56, 57] (see also [58]) and in [63] that the set defined in (31) is compatible with itself. Modifying suitably the previous formulas we can obtain examples of various kinds of compatibility of the set with itself, for example, polynomial compatibility or -compatibility considered in the next section. For details see [63] (see also [56–58]).

One can extend case to in various ways, for example, constructing sets contained in certain infinite spirals in (see, e.g., the spiral in Figure 3 for and ). Examples of a -dimensional set which is compatible with itself and unbounded in each direction of can be obtained in this way (see [63]).

Figure 3: Spiral.

6. Compatibility of Support and Existence of Convolution

The condition of compatibility (see Definition 16) of the supports of locally integrable functions is sufficient for existence of their convolution in (see, e.g., [7, p. 124]; see also [5, 6]).

Theorem 18. Let . If the supports of the functions and are compatible sets, then the convolution exists in .

The condition of compatibility of supports is also sufficient for existence of the convolution in the space of distributions. The following result is very well known (see, e.g., [7, p. 156]; see also [5, 6]).

Theorem 19. Let . If the supports of the distributions and are compatible sets, then exists in and .

However, it follows from Theorem 3 that the counterpart of Theorem 19 for the convolution in of two tempered distributions is not true under the assumption of compatibility of their supports. In particular, the compatibility of supports of two functions of the class does not guarantee that their convolution belongs to this class. This requires a suitable modification of the notion. In [56–58], the following modification of compatibility was introduced.

Definition 20. Two sets are polynomially compatible, if there is a positive polynomial on such that the following implication holds:

The assumption of polynomial compatibility of supports is sufficient for the convolution of two measurable slowly increasing functions to exist in and to represent a function of the same class (see [56–58]):

Theorem 21. Let . If the supports of the functions and are polynomially compatible sets, then exists in and .

Under the assumption of polynomial compatibility of supports the result concerning the convolution in is analogous to Theorem 19. Namely, the condition is sufficient for existence of the convolution of two tempered distributions in [56, 57] (see also [50, 58]).

Theorem 22. Let . If the supports of the tempered distributions and are polynomially compatible sets, then the convolution exists in and .

The last result was extended in different ways to the case of distributions in the Gelfand-Shilov spaces in [60–62] (see also [76]).

There exist results on the existence of the convolution in of Beurling ultradistributions and of the convolution in of Beurling tempered ultradistributions, analogous to Theorems 19 and 22. Let us recall the following existence result concerning the convolution in of Beurling ultradistributions, proved in [59].

Theorem 23. Let . If the supports of the Beurling ultradistributions and are compatible sets, then the convolution exists in and .

In [59], a modification of compatibility condition corresponding to the space of Beurling tempered ultradistributions was introduced via the associated function for the sequence . We present it below in a slightly relaxed form (see [77]).

Definition 24. Two sets are -compatible, if there is a constant such that for all and .

The following theorem is true (compare [59] and [77]).

Theorem 25. Let . If the supports the Beurling tempered ultradistributions and are -compatible sets, then the convolution exists in and .

7. Inverse Results

It should be noted that the conditions of compatibility and polynomial compatibility of supports of distributions and tempered distributions, assumed in Theorems 19 and 22, are in a sense optimal in terms of supports for the existence of the convolutions in and in , respectively, and they cannot be relaxed. The situation is precisely described by Theorems 26 and 27 formulated below which were proved in [78].

Theorem 26. Let . Suppose that the convolution exists in for each pair , of distributions such that and . Then and are compatible.

Theorem 27. Let . Suppose that the convolution exists in for each pair of tempered distributions such that and . Then and are polynomially compatible.

We prove next an analogue of Theorem 26 for the case of Beurling ultradistributions.

Theorem 28. Let . Assume that the convolution exists in for each pair of Beurling ultradistributions with the supports contained in and , respectively. Then and are compatible.

Proof. Assume that and are not compatible. This assumption implies that there exist sequences and of elements of sets and , respectively, such that

It is clear that we may assume, in addition, that

Let us define as measures of the form

where denotes the Dirac measure in concentrated at the point , that is, the Beurling ultradistribution defined by for all . Obviously, supports of and are contained in and , respectively.

Choose nonnegative functions and , identically equal to on the balls and , respectively. Then define the functions by

Hence

for , and and . Clearly, .

For sufficiently large , we have the inclusion . For such indices , due to (34)–(37), we get

where is the number of the indices for which . Since as , we conclude that the convolution does not exist in , which contradicts the assumption. Thus the assertion of the theorem is proved.

Whether Theorem 27 has its counterpart for Beurling tempered ultradistributions remains an open problem.

Problem 29. Let . Assume that the convolution exists in for each pair , of Beurling ultradistributions with the supports contained in and , respectively. Do the sets and have to be -compatible?

Let us return now to the case of the convolution in of locally integrable functions. We will modify for this case the notion of compatible sets expressed by means of certain equivalent conditions (cf. [79]).

Proposition 30. Let and be measurable sets in . The following conditions are equivalent.

(i)For each there exists an such that for all .(ii)For each there is an such that

(iii)For arbitrary sequences and of positively bounded subsets of and , respectively, the following implication holds:

(iv)For arbitrary sequences and of massive points of the sets and , respectively, the following implication holds:

Definition 31. One calls the sets quasi-compatible if one of conditions (i)–(iv) is satisfied.

It is clear that if measurable sets are compatible, then they are quasi-compatible, but not conversely.

The proof of Proposition 30 is based on the two following lemmas (see [79]).

Lemma 32. If a measurable set is positively bounded, then for every with there exists a massive point of such that .

Lemma 33. For any two measurable sets and in such that and there exists a pair , where is a point in and is a measurable subset of with , having the following property: for every there exists a measurable subset of with such that the inclusions hold for all .

The following assertion is an easy corollary from Theorem 18.

Theorem 34. If and the supports of the functions and are (contained in) quasi-compatible sets, then the convolution exists and represents a locally integrable function.

The condition of quasi-compatibility of measurable sets in can be described in the following way in the context of the existence of the convolutions in the space of locally integrable functions (cf. [79]).

Theorem 35. Let and be measurable sets in . Suppose that convolution exists and represents a locally integrable function in for any whose supports are contained in and , respectively. Then and are quasi-compatible.

Proof. Suppose that and are not quasi-compatible sets in . Then, by Proposition 30, one can find two sequences and of massive points of the sets and , respectively, such that

We may assume, in addition, that

Let and be the closed balls in of radius 1 with centers at and , respectively, for . Let be the closed ball with the center at and radius . Put

By (42), (43), and Proposition 30, we have for all and, moreover, and for ; in addition, and for all .

Define the functions , and in the following way:

where and for . According to (44) and (43), and are well defined nonnegative locally integrable functions with the supports contained in and , respectively. Moreover, and . Hence

which contradicts our assumption that exists in and thus the proof of our assertion is completed.

Acknowledgments

The authors thank the referees for valuable comments which helped in improving the paper. Special thanks are addressed to Mr. K. Majchrowski for preparing the figures placed in the text. This work was partly supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge.

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