Compatibility Conditions and the Convolution of Functions and Generalized Functions

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.

The theory developed by Colombeau (see [38]; see also [39]) and his followers (see, e.g., [40][41][42][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 R  meant in the classical sense of general equivalent definitions introduced independently by several authors (see [2,[44][45][46][47][48][49][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 R  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][57][58][59]; see also [60][61][62][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 R  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 R  .

Preliminaries
We will use the standard multidimensional notation concerning R  and N  0 , 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 R  and but we write also The standard notation will be also used for the known spaces of (complex-valued) functions on , and the known (sub) spaces of distributions on R  : C  (R  ), D  (R  ), and S  (R  ) (cf. [1,7]).
For a set  ⊆ R  we denote   := R  \ .It will be convenient to use the following notation for  ⊆ R  and a function  on R  : ) only in case  = 0.Moreover, the symbol ⊏ R  will be used in the following sense: The words "measure" and "measurable" concerning the sets in R  are considered in the sense of Lebesgue; the Lebesgue measure of a given Lebesgue measurable set  ⊆ R  will be denoted by ℓ  ().If  ⊆ R  is a measurable set, we use the following notation: That is, We call a measurable set for a certain polynomial  on R  (resp., for every polynomial  and some positive constant ()) and for almost all  ∈ R  .The sets of all measurable slowly increasing and rapidly decreasing functions on R  will be denoted by P(R  ) and R(R  ), respectively.
In our considerations concerning the convolution in the spaces D (  ) (R  ) of ultradistributions and S (  ) (R  ) of tempered ultradistributions of Beurling type (the respective definitions and notation are introduced in Section 4) we always assume that a given sequence (  ) = (  ) ∈N 0 of positive numbers satisfies the following three conditions (see [10-12, 18, 51, 52, 59]): where  > 0 and  > 0 are some constants.We extend the sequence (  ) ∈N 0 to its multidimensional version (  ) ∈N  0 as follows: By the associated function for the sequence (  ) we mean the function on [0, ∞) given by  () := sup where log + := max{log , 0} and (0) := 0.
However, in general, the existence of the convolution of functions in  1 loc 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  1  loc , but their convolution may be a function of arbitrarily fast increase at infinity.

Theorem 3. Let Φ be an arbitrary continuous (complexvalued) function on R 𝑑 . There exists a nonnegative 𝐶
the convolution  *  exists everywhere in R  , 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  ∈ R  ; the convergence in (9) can be slow: the faster the increase of the function Φ (i.e., the required increase of the convolution  * ) is at infinity the slower the approach of  to 0 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 D  and in D (  ) (see Definitions 5 and 12) of elements of the subspaces S  (R  ) and S (  ) (R  ) of the spaces D  (R  ) and D (  ) (R  ) 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][6][7]) that distributions are elements of the strong dual D  (R  ) of the basic space D(R  ) of test functions, that is, linear continuous functionals on the space D(R  ) defined as the inductive limit: where the symbol  ⊏⊏ R  means here that  are compact sets growing up to R  .Recall that D  (R  ), for a fixed  ⊏ R  , means the space of all  ∞ functions on R  whose support is contained in , with the topology given by the family {‖ ⋅ ‖ , :  ∈ N 0 } of the norms defined as follows: Locally integrable functions on R  may be treated as distributions (so-called regular distributions), because the space  1 loc (R  ) can be naturally embedded into D  (R  ).The known structural theorem describes every distribution as a derivative (in the distributional sense) locally, that is, on each relatively compact set in R  , of finite order of a continuous function (see [1,[5][6][7]).
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 D(R  ) is said to be a strong approximate unit on R  if for every  ⊏ R  there exists an  0 ∈ N such that   () = 1 for  ∈  and  ≥  0 (hence One denotes the set of all strong approximate units on R  by U(R  ).(15) whenever the above limit exists for every strong approximate unit (  ) ∈ U(R 2 ) and  ∈ D(R  ).One says then that the convolution  *  exists in D  .
By the Lebesgue theorem, if ,  ∈  1 loc (R  ) and the convolution  *  exists in  1 loc , then it exists in D  and represents a regular distribution in D  (R  ) (see [6, p. 63]).
Analogously to and independently of Definition 5, one may define the convolution in S  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 S  , namely, the respective counterpart of the aforementioned sequential definition of Vladimirov (cf.[6,46]).Definition 6.For given ,  ∈ S  (R  ) one defines the convolution  *  in S  by ⟨ * , ⟩ := lim (16) whenever the above limit exists for every strong approximate unit (  ) ∈ U(R 2 ) and  ∈ S(R  ).One says then that the convolution  *  exists in S  .
In [45], Shiraishi posed the problem of whether the assumption that the convolution  *  of two tempered distributions ,  ∈ S  (R  ) exists in D  implies the existence of the convolution  *  in S  (in particular, whether  *  ∈ S  (R  )).
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 R 1 , such that the convolution  *  exists in D  , but  *  ∉ S  (R 1 ).
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 Φ the function defined by (or any continuous function of even faster increase).Clearly, Φ represents a distribution which is not tempered, but from Theorem 3 it follows that there exists a bounded (even vanishing at infinity)  ∞ function  on R  representing a tempered distribution for which the convolution  *  exists everywhere and exceeds everywhere on R  the function Φ.Consequently,  *  represents a distribution but not a tempered distribution.

Convolution of Beurling Ultradistributions
We will recall the definitions of the space of D (  ) (R  ) of Beurling ultradistributions (see [9-12, 18, 51]) and the space S (  ) (R  ) 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 (.1)-(.3)(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 ℎ > 0 and a regular compact subset  of R  (see [10][11][12]), we define the space D (  ) ,ℎ (R  ), consisting of all functions  from E(R  ) with support contained in  such that         ,ℎ := sup with the topology induced by the above norm.Then the basic space D (  ) (R  ) of test functions is defined by means of the projective and inductive limits as follows: where the symbol  ⊂⊂ R  means that  are regular compact sets growing up to R  .In addition, for a fixed  > 0, we denote by S where the symbol ⟨⋅⟩ is defined in (1), equipped with the topology induced by the above norm  ,2 .Then we define The strong dual of D (  ) (R  ), denoted by D (  ) (R  ), 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  ∈ D (  ) (R  ).Then, for each open, relatively compact set  in R  , there are measures   ∈ C  () for  ∈ N  0 and positive constants  and  such that The space S (  ) (R  ) of all Beurling tempered ultradistributions is meant as the strong dual of the space S (  ) (R  ) defined previously; it was introduced by Pilipovi ć in [73] (see also [18,52,59]).Since D (  ) (R  ) is dense in S (  ) (R  ) 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 D (  ) of Beurling ultradistributions and S (  ) of Beurling tempered ultradistributions we refer to [18,51,52,59,60,[73][74][75].
There are various general definitions of the convolution in D (  ) of Beurling ultradistributions (see [51]) and of the convolution in S (  ) of Beurling tempered ultradistributions (see [52]).They are suitable counterparts of the known general definitions of the convolution in D  and the convolution in S  mentioned in Section 3. The fact that the mentioned definitions of the convolution in D (  ) of Beurling ultradistributions are equivalent and that the corresponding definitions of the convolution in S (  ) 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 D (  ) and in S (  ) which correspond to Vladimirov's definition of the convolution in D  and in S  , respectively.The definitions are based on the notions of strong D (  ) -approximate unit and strong S (  )approximate unit.Definition 9. A sequence (  ) of elements of D (  ) (R  ) is said to be a strong D (  ) -approximate unit on R  if for every  ⊏ R  there exists an  0 ∈ N such that   () = 1 for  ∈  and  ≥  0 (hence (R  ) of strong S (  ) -approximate units contains sufficiently many sequences, for example, all sequences of the form where  is a function of class D (  ) (R  ) or S (  ) (R  ), respectively, such that  = 1 in some neighbourhood of 0.
Vladimirov's version of the definition of the convolution in D (  ) of Beurling ultradistributions has the following form.
Definition 12.For given Beurling ultradistributions ,  ∈ D (  ) (R  ) the convolution  *  in D (  ) is defined by whenever the limit in (26) exists for every strong approximate Analogously, the convolution in S (  ) of Beurling tempered ultradistributions can be defined as follows.

Compatibility Conditions
Let us present various forms of the condition imposed on sets ,  ⊆ R  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 R  (see, e.g., [5, p. 383]).Proposition 14.Let ,  ⊆ R  be arbitrary closed sets.The following conditions are equivalent: The meaning of the conditions for  = 1 can be seen in Figure 1 (where  and  mean R  ).As proved in [5, pp. 383-384], conditions (Σ)-(Σ  ) are equivalent if the sets  and  are closed in R  .In general, we have the following equivalence (see [76]).
Proposition 15.Let ,  ⊆ R  be arbitrary sets.The following conditions are equivalent: If the sets ,  ⊆ R  are closed, then each of the above conditions is equivalent to each of conditions (Σ)-(Σ  ).
Definition 16.Two sets ,  ⊆ R  are called compatible if one of equivalent conditions (Σ  )-() is satisfied.
There are two well known particular cases of compatible sets  and  in R 1 : (1 ∘ ) at least one of the sets  and  is bounded; (2 ∘ ) both  and  are bounded from the left or both from the right.
Case (1 ∘ ) extends clearly to R  for  > 1 and case (2 ∘ ) can be described in R  in the following form: ,  ⊂ R  are (or are contained in) suitable cones with vertices at 0 such that  is an open convex cone and  ⊂  * , where A * means the cone dual to  (see [18,
Another, less obvious, case of compatible sets in R 1 is the following: (3 ∘ ) both sets  and  in R 1 are unbounded from both sides, that is, unbounded both from below and from above in R 1 .
In [63] (see also [56][57][58]) various pairs of compatible sets  and  in R 1 , both unbounded from both sides, are considered; for  =  in particular.They were constructed as countable sums of intervals (of length 1) situated in R 1 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 (  ) ∈N 0 be a sequence of positive integers.First denote the sets   of indices   := {0, 1, . . .,   − 1} for  ∈ N 0 .Consider the (doubly indexed) intervals  , and  , in (−∞, 0) and (0, ∞), respectively, defined in the following way: where , := for any  ∈   and  ∈ N 0 (the first sum on the right in ( 29) and ( 30) is meant to be equal to 0 for  = 0).Now define the set  ⊂ R 1 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][57][58]).One can extend case (3 ∘ ) to R  in various ways, for example, constructing sets contained in certain infinite spirals in R  (see, e.g., the spiral in Figure 3 for  = 2 and  = ).Examples of a -dimensional set which is compatible with itself and unbounded in each direction of R  can be obtained in this way (see [63]).

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  1 loc (see, e.g., [7, p. 124]; see also [5,6]).
Theorem 18.Let ,  ∈  1 loc (R  ).If the supports of the functions  and  are compatible sets, then the convolution  *  exists in  1 loc .
The condition of compatibility of supports is also sufficient for existence of the convolution in the space D  (R  ) of distributions.The following result is very well known (see, e.g., [7, p. 156]; see also [5,6]).However, it follows from Theorem 3 that the counterpart of Theorem 19 for the convolution in S  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 P(R  ) does not guarantee that their convolution belongs to this class.This requires a suitable modification of the notion.In [56][57][58], the following modification of compatibility was introduced.
The assumption of polynomial compatibility of supports is sufficient for the convolution of two measurable slowly increasing functions to exist in  1 loc and to represent a function of the same class (see [56][57][58]): Theorem 21.Let ,  ∈ P(R  ).If the supports of the functions  and  are polynomially compatible sets, then  *  exists in  1 loc and  *  ∈ P(R  ).
Under the assumption of polynomial compatibility of supports the result concerning the convolution in S  is analogous to Theorem 19.Namely, the condition is sufficient for existence of the convolution of two tempered distributions in S  [56,57] (see also [50,58]).
Theorem 22.Let ,  ∈ S  (R  ).If the supports of the tempered distributions  and  are polynomially compatible sets, then the convolution  *  exists in S  and  *  ∈ S  (R  ).
The last result was extended in different ways to the case of distributions in the Gelfand-Shilov spaces K  (  ) in [60][61][62] (see also [76]).
There exist results on the existence of the convolution in D (  ) of Beurling ultradistributions and of the convolution in S (  ) of Beurling tempered ultradistributions, analogous to Theorems 19 and 22.Let us recall the following existence result concerning the convolution in D (  ) of Beurling ultradistributions, proved in [59].
In [59], a modification of compatibility condition corresponding to the space S (  ) of Beurling tempered ultradistributions was introduced via the associated function  for the sequence (  ).We present it below in a slightly relaxed form (see [77]).

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 D  and in S  , respectively, and they cannot be relaxed.The situation is precisely described by Theorems 26 and 27 formulated below which were proved in [78].
For sufficiently large , we have the inclusion   (, 1) ⊂   (0, ).For such indices , due to (34)-(37), we get where   is the number of the indices  for which   ,   ∈   (0, ).Since   → ∞ as  → ∞, we conclude that the convolution  *  does not exist in D (  ) , 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 ,  ⊆ R  .Assume that the convolution  *  exists in S (  ) for each pair ,  ∈ S (  ) (R  ) 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  1 loc 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 R  .The following conditions are equivalent.
(ii) For each  ⊏ R  there is an  ⊏ R  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 ,  ⊂ R  quasi-compatible if one of conditions (i)-(iv) is satisfied.
It is clear that if measurable sets ,  ⊂ R  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  ⊂ R  is positively bounded, then for every  with 0 <  < || ∞ there exists a massive point  of  such that || ≥ || ∞ − .Lemma 33.For any two measurable sets  and  in R  such that ℓ  () > 0 and ℓ  () > 0 there exists a pair (, ), where  is a point in R  and  is a measurable subset of  with ℓ  () > 0, having the following property: for every  ∈  there exists a measurable subset   of  with ℓ  (  ) > 0 such that the inclusions  +  −   ⊆  hold for all  ∈ .
The following assertion is an easy corollary from Theorem 18.
Theorem 34.If ,  ∈  1 loc (R  ) 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 R  can be described in the following way in the context of the existence of the convolutions in the space  1 loc (R  ) of locally integrable functions (cf.[79]).
Theorem 35.Let  and  be measurable sets in R  .Suppose that convolution  *  exists and represents a locally integrable function in R  for any ,  ∈ Let (  ) and (  ) be the closed balls in R  of radius 1 with centers at   and   , respectively, for  ∈ N. Let  be the closed ball with the center at 0 and radius  + 2. Put   :=  ∩  (  ) ,   :=  ∩  (  ) , for  ∈ N. (44) By ( 42), (43), and Proposition 30, we have   +   ⊆  for all  ∈ N and, moreover,   ∩   = 0 and   ∩   = 0 for  ̸ = ; in addition, ℓ  (  ) > 0 and ℓ  (  ) > 0 for all  ∈ N. Define the functions , , and  in the following way: where   = [ℓ  (  )] −1 and   = [ℓ  (  )] −1 for  ∈ N. According to ( 44) and ( 43),  and  are well defined nonnegative locally integrable functions with the supports contained in  and , respectively.Moreover,  ∈  1 loc and supp  =  ⊏ R  .Hence which contradicts our assumption that  *  exists in  1 loc and thus the proof of our assertion is completed.

Theorem 19 .
Let ,  ∈ D  (R  ).If the supports of the distributions  and  are compatible sets, then  *  exists in D  and  *  ∈ D  (R  ).
)  ()   ( + )]   ≥ and, in addition, if there exists a positive constant ℎ such that sup One denotes the set of all strong D (  ) -approximate units on R  by U (  ) (R  ).If in the previous definition the assumption that   ∈ D (  ) (R  ) for  ∈ N is replaced by   ∈ S (  ) (R  ) for  ∈ N and the remaining assumptions are preserved, then the sequence (  ) is called a strong S (  ) -approximate unit.One denotes the set of all strong S (  ) -approximate units on R  by U Remark 11.According to the known Denjoy-Carleman-Mandelbrojt theorem, the previously defined class U (  ) (R  ) of strong D (  ) -approximate units as well as the class U (  )

Theorem 26 .
Let ,  ⊆ R  .Suppose that the convolution  *  exists in D  for each pair ,  ∈ D  (R  ) of distributions such that supp  ⊆  and supp  ⊆ .Then  and  are compatible.Let ,  ⊆ R  .Suppose that the convolution  *  exists in S  for each pair ,  ∈ S  (R  ) of tempered distributions such that supp  ⊆  and supp  ⊆ .Then  and  are polynomially compatible.Let ,  ⊆ R  .Assume that the convolution  *  exists in D (  ) for each pair ,  ∈ D (  ) (R  ) 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 lim (  +   ) =  in R  for some  ∈ R  .
1 loc (R  ) whose supports are contained in  and , respectively.Then  and  are quasicompatible.Proof.Suppose that  and  are not quasi-compatible sets in R  .Then, by Proposition 30, one can find two sequences (  ) and (  ) of massive points of the sets  and , respectively,           = lim           = ∞,  := sup ∈N       +       < ∞.We may assume, in addition, that      +1     −           > 2,      +1     −           > 2, for  ∈ N.