A sequential approach to the convolution of Roumieu ultradistributions

We consider several general sequential conditions for convolvability of two Roumieu ultradistributions on Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^d$$\end{document} in the space D′{Mp}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}'^{\{M_p\}}$$\end{document} and prove that they are equivalent to the convolvability of these ultradistributions in the sense of Pilipović and Prangoski. The discussed conditions, based on two classes U{Mp}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {U}}}^{\{M_p\}}$$\end{document} and U¯{Mp}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{{\mathbb {U}}}^{\{M_p\}}$$\end{document} of approximate units and corresponding sequential conditions of integrability of Roumieu ultradistributions, are analogous to the known convolvability conditions in the space D′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}'$$\end{document} of distributions and in the space D′(Mp)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}'^{(M_p)}$$\end{document} of ultradistributions of Beurling type. Moreover, the following property of the convolution and ultradifferential operator P(D) of class {Mp}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{M_p\}$$\end{document} is proved: if S,T∈D′{Mp}(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S, T \in \mathcal{D}'^{\{M_{p }\}}({\mathbb {R}}^d)$$\end{document} are convolvable, then P(D)(S∗T)=(P(D)S)∗T=S∗(P(D)T).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P(D)(S*T) = (P(D)S)*T = S*(P(D)T). \end{aligned}$$\end{document}


Introduction
Deep investigations of the convolution of two ultradistributions of Roumieu type (that we call shorter Roumieu ultradistributions) in the non-quasianalytic case were carried out via e-tensor product by Pilipović and Prangoski in [19] and, with important improvements, by Dimovski et al. in [7]. The authors gave there general functional definitions and proved fundamental results on convolvability and the convolution of Roumieu ultradistributions in a way analogous to the known general approaches of Chevalley and Schwartz in case of distributions. For other aspects of the theory, see, e.g., [1,2,4,6,8,20,24]. See also the recent article [21] for results concerning the quasianalytic case.
The aim of this paper is to discuss sequential conditions playing a similar role in the study of the convolution of Roumieu ultradistributions to those used in the sequential theories of the convolution of distributions (see [5,9,15,23]) and ultradistributions of Beurling type (see [1,10,11]). The conditions are based on two types of R-approximate units (Definitions 4.2 and 4.3), being the counterparts of the approximate units in the sense of Dierolf and Voigt (see [5]). The respective classes U fM p g and U fM p g of R-approximate units are used in a sequential characterization of integrable Roumieu ultradistributions (see [16]), analogous to that proved by Pilipović in [18] in case of integrable ultradistributions of Beurling type. As a consequence, we give in this paper several sequential definitions of the convolution of Roumieu ultradistributions (Definition 7.2). We prove in Theorem 7.1, that all our sequential definitions of the convolution of Roumieu ultradistributions are equivalent to those given in [19] and [7]. An important application of the notion of R-approximate units is presented in the proof of Theorem 8.1, describing a non-trivial property of the convolution of Roumieu ultradistributions and ultradifferential operators of the class fM p g.
It is worth to recall that Pilipović in [18] used a different class of approximate units to prove the same property in case of the convolution of ultradistributions of Beurling type. Our proof of Theorem 8.1 is based on similar ideas but discussions concerning the class R play an essential role in our case.

Preliminaries
We consider complex-valued C 1 -functions and Roumieu ultradistributions defined on R d (or on an open subset of R d ) using the standard multi-dimensional notation in R d .
To mark the dimension of R d , which is essential in some situations, we denote the considered spaces of test functions and the corresponding spaces of Roumieu ultradistributions simply by adding the index d at the end of the respective symbol. Moreover, if necessary, the constant function 1 on R d will be denoted by 1 d and the value of T 2 D 0 fM p g d on u 2 D fM p g d by hT; ui d .
The spaces of test functions and Roumieu ultradistributions are defined by a given sequence ðM p Þ p2N 0 of positive numbers. Usually some of the following conditions are imposed on the sequence ðM p Þ: for certain constants A [ 0 and H [ 0. We can and will assume that H ! 1. Clearly, conditions (M. For simplicity, we will assume in the whole paper that the sequence ðM p Þ satisfies the three conditions (M.1), (M.2) and (M.3), not discussing which of them can be weakened or omitted in the formulations of presented theorems.
It follows, by induction, from (M.1) that M p Á M q M 0 M pþq for p; q 2 N 0 (see [16]). Under the assumption that M 0 ¼ 1, which we adopt hereinafter for simplicity, the last inequality admits the form: It will be convenient to extend the sequence ðM p Þ p2N 0 to (M k Þ k2N d 0 by means of the formula: Due to the extended notation, we immediately get the extended version of inequality (2.1): The associated function of the sequence ðM p Þ is given by For an arbitrary k ¼ ðk 1 ; . . .; k d Þ 2 N d 0 denote by D k the differential operator of the form An essential role in our considerations will be played by Komatsu's lemma proved in [14] (see Lemma 3.4 and Proposition 3.5) in which numerical sequences monotonously increasing to infinity are involved. The class of such sequences ðr p Þ p2N 0 (with r 0 ¼ 1) was denoted by R in [19] and [7] and we preserve this notation in our paper.
For every ðr p Þ 2 R we call ðR p Þ the product sequence corresponding to ðr p Þ if its elements are of the form R p :¼ Q p i¼0 r i for p 2 N 0 (i.e., R 0 ¼ 1). Let us recall Komatsu's lemma in the following equivalent form which emphasizes the symmetry of two assertions: Lemma 2.1 Let ða k Þ k2N 0 be a sequence of nonnegative numbers.
(I) The following two conditions are equivalent: (II) the following two conditions are equivalent: where ðR k Þ is the product sequence corresponding to the sequence ðr k Þ 2 R: The above lemma can be easily extended to the d-dimensional version concerning sequences ða k Þ k2N d 0 of nonnegative numbers.
It is worth noticing that Lemma 2.1 delivers two simple characterizations (dual to each other): 1 of slowly increasing sequences (i.e., satisfying ðA 1 Þ), 2 of rapidly decreasing sequences (i.e., satisfying ðA 2 Þ) of nonnegative numbers. They are expressed through respective properties of sequences, described by product sequences corresponding to sequences of the class R.
In what follows we will also apply the following simple lemma (see [16]): Lemma 2.2 For every ðr p Þ 2 R, the following inequality holds: where ðR p Þ is the product sequence corresponding to ðr p Þ.
For a given sequence ðM p Þ, a regular compact set K in R d and h [ 0, the symbol E fM p g K;h;d will mean the locally convex space (l.c.s.) of all C 1 -functions u on R d such that with the topology defined by the semi-norm q K;h given above, while the symbol D fM p g K;h;d will mean the Banach space of all C 1 -functions u satisfying (3.1) and having supports contained in K, with the topology of the norm q K;h in (3.1).
For a fixed sequence ðM p Þ, we consider the following locally convex spaces of ultradifferentiable functions on R d : with the norm k Á k K;ðr p Þ defined above.
The following result is essentially due to Komatsu [14] (see also [4,16]), since it is a consequence of his Lemma 2.1 recalled above.
with the norm k Á k ðr p Þ defined in (3.6). For a given sequence ðM p Þ, the following projective description of the space D fM p g L 1 ;d follows from the results proved in [3,7,21]: where the equality holds in the sense of l.c.s.
In [16], the following assertion concerning the product of functions in D fM p g L 1 ;d is proved: Moreover, for every ðr p Þ 2 R such that r 1 [ 2 the inequality holds: where ðr p Þ=2 is meant in the sense of (2.4).   of Roumieu integrable ultradistributions is a subspace of the space D 0fM p g of Roumieu ultradistributions. Definition 4.2 By an R-approximate unit, we mean a sequence (P n ) of ultradifferentiable functions P n 2 D fM p g d converging to 1 in E fM p g d such that the following property holds for every sequence ðr p Þ 2 R: where ðR p Þ is the product sequence corresponding to ðr p Þ.

Definition 4.3
By a special R-approximate unit, we mean an R-approximate unit ðP n Þ such that for every compact set K & R d , there exists an index n 0 2 N such that P n ðxÞ ¼ 1 for all n ! n 0 and x 2 K.
We denote the class of all R-approximate units on R d by U

Integrability of Roumieu ultradistributions
We formulate below a characterization of integrable Roumieu ultradistributions, in the form of five equivalent conditions, which is an analog of the theorem of Dierolf and Voigt concerning integrable distributions (see [5]) and of the theorem of Pilipović concerning ultradistributions of Beurling type (see [18]). The proof of the theorem is given in [16].
The following conditions are equivalent:

Convolution of Roumieu ultradistributions
Pilipović and Prangoski made in [19] a deep study of the convolution of Roumieu ultradistributions. The study was based on the investigation of the tensor product of the respective spaces of test functions. Let us recall some results proved and observations made in [19].
The authors use the results on the e tensor product from [14] to prove that in the sense of an isomorphism. They consider, analogously to ideas applied in [17] to the convolution of measures, the following semi-norms in the space D fM p g L 1 ;d : Denote by e e D fM p g L 1 ;d the l.c.s. D fM p g L 1 ;d equipped with the topology defined by the family fq g;ðr p Þ : g 2 C 0 ; ðr p Þ 2 Rg of semi-norms and the strong dual of e e D fM p g . Relations between the strong dual of e e D fM p g L 1 ;d and the space D 0fM p g L 1 ;d of integrable Roumieu ultradistributions were studied in [19]. The results obtained in [19] were then improved in [21], where the equality e e D fM p g The obtained results allowed the authors to give in [19] the following definitions of convolvability and the convolution of two Roumieu ultradistributions, analogous to the Schwartz definition of the convolution of distributions (see [22]): If the following condition is satisfied: where u M is the function of the class E The following result on equivalence of convolvability conditions for Roumieu ultradistributions was proved in [19]: The following conditions are equivalent to condition ðSÞ of convolvability for S and T:  Each of the following two conditions is equivalent to condition ðSÞ of convolvability for S and T: Moreover, if any of the conditions ðSÞ, ðc 1 Þ, ðc 2 Þ holds, then for all u 2 D fM p g d .
In the next section, we are going to formulate several sequential conditions of convolvability of Roumieu ultradistributions, which are equivalent to conditions ðSÞ, ðc 1 Þ and ðc 2 Þ.

Sequential convolutions of Roumieu ultradistributions
The notion of R-approximate unit makes it possible to consider several sequential definitions of the convolution of Roumieu ultradistributions based on corresponding sequential conditions of convolvability. The conditions require that respective numerical sequences, corresponding to a given pair of Roumieu ultradistributions via certain approximate units, are Cauchy sequence (Cauchy s. in short) for all approximate units from a given class. The first definition of this kind was given for the convolution of distributions by Vladimirov in [23] and its equivalent versions were discussed in [5] and [9]. Their counterparts for ultradistributions of Beurling type were discussed in [10] (see also [1]).
We will prove in Theorem 7.1 that all the sequential definitions are equivalent to the definition of the general convolution of Roumieu ultradistributions in the sense of Pilipović and Prangoski [19]. Our proof of Theorem 7.1 will be based on the integrability result stated in the previous section.
respectively.  [5,9,23]). It follows from Theorem 5.1 that they are equivalent to all the conditions listed in Definition 7.1.

Remark 7.2
The convolution of Roumieu ultradistributions in D 0fM p g d investigated in [19] and [7] and its sequential versions discussed in this paper is a general notion.
It embraces various particular cases, e.g., expressed in terms of supports of given Roumieu ultradistributions. where P 2nÀ1 :¼ P 1 n and P 2n :¼ P 2 n for n 2 N.
Proof We will prove the equivalence of convolvability conditions given in Definitions 7.1 and 6.1 and in Theorem 6.2 according to the following scheme of implications:

hS Ã
for all u 2 D fM p g d . Consequently, we have shown the implication ðP 1 Þ ) ðc 1 Þ and that (7.6) is true under the assumption of condition ðP 1 Þ.
From the above, by symmetry, we conclude the implications ðPÞ ) ðP 2 Þ and ðP 2 Þ ) ðc 2 Þ as well as the equalities which are true under conditions ðPÞ and ðP 2 Þ, respectively. By Theorem 6.2, we have ðc 1 Þ , ðc 2 Þ , ðSÞ and the equalities in (6.5) hold true, if any of the three conditions is satisfied. Hence, by (7.6) and (7.8), we have under the assumption of condition (c j ) for j 2 f1; 2g. Notice that also conditions (S) and (V) are equivalent and each of these conditions implies

Ultradifferential property of the convolution
Let us consider an ultradifferential operator P(D) defined by Komatsu in [12] as follows: Definition 8.1 An operator of the form is called an ultradifferential operator of class fM p g if for every L [ 0 there is a constant C L such that Clearly, the condition in Definition 8.1 can be equivalently expressed as follows Then according to Lemma 2.1, part (II), there is a sequence ðu p Þ 2 R such that sup k U jkj M k jc k j À Á \1, where U k ¼ Q p k u p . In other words, an ultradifferential operator of the form (8.1) is of class fM p g if there are C [ 0 and ðu p Þ 2 R such that In Theorem 8.1 below we prove an important and non-trivial property of the convolution of Roumieu ultradistributions. In the proof, we will need the following very useful result from [19]: For every sequence ðs p Þ 2 R, there exists a sequence ðr p Þ 2 R such that r p s p for p 2 N and R pþq 2 pþq R p R q for all p; q 2 N 0 : ð8:3Þ . It suffices to prove that the ultradistributions P(D)S and T are convolvable in D 0fM p g d and that the first equality of (8.4) holds. The remaining part of the assertion follows then directly from Remark 7.5. To prove the convolvability of P(D)S and T, we have to show that the sequence hPðDÞS T; P n u M i ð Þ n2N is convergent. We have Applying (8.1) and Leibniz' rule and then changing the order of summation, we get, for all n 2 N, the equalities where N :¼ N d 0 n fð0; . . .; 0Þg. This means that equations (8.7) hold for m n defined in (8.9) and for all n 2 N. Clearly, ðm n Þ depends on the initial sequence ðP n Þ.
It suffices to show (8.8). Choose h 2 D fM p g d such that hðxÞ ¼ 1 for x 2 supp u. By (8.9), we have hS T; m n i ¼ hðS TÞh M ; m n i; n 2 N: The sequence hðS TÞh M ; P n i À Á n2N is convergent for every ðP n Þ 2 U fM p g 2d , by the assumption that S and T are convolvable in D 0fM p g d . To prove (8.8) it is enough to show that also ðP n þ m n Þ 2 U fM p g 2d . Since ðP n Þ; ðP n Þ 2 U fM p g 2d , for each compact set K in R 2d there exists an n 0 2 N such that D i P n ðx; yÞ ¼ 0 and P n ðx; yÞ ¼ 1 for ðx; yÞ 2 K; i 2 N and n [ n 0 . Consequently, in view of (8.9), P n ðx; yÞ þ m n ðx; yÞ ¼ 1 for ðx; yÞ 2 K; n [ n 0 : Therefore it remains to prove, for every ðt p Þ 2 R, that Fix an arbitrary ðt p Þ 2 R. The coefficients of the ultradifferential operator P(D) satisfy (8.2) for some ðu p Þ 2 R. Putting s k :¼ minft k ; u k g for k 2 N, we have ðs p Þ 2 R. By Lemma 8.1, there exists a sequence ðr p Þ 2 R such that r k s k and inequality (8.3) holds. In addition we assume, according to Remark  According to assumption (8.11), consider the sequences ðr p Þ and ðr p Þ of the class R defined by r p :¼ r p =8H and r p :¼ r p =16H 2 , respectively, for p 2 N. Clearly, 2 2jaþiÀjj ð2HÞ jaþiÀjj kD aþiÀj x P n k 1 R jaþiÀjj M aþiÀj kP n k ðr p Þ and 2 2jbþjj ð2HÞ 2jbþjj kD bþj x u M k 1 R jbþjj M bþj kuk ðr p Þ for all a; b; i; j 2 N d 0 , j a. We deduce from (8.17) and the above estimates that i.e., (8.10) is proved, as required. The assertion of Theorem 8.1 is proved. h Theorem 8.1 has also been shown in the quasianalytic case in the article [21]. The proof there is given via a completely different method (cf. [21,Cor. 5.10]).