Existence theorems on convolution of functions, distributions and ultradistributions

Compatibility Conditions and the Convolution of Functions and Generalized Functions

“…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.…”

“…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 Pilipović in [73] (see also [18,52,59]). Since D ( ) (R ) is dense in S ( ) (R ) and the inclusion mapping is continuous, we have S ( ) (R ) ⊂ D ( ) (R ).…”

“…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.…”

A sequential approach to the convolution of Roumieu ultradistributions

“…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]).…”

Equivalent Conditions for Integrability of Distributions

“…The first one says that if the supports of Roumieu ultradistributions satisfy the known condition of compatibility, known also as condition (Σ) (see [8], p. 383 and [1], p. 124), then the convolution of these ultradistributions exists in D ′{Mp} d . The proof is based on the known representation theorem fo Roumieu ultradistributions (see [14], Theorem 8.7) and on the idea in the proof of Theorem 2 from [12] concerning ultradistributions of Beurling type. It should be noticed that the condition of compatibility is optimal in terms of supports of Roumieu ultradistributions, according to the second theorem (which follows from Theorem 5.1 proved in [10].…”