On the convolution in the space of tempered ultradistributions of Beurling type

Existence theorems on convolution of functions, distributions and ultradistributions

“…In [14], a certain modification of compatibility condition, corresponding to the space S (M p ) (R d ), was given via the associated function M for the sequence (M p ). We present it here in a slightly relaxed form: [15]).…”

“…The space S (M p ) (R d ) of all Beurling tempered ultradistributions is meant as the strong dual of the space S (M p ) (R d ) defined above (see [22], [14], [2]). Since D (M p ) (R d ) is dense in S (M p ) (R d ) and the inclusion mapping is continuous, so S (M p ) (R d ) can be embedded into the space D (M p ) (R d ).…”

“…Since D (M p ) (R d ) is dense in S (M p ) (R d ) and the inclusion mapping is continuous, so S (M p ) (R d ) can be embedded into the space D (M p ) (R d ). For other properties of the space S (M p ) we refer to [22], [14] and [2].…”

“…There are various general definitions of the convolutions in D (M p ) of Beurling ultradistributions (see [12]) and in S (M p ) of Beurling tempered ultradistributions (see [14]). They are counterparts of the known general definitions of the convolutions in D and in S (see section 5).…”

“…That the mentioned definitions of the convolution in D (M p ) of Beurling ultradistributions are equivalent and that the corresponding definitions of the convolution in S (M p ) of Beurling tempered ultradistributions are equivalent was proved in [12] and [14], respectively (see also [2]).…”