Convolution of distributions in sequential approach

Abstract

We consider several general sequential conditions for convolvability of two Roumieu ultradistributions on R d in the space D ′{Mp} 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} and U {Mp} of approximate units and corresponding sequential conditions of integrability of Roumieu ultradistributions, are analogous to the known convolvability conditions in the space D ′ of distributions and in the space D ′(Mp) of ultradistributions of Beurling type. Moreover, the following property of the convolution and ultradifferential operator P (D) of class {M p } is proved: if S, T ∈ D ′{Mp} (R d ) are convolvable, then P (D)(S * T ) = (P (D)S) * T = S * (P (D)T ).

Equivalent Conditions for Integrability of Distributions

“…We prove in [18] that all the sequential definitions are equivalent to the definition of the general convolution of Roumieu ultradistributions in the sense of S. Pilipović and B. Prangoski [21]. Our proof is based on the integrability result proved in the previous section.…”

“…The approximate units are used in several sequential definitions of the convolvability and of the convolution of Roumieu ultradistributions (Definitions 6 and 7 in section 7). We prove their equivalence in [18]. Moreover, we consider in section 8 the notion of multi-convolution of Roumieu ultradistributions and formulate some results concerning associativity of convolution of Roumieu ultradistributions analogous to those given in [13] in the Beurling case.…”

“…The aim of this paper is to present a characterization of integrable Roumieu ultradistributions (Theorem 1), which allows us to formulate sequential conditions of convolvability of two Roumieu ultradistributions analogous to those used in the sequential theories of the convolution of distributions (see [26], [3], [9] and [17]) and of ultradistributions of Beurling type (see [11], [12] and [2]). The sequential conditions of integrability are based on two types of Rapproximate units (Definition 3 and Definition 4), being counterparts of the approximate units in the sense of Dierolf and Voigt (see [3]).…”

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]).…”

A sequential approach to ultradistribution spaces

“…The difference is intrinsic and requires distinct techniques: instead of polynomials we have to use functions of sub-exponential growth and apply their specific properties. In showing that the sequential approach is equivalent to the classical approach to ultradistribution theory (see [12][13][14]) one needs to know intrinsic structures of ultradistributions as well as of tempered ultradistributions (see [5][6][7][8][9][10][17][18][19][20][21][22][23][24][25]); the equivalence of the two approaches will be proved through Hermite expansions and certain structural properties of tempered ultradistributions.…”