TY - JOUR
AU - Pym, John S.
AB - By JOHN S. PYM [Received 5 March 1963] 1. Introduction THI S paper presents a generalization of th e accepted notions of convolution by rejecting all tha t is not algebraically essential. What remains is a semi- group E on which a translation-invariant linear space 3F of functions is given; convolution appears as a multiplication defined for certain of the linear functionals on SF,\ I t is a natural consequence of this structure that we have not one, but two, multiplications arising, corresponding to translation of elements of 3? on the left and on the right by points of E. The problem of when these multiplications coincide is connected with questions of interchanging the order of limit processes ((3.6) below), and the order of integration (3.3). The principal result (5.2) gives conditions, in terms of topological properties of convolution, for the two multiplications to be equal. 2. Definitions and algebraic consequences Throughout this paper, E will denote some semigroup, tha t is, a set with a binary associative operation. $F will be a linear space of real- or complex- valued functions on E which contains both all the left translates f (x e E) (defined by f (y) = 
TI - The Convolution of Linear Functionals
JF - Proceedings of the London Mathematical Society
DO - 10.1112/plms/s3-14.3.431
DA - 1964-07-01
UR - https://www.deepdyve.com/lp/wiley/the-convolution-of-linear-functionals-TjMLRQt0M9
SP - 431
EP - 444
VL - s3-14
IS - 3
DP - DeepDyve
ER -