The following definition of a non-commutative neutrix product of distributions on R'
Definition 1. Let f and g be arbitrary distributions and let gn = g * Sn , Sn = ne(nz) 12 = 1,2, . . .
Where Q be a fixed infinitely differentiable function having the properties (1) e(z) = 0 for 1x1 2 1, (2) e ( 4 2 0, (3) (4) e ( 4 = e( -4 9 j e ( x ) d r = 1. -1
The neutrix product f o g of f and g is defined by the neutrix limit (f o g, CJ) = N-lim (jgn, C J ) = N-lim ( j , g.@) n-w Il--fW for all test functions CJ with compact support contained in the open interval (a, a), where N is the neutrix having domain N' = {1,2, ..., n, .. .) and range N" the real numbers with negligible functions linear sums of the functions dln'-1n, lnrn for 1 > 0 and r = 1,2, ... and all functions f(n) for which lim f ( n ) = 0. n-M, The definition of the neutrix limit was given by J. G . VAN DER CORPUT [4] : Definition 2. A neutrix N is a commutative additive group of functions U(E) defined on a domain N' with values in an additive group N", where further if for some U in N , U(E) = y for all f in N', then y = 0. The functions in N are called negligible