On the Noncommutative Neutrix Product of Distributions

Let f and g be distributions and let gn = (g ∗ δn)(x), where δn(x) is a certain sequence converging to the Dirac-delta function δ(x). The noncommutative neutrix product f ◦ g of f and g is defined to be the neutrix limit of the sequence { f gn}, provided the limit h exists in the sense that N-limn→∞〈 f (x)gn(x),φ(x)〉 = 〈h(x),φ(x)〉, for all test functions in . In this paper, using the concept of the neutrix limit due to van der Corput (1960), the noncommutative neutrix products xr + lnx+ ◦ x−r−1 − lnx− and x−r−1 − lnx− ◦ xr + lnx+ are proved to exist and are evaluated for r = 1,2, . . . . It is consequently seen that these two products are in fact equal.


Introduction
Certain operations on smooth functions (such as addition, and multiplication by scalars) can be extended without difficulty to arbitrary distributions.Others (such as multiplication, convolution, and change of variables) can be defined only for particular distributions.We are obliged to impose certain restrictions on the distributions when we try to define a multiplicative operation for distributions.
The technique of neglecting appropriately defined infinite quantities was devised by Hadamard and the resulting finite value extracted from the divergent integral is usually referred to as the Hadamard finite part.In fact, Hadamard's method can be regarded as a particular application of the neutrix calculus developed by van der Corput, see [1,2].This is a very general principle for the discarding of unwanted infinite quantities from asymptotic expansions and has been widely exploited in the context of distributions, by Fisher in connection with the problem of distributional multiplication, see [3][4][5][6] or [7].Definition 1.1.Let f be a distribution in Ᏸ and let α be an infinitely differentiable function.The product α f is defined by for all functions φ in Ᏸ.
The first extension of the product of a distribution and an infinitely differentiable function is the following, see, for example, [10].Definition 1.2.Let f and g be distributions in Ᏸ for which on the interval (a,b), f is the rth derivative of a locally summable function F in L p (a,b) and g (r) is a locally summable function in L q (a,b) with 1/ p + 1/q = 1.Then the product f g = g f of f and g is defined on the interval (a,b) by Now let ρ be a fixed infinitely differentiable function having the following properties: We define the function δ n (x) = nρ(nx) for n = 1,2,....It is obvious that {δ n } is a regular sequence of infinitely differentiable functions converging to the Dirac delta function δ(x).Now let f be an arbitrary distribution and define the function f n by Then f n is a sequence of infinitely differentiable functions converging to the distribution f .The next definition for the product of two distributions, given in [11], is in general noncommutative and generalizes Definition 1.However, there are still many pairs of distributions whose products do not exist by Definition 1.3.
We need the following definitions of the neutrix and the neutrix limit to define the product for a considerably larger class of pairs of distributions, see [1,4,2].Definition 1.4.A neutrix N is defined as a commutative additive group of functions ν(ξ) defined on a domain N with values in an additive group N , where further if for some ν in N,ν(ξ) = γ for all ξ ∈ N , then γ = 0.The functions in N are called negligible functions.Definition 1.5.Let N be a set contained in a topological space with a limit point b which does not belong to N .If f (ξ) is a function defined on N with values in N and it is possible to find a constant c such that f (ξ) − c ∈ N, then c is called the neutrix limit of f as ξ tends to b and write N-lim ξ→b f (ξ) = c.
The following definition for the noncommutative product of two distributions was given in [4] and generalizes Definitions 1.2 and 1.3.Definition 1.6.let f and g be arbitrary distributions and let g n = g * δ n .The neutrix product f • g of f and g exists and is equal to for all φ ∈ Ᏸ, where N is the neutrix having domain N = {1, 2,...,n,...} and range N the real numbers, with negligible functions finite linear sums of the functions and all functions which converge to zero in the usual sense as n tends to infinity, see [10,11] or [2].
The next theorem shows that Definition 1.6 is really the extension of Definition 1.3 and the proof of theorem is immediate.
Theorem 1.7.Let f and g be distributions for which the product f •g exists.Then the neutrix product f • g exists and defines the same distribution (see [11]).

Results on the neutrix product
In many elementary applications a relatively small class of distributions is sufficient.This consists of the regular distributions, delta functions, derivatives of delta functions, and linear combinations of these.This may give the impression that the delta and its derivatives are the only singular distributions which really matter.However, there are many examples of singular distributions other than these which are of immediate practical interest.We here define the neutrix product of a singular distribution x −r−1 − lnx − and the locally summable function x r + lnx + .
In the following theorem, which was proved in [12], the locally summable function x r + lnx + and the distribution x −r − are defined by where Further the distribution x −r − lnx − , which is distinct from the definition given by Gel'fand and Shilov [13], is defined by (see [14]) for r = 1,2,..., where where H(x) denotes the Heaviside function.It can be easily shown by induction that the distribution x −r − lnx − is also defined by an equation The following lemma is easily proved.
Theorem 2.3.The neutrix products 6 Abstract and Applied Analysis for r = 1,2,..., where (2.12) Proof.We put on the interval [0,1/n], the intersection of the supports of x r + lnx + and (x −r−1 − lnx − ) n .The neutrix limit of the sequence x r ) n obviously converges, as n tends to infinity, to the neutrix product given as in (2.3).Thus it is sufficient to evaluate the neutrix product of the distributions x r + lnx + and [ln 2 x − ] (r+1) so as to complete the proof of the theorem.Now we have on the interval [0,1/n] that Making the substitutions nx = v and nt = u, (2.16) (2.17) Emin Özc ¸aḡ et al. 7 For k = 0, we have Further when k = 1, we have (2.20) Let φ(x) be an arbitrary function in Ᏸ.Then by the mean value theorem φ(x) = φ(0) + xφ (ξx) where 0 < ξ < 1.It follows that (2.21) and so N-lim for k = 0,1,2,.... Then we have on the interval [−1/n,0], the intersection of the supports of ln 2 x − and ( on making the substitutions −nt = u and −nx = v.Thus N-lim (2.27) Emin Özc ¸aḡ et al. 9 We have, on making the substitution v = uy,