Non-commutative Neutrix Product Research Articles

The noncommutative neutrix product of $x_{-}^{-s}\ln^mx_{-}$ and $x_{+}^{r}$

In distribution theory, the product of two distributions can only be defined under certain conditions. In this paper, by using Fisher's definition, it is proved that if Gm,s(x) denotes the distribution ( ln m+1x-)(s), then the noncommutative neutrix product [Formula: see text] exists and [Formula: see text] for m, s ≥ 1, where [Formula: see text]. Related neutrix products are then deduced.

Some results on the non-commutative neutrix product of distributions

It is proved that the non-commutative neutrix product of the distributions x −r and x s ln q |x| exists and for r, q=1, 2, …, s=0,±1,±2, … and r−s>1.

On the non-commutative neutrix product of the distributions x r ln p |x| and x s ln q |x|

It is proved that the non-commutative neutrix product of the distributions x r ln p |x| and x s ln q |x| exists and r, s=0,±1,±2, …, p, q=1, 2, … and r+s<−1.

On the non-commutative neutrix product of Dirac-delta distribution with an infinitely differentiable function

In this article, we first consider the non-commutative neutrix product of f +(x) and δ(r)(x) and also the product of f −(x) and δ(r)(x) for r=0, 1, 2, …, where f is an infinitely differentiable function and f +(x)=H(x)f(x) and f −(x)=H(−x)f(x), H denoting Heaviside’s function. Then we generalize some results obtained by Fisher [Fisher, B., 1971, The product of distributions. Quarterly Journal of Mathematics Oxford, 22, 291–298 and Fisher, B., 1982, A non-commutative neutrix product of distributions. Mathematische Nachrichten, 108, 117–127.].

On the non-commutative neutrix product of the distributions and

Let f and g be distributions and g n = (g*δ n )(x), where δ n (x) is a certain sequence converging to the Dirac delta-function. The non-commutative neutrix product f ° g of f and g is defined to be the neutrix limit of the sequence {fg n }, provided its limit h exists in the sense that for all functions ϕ in 𝒟. It is proved that for μ < r−1; μ≠0,±1,±2, …, r = 1, 2, … and p, q = 0, 1, 2….

A note on the product*

The difficulties inherent in defining products, powers or nonlinear operations of generalized functions have not prevented their appearance in the literature (see e.g. [1, 2, 3]). In [4], a definition for a product of distributions is given using a δ-sequence. However, they show that δ2 does not exist. Extending definitions of products from one-dimensional space to m-dimensional by using appropriate delta-sequences has recently been an interesting topic in distribution theory. From a single variable function p(s) (see [5]) defined on R +, Li and Fisher introduce a ‘workable’ delta sequence in Rm to deduce a non-commutative neutrix product of r−k and Δδ (Δ denotes the Laplacian) for any positive integer k between 1 and m – 1 inclusive. In [6], Li provides a modified δ-sequence and defines a ‘bridge’ distribution which can be used to obtain the more general product of r−k and Δlδ. The object of this paper is to utilize the fact that is an elementary solution of Laplacian equation Δu(x) = δ(x) and to apply a much simpler version of the main result in [7] to derive the product r−k. ▽(Δr 2-m ) where m is any dimension greater than 2. Finally, we also compute the products r −k . Δ In r and r −k . ▽(Δ ln r) based on the equation for m = 2

Some results on the non-commutative neutrix product of distributions

In this paper, using the concept of the neutrix limit due to J.G. van der Corput [3], the non-commutative neutrix products are proved to exist and are evaluated for s=0,1,...and r=1,2,...

The sequential approach to the product of distribution

It is well known that the sequential approach is one of the main tools of dealing with product, power, and convolution of distribution (cf. Chen (1981), Colombeau (1985), Jones (1973), and Rosinger (1987)). Antosik, Mikusiński, and Sikorski in 1972 introduced a definition for a product of distributions using a delta sequence. However,δ2as a product ofδwith itself was shown not to exist (see Antosik, Mikusiński, and Sikorski (1973)). Later, Koh and Li (1992) chose a fixedδ-sequence without compact support and used the concept of neutrix limit of van der Corput to defineδkand(δ′)kfor some values ofk. To extend such an approach from one-dimensional space tom-dimensional, Li and Fisher (1990) constructed a delta sequence, which is infinitely differentiable with respect tox1,x2,…,xmandr, to deduce a non-commutative neutrix product ofr−kandΔδ. Li (1999) also provided a modifiedδ-sequence and defined a new distribution(dk/drk)δ(x), which is used to compute the more general product ofr−kandΔlδ, wherel≥1, by applying the normalization procedure due to Gel'fand and Shilov (1964). We begin this paper by distributionally normalizingΔr−kwith the help of distributionx+−n. Then we utilize several nice properties of theδ-sequence by Li and Fisher (1990) and an identity ofδdistribution to derive the productΔr−k⋅δbased on the results obtained by Li (2000), and Li and Fisher (1990).

The product of r−k and ∇δ on ℝm

In the theory of distributions, there is a general lack of definitions for products and powers of distributions. In physics (Gasiorowicz (1967), page 141), one finds the need to evaluateδ2when calculating the transition rates of certain particle interactions and using some products such as(1/x)⋅δ. In 1990, Li and Fisher introduced a “computable” delta sequence in anm-dimensional space to obtain a noncommutative neutrix product ofr−kandΔδ(Δdenotes the Laplacian) for any positive integerkbetween 1 andm−1inclusive. Cheng and Li (1991) utilized a netδϵ(x)(similar to theδn(x)) and the normalization procedure ofμ(x)x+λto deduce a commutative neutrix product ofr−kandδfor any positive real numberk. The object of this paper is to apply Pizetti's formula and the normalization procedure to derive the product ofr−kand∇δ(∇is the gradient operator) onℝm. The nice properties of theδ-sequence are fully shown and used in the proof of our theorem.

A note on the non-commutative neutrix product of distributions

The distributionF(x +, −r) Inx+ andF(x −, −s) corresponding to the functionsx + −r lnx+ andx − −s respectively are defined by the equations $$\left\langle {F(x_ + , - r)\ln x_ + ,\phi (x)} \right\rangle = \int_0^\infty {x^{ - r} \ln x\left[ {\phi (x) - \sum\limits_{i = 0}^{r - 2} {\frac{{\phi ^{(i)} (0)}}{{i!}}x^i \frac{{\phi ^{(i)} (0)}}{{(r - 1)!}}H(1 - x)x^{r - 1} } } \right]dx} $$ (1) and $$\left\langle {F(x_ + , - s),\phi (x)} \right\rangle = \int_0^\infty {x^{ - s} \left[ {\phi (x) - \sum\limits_{i = 0}^{s - 2} {\frac{{\phi ^{(i)} (0)}}{{i!}}( - x^i )\frac{{\phi ^{(s - 1)} (0)}}{{(s - 1)!}}H(1 - x)x^{s - 1} } } \right]dx} $$ (2) whereH(x) denotes the Heaviside function. In this paper, using the concept of the neutrix limit due to J G van der Corput [1], we evaluate the non-commutative neutrix product of distributionsF(x +, −r) lnx+ andF(x −, −s). The formulae for the neutrix productsF(x +, −r) lnx + ox − −s, x+ −r lnx+ ox − −s andx − −s o F(x+, −r) lnx+ are also given forr, s = 1, 2, ...

On the Non-Commutative Neutrix Product

Abstract The non-commutative neutrix product of the distributions and is evaluated for r = 0, 1, 2, . . . and s = 1, 2, . . . . Further neutrix products are then deduced.

On the non-commutative neutrix product (x + r lnx +) ox − −s

The non-commutative neutrix product of the distributionsx+rlnx+ andx−−s is evaluated forr=0, 1, 2, … ands=1, 2, …. Further neutrix products are then deduced.

On the product of the function [image omitted] and the distribution [image omitted

It is proved that the non-commutative neutrix product of the function [image omitted] and the distribution (x+i0)-s is equal to([image omitted]

A non-commutative neutrix product of distributions

A non-commutative neutrix product of distributions