Further Results on Colombeau Product of Distributions

In quantum physics one finds the need to evaluate δ, when calculating the transition rates of certain particle interactions; see [1]. The problem of defining products of distributions is also closely connected with the problem of renormalization in quantum field theory. Due to the large use of distributions in the natural sciences and other mathematical fields, the problem of the product of distributions [2] is an objective of many research studies. Starting with the historically first construction of distributional multiplication by König [3], and the sequential approach developed in [4] by Antosik et al., there have been numerous attempts to define products of distributions, see [5–7], or rather to enlarge the number of existing products. Having a look at the theory of distributions [8, 9] we realize that there are two complementary points of view.


Introduction
In quantum physics one finds the need to evaluate  2 , when calculating the transition rates of certain particle interactions; see [1].The problem of defining products of distributions is also closely connected with the problem of renormalization in quantum field theory.Due to the large use of distributions in the natural sciences and other mathematical fields, the problem of the product of distributions [2] is an objective of many research studies.Starting with the historically first construction of distributional multiplication by König [3], and the sequential approach developed in [4] by Antosik et al., there have been numerous attempts to define products of distributions, see [5][6][7], or rather to enlarge the number of existing products.
Having a look at the theory of distributions [8,9] we realize that there are two complementary points of view.
(1) A distribution  ∈ D  (R  ) is a continuous functional on the space D(R  ) of space functions (compactly supported smooth functions).Here we have a linear action of  on a test function .
(2) Letting {  } be a sequence of smooth functions converging to the Dirac measure, a family of regularization {  } can be produced by convolution, which converges weakly to the original distribution  ∈ D  (R  ).Identifying two sequences {  } and {}  if they have the same limit, we obtain a sequential representation of the space of distributions.Other authors use the equivalence classes of nets of regularization.The delta-net {  } >0 is defined by When we work with regularization the nonlinear structure is lost by identifying sequences (nets) with the same limit.So we have to get nonlinear theory of generalized functions that will work with regularization, but identify less.The actual construction of such algebras enjoying these optimal properties is due to Colombeau [10].His theory of algebras of generalized functions offers the possibility of applying large classes of nonlinear operations to distributional objects.Some of them are used on solving differential equations with nonstandard coefficients [11].The "association process" in International Journal of Mathematics and Mathematical Sciences Colombeau algebra providing a faithful generalization of the equality of distributions in D(R  ) enables us to obtain results in terms of distributions, the so-called Colombeau products.In fact, we evaluate particular product of distributions with coinciding singularities, as embedded in Colombeau algebra, in terms of associated distributions; see [12][13][14].Therefore, the results obtained can be reformulated as regularized model products in the classical distribution theory.
In 1966, Mikusinski published his famous result [15]: where neither of the products on the left-hand side here exists, but their difference still has a correct meaning in distribution space D  (R).Formula including balanced products of distributions with coinciding singularities can be found in mathematical and physical literature.In this paper results on product of distributions  −−1/2 + and  −−1/2 − are derived.

Colombeau Algebra
The basic idea underlying Colombeau's theory in its simplest form is that of embedding the space of distributions into a factor algebra of C ∞ (R  )  ,  = (0, 1) with regularization by convolution with a fixed "mollifier" .
The space of test function is Functional act on test functions and points is where Taking this sequence as a representative of  we obtain an embedding of ) into this algebra via convolution as above will not yield a subalgebra since of course ( *   )( *   ) ̸ = () *   in general.The idea, therefore, is to factor out an ideal N(R  ) such that this difference vanishes in the resulting quotient.In order to construct N(R  ) it is obviously sufficient to find an ideal containing all differences ( *   ) − () ∈ .Taylor expansion of ( *   ) −  shows that this term will vanish faster than any power of , uniformly on compact sets, in all derivatives.We use the following space: moderate functionals, denoted by E  (R  ), are defined by the property: Null functionals, denoted by N(R  ), are defined by the property: In other words, moderate functionals satisfy a locally uniform polynomial estimate as  → 0 when acting on   , together with all derivatives, while null functionals vanish faster than any power of  in the same situation.The null functionals form a differential ideal in the collection of moderate functionals.
We define space of functions  : Moderate functionals, denoted by E 0 (R  ), are defined by the property: Null functionals, denoted by N 0 (R  ), are defined by the property: Definition 1. Space of generalized functions G(R  ), generalized complex numbers, and generalized real numbers is the factor algebra defined as The space of distributions is imbedded by convolution: Equivalence classes of sequences (  ) ∈ in G(R  ) will be denoted by  = class [(  ) ∈ ].If  is a generalized function with compact support  ⊂⊂ R  and   () is a representative of , then its integral is defined by Let ,  ∈ G(R  ).Then (16)

Results on Some Products of Distributions
It was proved in [12] that for any  ∈ R/Z the product of the generalized functions x + and x−−1 − in G(R) admits associated distributions and it holds with the particular cases for  = 0, ±1, ±2, . . .and Here we will make a generalization of (19).In order to prove the main theorem we need the following lemmas, easily proved by induction.
Proof.For given  ∈  0 (R) we suppose that supp () ⊆ [, ], without loss of generality.Then using the embedding rule and the substitution  = ( − )/ we have the representatives of the distribution  −−1/2 + in Colombeau algebra: Similar, using the embedding rule and the substitution  = ( − )/ we have the representatives of the distribution Then, for any () ∈ D(R) we have using the substitution  = −/.
By the Taylor theorem we have that International Journal of Mathematics and Mathematical Sciences for  ∈ (0, 1).Using this for (25) we have where for  = 0, 1, . . ., 2 and we have changed the order of integration.Putting  −  = ( − ) we have Thus Next, suppose that  is even, less than 2 and that  is less than .It follows from Lemma 2 that ∫      () () is an even or odd function accordingly as  +  is odd or even.
We thus have that  −  () () ∫      () () is an odd function and  , = 0.If  ≥  then − <  and by changing the order of integration we can prove that again  , = 0.If we suppose that  is odd and less than 2 we can prove in a similar manner that  , = 0. (34) Therefore passing to the limit, as  → 0, we obtain (22) proving the theorem.