Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2013, Article ID 918905, 5 pages
http://dx.doi.org/10.1155/2013/918905
Research Article
Further Results on Colombeau Product of Distributions
Biljana Jolevska-Tuneska1 and Tatjana Atanasova-Pacemska2
1
Faculty of Electrical Engineering and Informational Technologies, Saints Cyril and Methodius University of Skopje,
Karpos II bb, 1000 Skopje, Macedonia
2
Faculty of Computer Sciences, Goce Delčev University of Štip, Krste Misirkov, Br 10A, 2000 Štip, Macedonia
Correspondence should be addressed to Biljana Jolevska-Tuneska; biljanaj@feit.ukim.edu.mk
Received 10 August 2012; Revised 28 November 2012; Accepted 10 December 2012
Academic Editor: Shyam Kalla
Copyright © 2013 B. Jolevska-Tuneska and T. Atanasova-Pacemska. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the
original work is properly cited.
Results on Colombeau product of distributions 𝑥+−𝑟−1/2 and 𝑥−−𝑟−1/2 are derived. They are obtained in Colombeau differential algebra
G(R) of generalized functions that contains the space D󸀠 (R) of Schwartz distributions as a subspace and has a notion of “association”
that is a faithful generalization of the weak equality in D󸀠 (R).
1. Introduction
In quantum physics one finds the need to evaluate 𝛿2 , 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.
(1) A distribution 𝑓 ∈ D󸀠 (R𝑛 ) is a continuous functional
on the space D(R𝑛 ) of space functions (compactly
supported smooth functions). Here we have a linear
action
𝜑 󳨀→ ⟨𝑓, 𝜑⟩
of 𝑓 on a test function 𝜙.
(1)
(2) Letting {𝜑𝑛 } be a sequence of smooth functions
converging to the Dirac measure, a family of regularization {𝑓𝑛 } can be produced by convolution,
𝑓𝑛 (𝑥) = 𝑓 ∗ 𝜑𝑛 = ⟨𝑓 (𝑦) , 𝜑𝑛 (𝑥 − 𝑦)⟩ ,
(2)
which converges weakly to the original distribution
𝑓 ∈ D󸀠 (R𝑛 ). Identifying two sequences {𝜑𝑛 } and {𝜑}𝑛
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 {𝜑𝜀 }𝜀>0 is defined by
𝑥
𝜑𝜀 (𝑥) = 𝜀−𝑛 𝜑 ( ) .
𝜀
(3)
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
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International Journal of Mathematics and Mathematical Sciences
Colombeau algebra providing a faithful generalization of the
equality of distributions in D(R𝑛 ) 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 [12–14]. Therefore,
the results obtained can be reformulated as regularized model
products in the classical distribution theory.
In 1966, Mikusinski published his famous result [15]:
𝑥−1 ⋅ 𝑥−1 − 𝜋2 𝛿 (𝑥) 𝛿 (𝑥) = 𝑥−2 ,
(4)
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 𝑥+−𝑟−1/2 and 𝑥−−𝑟−1/2 are derived.
2. 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∞ (R𝑛 )𝐼 , 𝐼 = (0, 1) with regularization by
convolution with a fixed “mollifier” 𝜑.
The space of test function is
A0 (R𝑛 ) = {𝜑 ∈ D (R𝑛 ) : ∫ 𝜑 (𝑥) 𝑑𝑥 = 1} .
(𝜑, 𝑥) 󳨀→ 𝑢 (𝜑, 𝑥) ,
∀𝐾 ⊂⊂ R𝑛 ∀𝛼 ∈ N𝑛0 ∃𝑝 ≥ 0 such that ∀𝜑 ∈ A𝑞 (R𝑛 ) ,
󵄨
󵄨
sup 󵄨󵄨󵄨𝜕𝛼 𝑢 (𝜑𝜀 , 𝑥)󵄨󵄨󵄨 = O (𝜀−𝑝 )
𝑥∈𝐾
as 𝜀 󳨀→ 0.
(8)
Null functionals, denoted by N(R𝑛 ), are defined by the
property:
∀𝐾 ⊂⊂ R𝑛 ∀𝛼 ∈ N𝑛0 ∃𝑝 ≥ 0
such that ∀𝑞 ≥ 𝑝, ∀𝜑 ∈ A𝑞 (R𝑛 ) ,
󵄨
󵄨
sup 󵄨󵄨󵄨𝜕𝛼 𝑢 (𝜑𝜀 , 𝑥)󵄨󵄨󵄨 = O (𝜀𝑞−𝑝 )
𝑥∈𝐾
(9)
as 𝜀 󳨀→ 0.
In other words, moderate functionals satisfy a locally uniform
polynomial estimate as 𝜀 → 0 when acting on 𝜑𝜀 , together
with all derivatives, while null functionals vanish faster than
any power of 𝜀 in the same situation. The null functionals
form a differential ideal in the collection of moderate functionals.
We define space of functions 𝑢 : A0 → C and denote
by E0 (R𝑛 ). It is a subalgebra in E(R𝑛 ) in sense of natural
identification.
Moderate functionals, denoted by E0𝑀(R𝑛 ), are defined
by the property:
∃𝑝 ≥ 0 such that ∀𝜑 ∈ A𝑝 (R𝑛 ) ,
(5)
Functional act on test functions and points is
𝑢 : A0 (R𝑛 ) × R𝑛 󳨀→ C,
moderate functionals, denoted by E𝑀(R𝑛 ), are defined by the
property:
󵄨󵄨
󵄨
−𝑝
󵄨󵄨𝑢 (𝜑𝜀 )󵄨󵄨󵄨 = O (𝜀 )
(10)
as 𝜀 󳨀→ 0.
Null functionals, denoted by N0 (R𝑛 ), are defined by the
property:
(6)
where 𝑢(𝜑, 𝑥) is required to belong to C∞ (R𝑛 ) with respect
to the second variable 𝑥 ∈ R𝑛 . Now let 𝜑𝜀 (𝑥) =
𝜀−𝑛 𝜑(𝑥/𝜀) for 𝜑 ∈ A0 (R𝑛 ). The sequence (𝑢 ∗ 𝜑𝜀 )𝜀∈𝐼 converges to 𝑢 in D󸀠 (R𝑛 ). Taking this sequence as a representative
of 𝑢 we obtain an embedding of D󸀠 (R𝑛 ) into the algebra
C∞ (R𝑛 ). However, embedding C∞ (R𝑛 ) ⊂ D󸀠 (R𝑛 ) into this
algebra via convolution as above will not yield a subalgebra
since of course (𝑓 ∗ 𝜑𝜀 )(𝑔 ∗ 𝜑𝜀 ) ≠ (𝑓𝑔) ∗ 𝜑𝜀 in general. The
idea, therefore, is to factor out an ideal N(R𝑛 ) such that
this difference vanishes in the resulting quotient. In order
to construct N(R𝑛 ) it is obviously sufficient to find an ideal
containing all differences (𝑓 ∗ 𝜑𝜀 ) − (𝑓)𝜀∈𝐼 . Taylor expansion
of (𝑓 ∗ 𝜑𝜀 ) − 𝑓 shows that this term will vanish faster than any
power of 𝜀, uniformly on compact sets, in all derivatives. We
use the following space:
A𝑞 (R𝑛 )
= {𝜑 ∈ A0 (R𝑛 ) : ∫ 𝑥𝛼 𝜑 (𝑥) 𝑑𝑥 = 0 for 1 ≤ |𝛼| ≤ 𝑞} ;
(7)
∃𝑝 ≥ 0 such that ∀𝑞 ≥ 𝑝, ∀𝜑 ∈ A𝑞 (R𝑛 ) ,
󵄨󵄨
󵄨
𝑞−𝑝
󵄨󵄨𝑢 (𝜑𝜀 )󵄨󵄨󵄨 = O (𝜀 )
as 𝜀 󳨀→ 0.
(11)
Definition 1. Space of generalized functions G(R𝑛 ), generalized complex numbers, and generalized real numbers is the
factor algebra defined as
G (R𝑛 ) =
E𝑀 (R𝑛 )
,
N (R𝑛 )
C=
E0𝑀 (C)
,
N0 (C)
R=
E0𝑀 (R)
.
N0 (R)
(12)
The space of distributions is imbedded by convolution:
𝑖 : D󸀠 (R𝑛 ) 󳨀→ G (R𝑛 ) ,
𝑖 (𝑢) = class of [(𝜑, 𝑥) 󳨀→ 𝑢 ∗ 𝜑 (𝑥)] .
(13)
Equivalence classes of sequences (𝑢𝜀 )𝜀∈𝐼 in G(R𝑛 ) will be
denoted by 𝑈 = class [(𝑢𝜀 )𝜀∈𝐼 ]. If 𝐺 is a generalized function
with compact support 𝐾 ⊂⊂ R𝑛 and 𝐺𝜀 (𝑥) is a representative
of 𝐺, then its integral is defined by
∫ 𝐺 𝑑𝑥 = class [∫ 𝜑 (𝑥) 𝐺𝜀 (𝑥) 𝑑𝑥] .
Let 𝐹, 𝐺 ∈ G(R𝑛 ). Then
(14)
International Journal of Mathematics and Mathematical Sciences
󸀠
(i) they are equal in the distribution sense, 𝐺 =D 𝐹 if
∫ (𝐺 − 𝐹) 𝜑 𝑑𝑥 = 0 ∈ C,
(15)
𝑛
for any 𝜑 ∈ D (R ) ;
(ii) they are associated 𝐺 ≈ 𝐹 if there exist a representative 𝐺𝜀 and 𝐹𝜀 of 𝐺 and 𝐹, respectively, such that
lim ∫ (𝐺𝜀 (𝑥) − 𝐹𝜀 (𝑥)) 𝜑 (𝑥) 𝑑𝑥 = 0,
𝜀→0
3
Proof. For given 𝜑 ∈ 𝐴 0 (R) we suppose that supp 𝜑(𝑥) ⊆
[𝑐, 𝑑], without loss of generality. Then using the embedding
rule and the substitution 𝑡 = (𝑦 − 𝑥)/𝜀 we have the
representatives of the distribution 𝑥+−𝑟−1/2 in Colombeau
algebra:
̃
𝑥+−𝑟−1/2 (𝜑𝜀 , 𝑥) = (−1)𝑟
3. Results on Some Products of Distributions
It was proved in [12] that for any 𝑎 ∈ R/Z the product of the
−𝑎−1 in G(R) admits associated
generalized functions 𝑥̃+𝑎 and 𝑥̃
−
distributions and it holds
̃
−𝑎−1 = 𝑥
−𝑎−1 ⋅ 𝑥
̃+𝑎 ≈ Γ (1 + 𝑎) Γ (−𝑎) 𝛿 (𝑥)
𝑥̃+𝑎 ⋅ 𝑥̃
−
−
2
𝜋
= − csc (𝜋𝑎) 𝛿 (𝑥) ,
2
=
1 ∞ −1/2 (𝑟) 𝑦 − 𝑥
2𝑟
) 𝑑𝑦
∫ 𝑦 𝜑 (
𝜀
(2𝑟 − 1)!! 𝜀𝑟+1 0
=
1 𝑑
2𝑟
∫ (𝑥 + 𝜀𝑡)−1/2 𝜑(𝑟) (𝑡) 𝑑𝑡.
(2𝑟 − 1)!! 𝜀𝑟 −𝑥/𝜀
(16)
for any 𝜑 ∈ D (R𝑛 ) .
(23)
Similar, using the embedding rule and the substitution
𝑠 = (𝑦 − 𝑥)/𝜀 we have the representatives of the distribution
𝑥−−𝑟−1/2 in Colombeau algebra:
̃
𝑥−−𝑟−1/2 (𝜑𝜀 , 𝑥) = (−1)𝑟
(17)
×∫
=
(18)
∫ 𝑠𝑝 𝜑(𝑟) (𝑠) 𝑑𝑠 = ∑ (−1)𝑘
𝑡
𝑘=0
𝑝!
𝑡𝑝−𝑘 𝜑(𝑟−𝑘−1)
(𝑝 − 𝑘)!
× (𝑡) 𝑑𝑡,
=
(19)
Lemma 2. For 𝑝 < 𝑟 one has
𝑝
𝑑
𝑟
𝑡
𝑘=0
∞
−∞
=
for p < r.
𝑐
(
𝑦−𝑥
) 𝑑𝑦
𝜀
1 −𝑥/𝜀
2𝑟
∫
(−𝑥 − 𝜀𝑠)−1/2 𝜑(𝑟) (𝑠) 𝑑𝑠.
(2𝑟 − 1)!! 𝜀𝑟 𝑐
(24)
=
(𝑥 + 𝜀𝑡)−1/2 (−𝑥 − 𝜀𝑠)−1/2 𝜑(𝑟) (𝑠) 𝑑𝑠 𝑑𝑡 𝑑𝑥
𝑑
1 𝑑
22𝑟
𝜓
𝜑(𝑟) (𝑡)
∫
∫
(−𝜀𝜔)
𝜔
((2𝑟 − 1)!!)2 𝜀2𝑟 𝑐
× ∫ (𝑡 − 𝜔)−1/2 (𝜔 − 𝑠)−1/2 𝜑(𝑟) (𝑠) 𝑑𝑠 𝑑𝑡 𝑑𝜔,
𝑐
̃
Theorem 4. The product of the generalized functions 𝑥+−𝑟−1/2
̃
and 𝑥−−𝑟−1/2 for 𝑟 = 0, 1, 2, . . . in G(R) admits associated
distributions and it holds
𝜋
𝛿(2𝑟) (𝑥) .
2 (2𝑟)!
𝜑
𝜔
𝑑
and 𝜑(−1) (𝑡) stands for ∫𝑡 𝜑(𝑠)𝑑𝑠.
̃ ̃
𝑥+−𝑟−1/2 ⋅ 𝑥−−𝑟−1/2 ≈
−1/2 (𝑟)
(−𝑦)
𝑑
1 𝑐𝜀
22𝑟
∫ 𝜓 (𝑥) ∫ 𝜑(𝑟) (𝑡)
2 𝜀2𝑟
−𝑑𝜀
−𝑥/𝜀
((2𝑟 − 1)!!)
−𝑥/𝜀
(21)
𝑦−𝑥
) 𝑑𝑦
𝜀
̃
̃
𝑥+−𝑟−1/2 (𝜑𝜀 , 𝑥) 𝑥−−𝑟−1/2 (𝜑𝜀 , 𝑥) 𝜓 (𝑥) 𝑑𝑥
×∫
𝑟!
𝑡𝑟−𝑘 𝜑(𝑟−𝑘−1) (𝑡) 𝑑𝑡,
(𝑟 − 𝑘)!
𝜑(
̃
̃
⟨𝑥+−𝑟−1/2 (𝜑𝜀 , 𝑥) ⋅ 𝑥−−𝑟−1/2 (𝜑𝜀 , 𝑥) , 𝜓 (𝑥)⟩
=∫
(20)
−1/2
(−𝑦)
Then, for any 𝜓(𝑥) ∈ D(R) we have
Lemma 3. For 𝑝 = 𝑟 one has
∫ 𝑠𝑟 𝜑(𝑟) (𝑠) 𝑑𝑠 = ∑ (−1)𝑘
0
−∞
Here we will make a generalization of (19). In order to prove
the main theorem we need the following lemmas, easily
proved by induction.
𝑑
0
1
2𝑟
(2𝑟 − 1)!! 𝜀𝑟+1
×∫
for 𝑟 = 0, ±1, ±2, . . . and
̃ ̃ 𝜋
𝑥+−1/2 ⋅ 𝑥−−1/2 = 𝛿 (𝑥) .
2
1 𝜕𝑟
2𝑟
(2𝑟 − 1)!! 𝜀 𝜕𝑥
−∞
with the particular cases
𝑟
̃ ̃ (−1) 𝜋
𝑥+−𝑟−1/2 ⋅ 𝑥−𝑟−1/2 =
𝛿 (𝑥) ,
2
1 𝜕𝑟 ∞ −1/2 𝑦 − 𝑥
2𝑟
) 𝑑𝑦
∫ 𝑦 𝜑(
𝜀
(2𝑟 − 1)!! 𝜀 𝜕𝑥 0
(22)
(25)
using the substitution 𝜔 = −𝑥/𝜀.
By the Taylor theorem we have that
𝜓(2𝑟+1) (𝜂𝜔)
𝜓(𝑘) (0)
2𝑟+1
(−𝜀𝜂) , (26)
(−𝜀𝜔)𝑘 +
+
1)!
𝑘!
(2𝑟
𝑘=0
2𝑟
𝜓 (−𝜀𝜔) = ∑
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International Journal of Mathematics and Mathematical Sciences
for 𝜂 ∈ (0, 1). Using this for (25) we have
For the case 𝑘 = 2𝑟 if 𝑝 ≠ 𝑟 again we have 𝐽𝑘,𝑝 = 0. For
𝑘 = 2𝑟 and 𝑝 = 𝑟 using Lemma 3 and changing the order of
integration we have
̃
̃
⟨𝑥+−𝑟−1/2 (𝜑𝜀 , 𝑥) ⋅ 𝑥−−𝑟−1/2 (𝜑𝜀 , 𝑥) , 𝜓 (𝑥)⟩
2𝑟
(−1)𝑘 𝜓(𝑘) (0)
22𝑟
=
𝐼𝑘 + 𝑂 (𝜀) ,
∑
((2𝑟 − 1)!!)2 𝑘=0 𝑘!𝜀2𝑟−𝑘
𝑑
𝑑
𝐽2𝑟,𝑟 = ∫ 𝑡𝑟 𝜑(𝑟) (𝑡) ∫ 𝑠𝑟 𝜑(𝑟) (𝑠) 𝑑𝑠 𝑑𝑡
(27)
𝑐
𝑡
𝑟
𝑖
(−1) 𝑟! 𝑑 2𝑟−𝑖 (𝑟)
∫ 𝑡 𝜑 (𝑡) 𝜑(𝑟−𝑖−1) (𝑡) 𝑑𝑡
−
𝑖)!
(𝑟
𝑐
𝑖=0
=∑
where
𝑑
𝑑
𝑐
𝑡
= (−1)𝑟 𝑟! ∫ 𝑡𝑟 𝜑(𝑟) (𝑡) ∫ 𝜑 (𝑠) 𝑑𝑠 𝑑𝑡
𝑑
𝑑
𝐼𝑘 = ∫ 𝜑(𝑟) (𝑡) ∫ 𝜑(𝑟) (𝑠)
𝑐
𝑡
𝑡
𝑠
(𝜔 − 𝑠)
𝑐
(𝑡) 𝑑𝑡 𝑑𝑠
𝑟
(−1)𝑖 𝑟! 𝑑 𝑟−𝑖 (𝑟−𝑖−1)
∫ 𝑠 𝜑
(𝑠) 𝜑 (𝑠) 𝑑𝑠
𝑖=0 (𝑟 − 𝑖)! 𝑐
for 𝑘 = 0, 1, . . . , 2𝑟 and we have changed the order of
integration.
Putting 𝜔 − 𝑠 = (𝑡 − 𝑠)𝑣 we have
∫ (𝑡 − 𝜔)
𝑐
(31)
𝑟 (𝑟)
= (−1)𝑟 𝑟!∑
𝑠
−1/2
𝑠
= (−1) 𝑟! ∫ 𝜑 (𝑠) ∫ 𝑡 𝜑
(28)
× ∫ (𝑡 − 𝜔)−1/2 (𝜔 − 𝑠)−1/2 𝜔𝑘 𝑑𝜔 𝑑𝑠 𝑑𝑡,
𝑡
𝑑
𝑟
𝑑
𝑠
𝑐
𝑐
= (𝑟!)2 ∫ 𝜑 (𝑠) (∫ 𝜑 (𝑡) 𝑑𝑡) 𝑑𝑠.
Further,
𝑑
𝑠
𝑑
𝑠
𝑐
𝑐
𝑐
𝑐
𝑠
∫ 𝜑 (𝑠) (∫ 𝜑 (𝑡) 𝑑𝑡) 𝑑𝑠 = ∫ (∫ 𝜑 (𝑡) 𝑑𝑡) 𝑑 (∫ 𝜑 (𝑡) 𝑑𝑡)
−1/2 𝑘
𝜔 𝑑𝜔
1
= ∫ 𝑣−1/2 (1 − 𝑣)−1/2 [𝑠𝑣 + (1 − 𝑣)𝑡]𝑘 𝑑𝑣
=
0
𝑘
1
𝑘!
=∑
∫ 𝑣𝑝−1/2 (1 − 𝑣)𝑘−𝑝−1/2 𝑠𝑝 𝑡𝑘−𝑝 𝑑𝑣
𝑝=0 𝑝! (𝑘 − 𝑝)! 0
1 𝑠
(∫ 𝜑 (𝑡) 𝑑𝑡)
2 𝑐
𝑐
2 󵄨𝑑
󵄨󵄨
󵄨󵄨 = 1 ,
󵄨󵄨
󵄨󵄨𝑐 2
(29)
(32)
and 𝐽2𝑟,𝑟 = (𝑟!)2 /2. So, 𝐼𝑘 = 0 for 𝑘 = 0, 1, . . . , 2𝑟 − 1 and
𝑘
𝐼2𝑟 =
𝑘!
1
1
=∑
𝐵 (𝑝 + , 𝑘 − 𝑝 + ) 𝑠𝑝 𝑡𝑘−𝑝 .
2
2
𝑝!
(𝑘
−
𝑝)!
𝑝=0
1
1
(2𝑟)!
((2𝑟 − 1)!!)2 𝜋
.
𝐵 (𝑟 + , 𝑟 + ) =
2
2
2
22𝑟+1
(33)
Finally we have
̃
̃
⟨𝑥+−𝑟−1/2 (𝜑𝜀 , 𝑥) ⋅ 𝑥−−𝑟−1/2 (𝜑𝜀 , 𝑥) , 𝜓 (𝑥)⟩
Thus
𝑘
𝑘!
1
1
𝐵 (𝑝 + , 𝑘 − 𝑝 + )
2
2
𝑝!
(𝑘
−
𝑝)!
𝑝=0
𝐼𝑘 = ∑
𝑑
𝑑
𝑐
𝑡
× ∫ 𝑡𝑘−𝑝 𝜑(𝑟) (𝑡) ∫ 𝑠𝑝 𝜑(𝑟) (𝑠) 𝑑𝑠 𝑑𝑡
(30)
𝑘
𝑘!
1
1
𝐵 (𝑝 + , 𝑘 − 𝑝 + ) 𝐽𝑘,𝑝 .
2
2
𝑝!
(𝑘
−
𝑝)!
𝑝=0
=
𝜋 𝜓(2𝑟) (0)
+ 𝑂 (𝜀)
2 (2𝑟)!
=
𝜋
⟨𝛿(2𝑟) (𝑥) , 𝜓 (𝑥)⟩ + 𝑂 (𝜀) .
2 (2𝑟)!
(34)
Therefore passing to the limit, as 𝜀 → 0, we obtain (22)
proving the theorem.
= ∑
Next, suppose that 𝑘 is even, less than 2𝑟 and that 𝑝 is
𝑑
less than 𝑟. It follows from Lemma 2 that ∫𝑡 𝑠𝑝 𝜑(𝑟) (𝑠)𝑑𝑠 is an
even or odd function accordingly as 𝑟 + 𝑝 is odd or even.
𝑑
We thus have that 𝑡𝑘−𝑝 𝜑(𝑟) (𝑡) ∫𝑡 𝑠𝑝 𝜑(𝑟) (𝑠)𝑑𝑠 is an odd function
and 𝐽𝑘,𝑝 = 0. If 𝑝 ≥ 𝑟 then 𝑘−𝑝 < 𝑟 and by changing the order
of integration we can prove that again 𝐽𝑘,𝑝 = 0. If we suppose
that 𝑘 is odd and less than 2𝑟 we can prove in a similar manner
that 𝐽𝑘,𝑝 = 0.
References
[1] S. Gasiorawics, Elementary Particle Physics, John Wiley & Sons,
New York, NY, USA, 1966.
[2] M. Oberguggenberger, Multiplication of Distributions and
Application to Partial Differential Equations, vol. 259 of Pitman
Research Notes in Mathematics Series, Longman, 1992.
[3] H. König, “Neue Begründung der theorie der distributions,”
Mathematische Nachrichten, vol. 9, pp. 129–148, 1953.
[4] P. Antosik, J. Mikusinski, and R. Sigorski, theory of Distributions.
The Sequential Approach, Elsevier, Amsterdam, The Netherlands, 1973.
International Journal of Mathematics and Mathematical Sciences
[5] B. Fisher, “The product of distributions,” The Quarterly Journal
of Mathematics, vol. 22, pp. 291–298, 1971.
[6] B. Fisher, “A noncommutative neutrix product of distributions,”
Mathematische Nachrichten, vol. 108, pp. 117–127, 1982.
[7] L. Z. Cheng and B. Fisher, “Several products of distributions on
R𝑚 ,” Proceedings of the Royal Society London A, vol. 426, no. 1871,
pp. 425–439, 1989.
[8] A. H. Zemanian, Distribution Theory and Transform Analysis,
Dover, 1965.
[9] I. M. Gel’fand and G. E. Shilov, Generalized Functions, vol. 1,
chapter 1, Academic Press, New York, NY, USA, 1964.
[10] J.-F. Colombeau, New Generalized Functions and Multiplication
of Distributions, vol. 84 of North-Holland Mathematics Studies,
North-Holland, Amsterdam, The Netherlands, 1984.
[11] B. Jolevska-Tuneska, A. Takači, and E. Ozcag, “On differential
equations with nonstandard coefficients,” Applicable Analysis
and Discrete Mathematics, vol. 1, no. 1, pp. 276–283, 2007.
[12] B. Damyanov, “Results on Colombeau product of distributions,”
Commentationes Mathematicae Universitatis Carolinae, vol. 38,
no. 4, pp. 627–634, 1997.
[13] B. Damyanov, “Some distributional products of Mikusiński type
in the Colombeau algebra 𝐺(𝑅𝑚 ),” Journal of Analysis and its
Applications, vol. 20, no. 3, pp. 777–785, 2001.
[14] B. Damyanov, “Modelling and products of singularities in
Colombeau’s algebra 𝐺(𝑅),” Journal of Applied Analysis, vol. 14,
no. 1, pp. 89–102, 2008.
[15] J. Mikusiński, “On the square of the Dirac delta-distribution,”
Bulletin de l’Académie Polonaise des Sciences, vol. 14, pp. 511–513,
1966.
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