GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 5, 1997, 413-419
GENERALIZATIONS OF NON-COMMUTATIVE NEUTRIX
CONVOLUTION PRODUCTS OF FUNCTIONS
BRIAN FISHER AND YONGPING CHEN
Abstract. The non-commutative neutrix convolution product of the
functions xr cos− (λx) and xs cos+ (µx) is evaluated. Further similar
non-commutative neutrix convolution products are evaluated and deduced.
In the following we let D be the space of infinitely differentiable functions
with compact support and let D0 be the space of distributions defined on
D. The convolution product f ∗ g of two distributions f and g in D0 is then
usually defined by the equation
h(f ∗ g)(x), φi = hf (y), hg(x), φ(x + y)ii
for arbitrary φ in D, provided f and g satisfy either of the conditions
(a)
either f or g has bounded support,
(b)
the supports of f and g are bounded on the same side;
see Gel’fand and Shilov [1].
Note that if f and g are locally summable functions satisfying either of
the above conditions then
Z ∞
Z ∞
(f ∗ g)(x) =
f (t)g(x − t) dt =
f (x − t)g(t) dt.
(1)
−∞
−∞
It follows that if the convolution product f ∗ g exists by this definition
then
f ∗ g = g ∗ f,
0
0
(2)
0
(f ∗ g) = f ∗ g = f ∗ g.
(3)
This definition of the convolution product is rather restrictive and so a
neutrix convolution product was introduced in [2]. In order to define the
1991 Mathematics Subject Classification. 46F10.
Key words and phrases. Neutrix, neutrix limit, neutrix convolution product.
413
c 1997 Plenum Publishing Corporation
1072-947X/97/0900-0413$12.50/0 
414
BRIAN FISHER AND YONGPING CHEN
neutrix convolution product we first of all let τ
the following properties:
(i) τ (x) = τ (−x),
(ii) 0 ≤ τ (x) ≤ 1,
(iii) τ (x) = 1 for |x| ≤ 21 ,
(iv) τ (x) = 0 for |x| ≥ 1.
The function τν is now defined by

1,

τ (ν ν x − ν ν+1 ),
τν (x) =

τ (ν ν x + ν ν+1 ),
be a function in D satisfying
|x| ≤ ν,
x > ν,
x < −ν,
for ν > 0.
We now give a new neutrix convolution product which generalizes the
one given in [2].
Definition 1. Let f and g be distributions in D0 and let fν = f τν for
ν > 0. Then the neutrix convolution product f 
∗ g is defined as the neutrix
limit of the sequence {fν ∗ g}, provided that the limit h exists in the sense
that
N- limhfν ∗ g, φi = hh, φi,
ν→∞
for all φ in D, where N is the neutrix (see van der Corput [3]), having
domain N 0 the positive reals and range N 00 the complex numbers, with
negligible functions finite linear sums of the functions
ν λ lnr−1 ν, lnr ν, ν r−1 eµν
(real λ > 0, complex µ 6= 0, r = 1, 2, . . . )
and all functions which converge to zero in the usual sense as ν tends to
infinity.
Note that in this definition the convolution product fν ∗ g is defined in
Gel’fand and Shilov’s sense, the distribution fν having bounded support.
In the original definition of the neutrix convolution product, the domain
of the neutrix N was the set of positive integers N 0 = {1, 2, . . . , n, . . . }, the
range was the set of real numbers, and the negligible functions were finite
linear sums of the functions
nλ lnr−1 n, lnr n
(λ > 0, r = 1, 2, . . . )
and all funtions which converge to zero in the usual sense as n tends to
infinity. In [4], the set of negligible functions was extended to include finite
linear sums of the functions nλ eµn (µ > 0). In [5], the domain of the
neutrix N was replaced by the set of real numbers, and the set of negligible
functions was extended to include finite linear sums of the functions
ν µ cos λν, ν µ sin λν
(λ 6= 0).
NEUTRIX CONVOLUTION PRODUCTS OF FUNCTIONS
415
It is easily seen that any results proved with the earlier definitions hold
with this latest definition. The following theorems, proved in [2], therefore
hold, the first showing that the neutrix convolution product is a generalization of the convolution product.
Theorem 1. Let f and g be distributions in D0 satisfying either condition (a) or condition (b) of Gel’fand and Shilov’s definition. Then the
∗ g exists and
neutrix convolution product f 
f
∗ g = f ∗ g.
Theorem 2. Let f and g be distributions in D0 and suppose that the
∗ g exists. Then the neutrix convolution
neutrix convolution product f 
product f 
∗ g 0 exists and
(f 
∗ g)0 = f 
∗ g0 .
Note, however, that equation (1) does not necessarily hold for the neutrix
convolution product and that (f 
∗ g)0 is not necessarily equal to f 0 
∗ g.
λx
We now define the locally summable functions eλx
+ , e− , cos+ (λx),
cos− (λx), sin+ (λx) and sin− (λx) by
š
š λx
0, x > 0,
e , x > 0,
λx
eλx
=
e+
=
−
eλx , x < 0,
0, x < 0,
š
š
0,
cos(λx), x > 0,
x > 0,
cos+ (λx) =
cos− (λx) =
0,
x < 0,
cos(λx), x < 0,
š
š
sin(λx), x > 0,
x > 0,
0,
sin+ (λx) =
sin− (λx) =
0,
x < 0,
sin(λx), x < 0.
It follows that
cos− (λx) + cos+ (λx) = cos(λx),
sin− (λx) + sin+ (λx) = sin(λx).
The following two theorems were proved in [4] and [5], respectively.
µx
Theorem 3. The neutrix convolution product (xr eλx
∗ (xs e+
) exists
− )
and
µx
(xr eλx
∗ (xs e+
) = Dλr Dµs
− )
λx
eµx
+ + e−
,
λ−µ
(4)
where Dλ = ∂/∂λ and Dµ = ∂/∂µ, for λ 6= µ and r, s = 0, 1, 2, . . . , these
neutrix convolution products existing as convolution products if λ > µ (or
<λ > <µ for complex λ, µ) and
r+s+1 λx
λx
(xr e−
)
∗ (xs eλx
e− ,
+ ) = B(r + 1, s + 1)x
where B denotes the Beta function, for all λ and r, s = 0, 1, 2, . . . .
(5)
416
BRIAN FISHER AND YONGPING CHEN
Note that for complex λ = λ1 + iλ2 and µ = µ1 + iµ2 , equation (4) can
be replaced by the equation
λx
eµx
+ + e−
λ−µ
(6)
µx
λx
+ e−
e+
.
λ−µ
(7)
s
r
(xr eλx
∗ (xs eµx
− )
+ ) = Dλ1 Dµ1
or by the equation
r
s
∗ (xs eµx
(xr eλx
− )
+ ) = Dλ2 Dµ2
Theorem 4. The neutrix convolution products cos− (λx) 
∗ cos+ (µx),
∗ sin+ (µx) exist
cos− (λx) 
∗ cos+ (µx), and sin− (λx) 
∗ sin+ (µx), sin− (λx) 
and
λ sin− (λx) + µ sin+ (µx)
,
λ2 − µ2
µ cos− (λt) + µ cos+ (µt)
∗ sin+ (µx) = −
,
cos− (λx) 
λ2 − µ2
λ cos− (λx) + λ cos+ (µx)
sin− (λx) 
,
∗ cos+ (µx) = −
λ2 − µ2
µ sin− (λx) + λ sin+ (µx)
,
sin− (λx) 
∗ sin+ (µx) = −
λ2 − µ2
cos− (λx) 
∗ cos+ (µx) =
(8)
(9)
(10)
(11)
for λ 6= ±µ.
We now give some generalizations of Theorems 3 and 4.
∗
Theorem 5. The neutrix convolution product [xr eλ1 x cos− (λ2 x)] 
cos+ (µ2 x)] exists and
[x e
s µ1 x
[xr eλ1 x cos− (λ2 x)] 
∗ [xs eµ1 x cos+ (µ2 x)] =
š
(λ1 − µ1 )[eµ1 x cos+ (µ2 x) + eλ1 x cos− (λ2 x)]
+
= Dλr 1 Dµs 1
2|λ − µ|2
(λ2 − µ2 )[eµ1 x sin+ (µ2 x) + eλ1 x sin− (λ2 x)]
+
2|λ − µ|2
(λ1 − µ1 )[eµ1 x cos+ (µ2 x) + eλ1 x cos− (λ2 x)]
+
−
2|λ − µ̄|2
›
(λ2 + µ2 )[eµ1 x sin+ (µ2 x) − eλ1 x sin− (λ2 x)]
−
2|λ − µ̄|2
+
6 µ = µ1 + iµ2 , λ 6= µ̄ and r, s = 0, 1, 2, . . . and
for λ = λ1 + iλ2 =
[xr eλ1 x cos− (λ2 x)] 
∗ [xs eλ1 x cos+ (λ2 x)] =
=
1
2
B(r + 1, s + 1)xr+s+1 eλ1 x cos− (λ2 x) +
(12)
NEUTRIX CONVOLUTION PRODUCTS OF FUNCTIONS
2(λ1 − µ1 )[eµ1 x cos+ (λ2 x) + eλ1 x cos− (λ2 x)]
−
(λ1 − µ1 )2 + 4λ22
›
2λ2 [eµ1 x sin+ (λ2 x) − eλ1 x sin− (λ2 x)]
,
−
(λ1 − µ1 )2 + 4λ22
µ=λ1 −iλ2
+ Dλr 1 Dµs 1
417
š
(13)
for all λ1 , λ2 =
6 0 and r, s = 0, 1, 2, . . . .
In particular
[eλ1 x cos− (λ2 x)] 
∗ [eλ1 x cos+ (λ2 x)] =
=
λ2 xeλ1 x cos− (λ2 x) − eλ1 x sin+ (λ2 x) + eλ1 x sin− (λ2 x)]
,
2λ2
for λ2 =
6 0.
Proof. Using equation (4) with λ 6= µ, µ̄, we have
4[eλ1 x cos− (λ2 x)] 
∗ [eµ1 x cos+ (µ2 x)] =
−iλ2 x
2x
2x
2x
+ e−iµ
+ e−
∗ [eµ1 x (eiµ
)] 
)] =
= [eλ1 x (eiλ
+
+
−
λ̄x
λ̄x
λx
= eλx
∗ eµ̄x
∗ eµ̄x
∗ eµx
∗ eµx
− 
+ + e− 
+ + e− 
+ + e− 
+ =
=
=
λx
λ̄x
eµ̄x + eλx
eµ̄x + e−
eµx
eµx + eλ̄x
−
−
+ + e−
+ +
=
+ +
+ +
λ−µ
λ − µ̄
λ̄ − µ
λ̄ − µ̄
µx
µ̄x
µ̄x
λ̄x
λ̄x
λx
λx
(λ1 −µ1 )(eµx
+ + e+ + e− + e− ) + i(λ2 − µ2 )(e+ − e+ + e− − e− )
+
|λ − µ|2
µx
µ̄x
µx
µ̄x
λ̄x
λ̄x
λx
+ e+
(λ1 − µ1 )(e+
+ eλx
− + e− ) + i(λ2 + µ2 )(e+ − e+ + e− − e− )
2
|λ − µ̄|
µ1 x
λ1 x
cos+ (µ2 x) + e
2(λ1 − µ1 )[e
cos− (λ2 x)]
=
+
|λ − µ|2
2(λ2 − µ2 )[eµ1 x sin+ (µ2 x) + eλ1 x sin− (λ2 x)]
+
+
|λ − µ|2
2(λ1 − µ1 )[eµ1 x cos+ (µ2 x) + eλ1 x cos− (λ2 x)]
−
+
|λ − µ̄|2
2(λ2 + µ2 )[eµ1 x sin+ (µ2 x) − eλ1 x sin− (λ2 x)]
−
|λ − µ̄|2
+
and equation (12) follows.
Similarly, using equations (4) and (5) with λ 6= λ̄ we have
4[xr eλ1 x cos− (λ2 x)] 
∗ [xs eλ1 x cos+ (λ2 x)] =
−iλ2 x
−iλ2 x
iλ2 x
2x
+ e+
= [xr eλ1 x (e−
)] 
∗ [xs eλ1 x (eiλ
)] =
+ e−
+
r λx
r λ̄x
= (xr eλx
∗ (xs eλx
∗ (xs eλ̄x
∗ (xs eλx
− )
+ ) + (x e− ) 
+ ) + (x e− ) 
+ )+
418
BRIAN FISHER AND YONGPING CHEN
+ (xr eλ̄x
∗ (xs eλ̄x
− )
+ )=
λ̄x
λ̄x
r λx
= B(r + 1, +1)xr+s+1 (eλx
∗ (xs e+
)+
− + e− ) + (x e− ) 
∗ (xs eλx
+ (xr eλ̄x
+ )=
− )
∗ (xs eλ̄x
= 2B(r + 1, s + 1)xr+s+1 eλ1 x cos− (λ2 x) + (xr eλx
+ )+
− )
+ (xr eλ̄x
∗ (xs eλx
+ ).
− )
λ̄x
∗ (xs eλx
) + (xr eλ̄x
∗ (xs e+
In order to evaluate (xr eλx
+ ) we put µ =
− )
− )
µ1 − iλ2 and consider
λx
λ̄x
e−
∗ eµ̄x

∗ eµx
+ + e− 
+ =
λx
eµx
eµ̄x + eλ̄x
+ + e−
−
+ +
=
λ−µ
λ̄ − µ̄
µ̄x
µx
µ̄x
λ̄x
λx
λx
λ̄x
(λ1 − µ1 )(eµx
+ + e+ + e− + e− ) + 2iλ2 (e+ − e+ + e− − e− )
=
2
|λ − µ|
2(λ1 − µ1 )[eµ1 x cos+ (λ2 x) + eλ1 x cos− (λ2 x)]
−
=
(λ1 − µ1 )2 + 4λ22
=
−
Then
2λ2 [eµ1 x sin+ (λ2 x) − eλ1 x sin− (λ2 x)]
.
(λ1 − µ1 )2 + 4λ22
λ̄x
∗ (xs e+
(xr eλx
) + (xr eλ̄x
∗ (xs eλx
+ )=
− )
− )
λ̄x
= Dλr 1 Dµs 1 [eλx
∗ eµ̄x
∗ eµx
− 
+ ]µ=λ1 −iλ2
+ + e− 
and equation (13) follows.
Note that by replacing x by −x in equation (12) gives an expression for
[xr eλ1 x cos+ (λ2 x)] 
∗ [xs eµ2 x cos− (µ2 x)].
Expressions for
[xr eλ1 x cos± (λ2 x)] 
∗ [xs eµ2 x sin∓ (µ2 x)],
[xr eλ1 x sin± (λ2 x)] 
∗ [xs eµ2 x cos∓ (µ2 x)],
∗ [xs eµ2 x sin∓ (µ2 x)]
[xr eλ1 x sin± (λ2 x)] 
can be obtained similarly.
References
1. I. M. Gel’fand and G. E. Shilov, Generalized functions, vol. I. Academic Press, 1964.
NEUTRIX CONVOLUTION PRODUCTS OF FUNCTIONS
419
2. B. Fisher, Neutrices and the convolution of distributions. Univ.
Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 17(1987), 119–135.
3. J. G. van der Corput, Introduction to the neutrix calculus. J. Analyse
Math. 7(1959–60), 291–398.
4. B. Fisher and Y. Chen, Non-commutative neutrix convolution products of functions. Math. Balkanica (to appear).
5. B. Fisher, S. Yakubovich, and Y. Chen, Some non-commutative neutrix convolution products of functions. Math. Balkanica (to appear).
(Received 16.08.1995)
Authors’ addresses:
B. Fisher
Department of Mathematics and Computer Science
University of Leicester
Leicester, LE1 7RH,
England
Y. Chen
Department of Mathematics and Statistics
University of Brunel
Uxbridge, UB8 3PH
England