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 x^{r} cos_{−}(λx) and x^{s} 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 D′ be the space of distributions defined on D. The convolution product f ∗ g of two distributions f and g in D′ is then usually defined by the equation

⟨(f ∗ g)(x), ϕ⟩ = ⟨f(y), ⟨g(x), ϕ(x + y)⟩⟩

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

(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,  (2)
(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

413 neutrix convolution product we first of all let τ be a function in D satisfying the following properties:

1991 Mathematics Subject Classification. 46F10.

Key words and phrases. Neutrix, neutrix limit, neutrix convolution product.

1072-947X/97/0900-0413$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

(i) τ(x) = τ(−x),

(ii) 0 ≤ τ(x) ≤ 1,

(iii) τ(x) = 1 for |x| ≤ \frac{1}{2},

(iv) τ(x) = 0 for |x| ≥ 1. The function τ_{ν} is now defined by

τ_{ν}(x) = \{ \begin{matrix}1, & |x| ≤ ν, \\ τ(ν^{ν}x − ν^{ν+1}), & x > ν, \\ τ(ν^{ν}x + ν^{ν+1}), & x < −ν, \end{matrix}

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 D′ and let f_{ν} = fτ_{ν} for ν > 0. Then the neutrix convolution product f ◯ ∗ g is defined as the neutrix limit of the sequence {f_{ν} ∗ g}, provided that the limit h exists in the sense that

N- lim_{ν→∞}⟨f_{ν} ∗ g, ϕ⟩ = ⟨h, ϕ⟩,

for all ϕ in D, where N is the neutrix (see van der Corput [3]), having domain N′ the positive reals and range N′′ the complex numbers, with negligible functions finite linear sums of the functions

ν^{λ} ln^{r−1} ν, ln^{r} ν, ν^{r−1}e^{μν} (real λ > 0, complex μ ≠ 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 f_{ν} ∗ g is defined in Gel’fand and Shilov’s sense, the distribution f_{ν} 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′ = {1, 2, …, n, … }, the range was the set of real numbers, and the negligible functions were finite linear sums of the functions

n^{λ} ln^{r−1} n, ln^{r} 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 n^{λ}e^{μn} (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 ≠ 0).

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 D′ satisfying either condition (a) or condition (b) of Gel’fand and Shilov’s definition. Then the neutrix convolution product f ◯ ∗ g exists and

f ◯ ∗ g = f ∗ g.

Theorem 2. Let f and g be distributions in D′ and suppose that the neutrix convolution product f ◯ ∗ g exists. Then the neutrix convolution product f ◯ ∗ g′ exists and

(f ◯ ∗ g)′ = f ◯ ∗ g′.

Note, however, that equation (1) does not necessarily hold for the neutrix convolution product and that (f ◯ ∗ g)′ is not necessarily equal to f′ ◯ ∗ g.

We now define the locally summable functions e_{+}^{λx}, e_{−}^{λx}, cos_{+}(λx), cos_{−}(λx), sin_{+}(λx) and sin_{−}(λx) by

e_{+}^{λx} = \{ \begin{matrix}e^{λx}, & x > 0, \\ 0, & x < 0, \end{matrix} e_{−}^{λx} = \{ \begin{matrix}0, & x > 0, \\ e^{λx}, & x < 0, \end{matrix}
cos_{+}(λx) = \{ \begin{matrix}cos(λx), & x > 0, \\ 0, & x < 0, \end{matrix} cos_{−}(λx) = \{ \begin{matrix}0, & x > 0, \\ cos(λx), & x < 0, \end{matrix}
sin_{+}(λx) = \{ \begin{matrix}sin(λx), & x > 0, \\ 0, & x < 0, \end{matrix} sin_{−}(λx) = \{ \begin{matrix}0, & x > 0, \\ sin(λx), & x < 0. \end{matrix}

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.

Theorem 3. The neutrix convolution product (x^{r}e_{−}^{λx}) ◯ ∗ (x^{s}e_{+}^{μx}) exists and

(x^{r}e_{−}^{λx}) ◯ ∗ (x^{s}e_{+}^{μx}) = D_{λ}^{r}D_{μ}^{s} \frac{e_{+}^{μx} + e_{−}^{λx}}{λ − μ} ,  (4)

where D_{λ} = ∂/∂λ and D_{μ} = ∂/∂μ, for λ ≠ μ and r, s = 0, 1, 2, …, these neutrix convolution products existing as convolution products if λ > μ (or ℜλ > ℜμ for complex λ, μ) and

(x^{r}e_{−}^{λx}) ◯ ∗ (x^{s}e_{+}^{λx}) = B(r + 1, s + 1)x^{r+s+1}e_{−}^{λx},  (5)

where B denotes the Beta function, for all λ and r, s = 0, 1, 2, ….

Note that for complex λ = λ_{1} + iλ_{2} and μ = μ_{1} + iμ_{2}, equation (4) can be replaced by the equation

(x^{r}e_{−}^{λx}) ◯ ∗ (x^{s}e_{+}^{μx}) = D_{λ_{1}}^{r}D_{μ_{1}}^{s} \frac{e_{+}^{μx} + e_{−}^{λx}}{λ − μ}  (6)

or by the equation

(x^{r}e_{−}^{λx}) ◯ ∗ (x^{s}e_{+}^{μx}) = D_{λ_{2}}^{r}D_{μ_{2}}^{s} \frac{e_{+}^{μx} + e_{−}^{λx}}{λ − μ} .  (7)

Theorem 4. The neutrix convolution products cos_{−}(λx) ◯ ∗ cos_{+}(μx), cos_{−}(λx) ◯ ∗ sin_{+}(μx), sin_{−}(λx) ◯ ∗ cos_{+}(μx), and sin_{−}(λx) ◯ ∗ sin_{+}(μx) exist and

cos_{−}(λx) ◯ ∗ cos_{+}(μx) = \frac{λ sin_{−}(λx) + μ sin_{+}(μx)}{λ^{2} − μ^{2}},  (8)
cos_{−}(λx) ◯ ∗ sin_{+}(μx) = −\frac{μ cos_{−}(λt) + μ cos_{+}(μt)}{λ^{2} − μ^{2}},  (9)
sin_{−}(λx) ◯ ∗ cos_{+}(μx) = −\frac{λ cos_{−}(λx) + λ cos_{+}(μx)}{λ^{2} − μ^{2}},  (10)
sin_{−}(λx) ◯ ∗ sin_{+}(μx) = −\frac{μ sin_{−}(λx) + λ sin_{+}(μx)}{λ^{2} − μ^{2}},  (11)

for λ ≠ ±μ.

We now give some generalizations of Theorems 3 and 4.

Theorem 5. The neutrix convolution product [x^{r}e^{λ_{1}x} cos_{−}(λ_{2}x)] ◯ ∗ [x^{s}e^{μ_{1}x} cos_{+}(μ_{2}x)] exists and

[x^{r}e^{λ_{1}x} cos_{−}(λ_{2}x)] ◯ ∗ [x^{s}e^{μ_{1}x} cos_{+}(μ_{2}x)] =

= D_{λ_{1}}^{r}D_{μ_{1}}^{s} \{\frac{(λ_{1} − μ_{1})[e^{μ_{1}x} cos_{+}(μ_{2}x) + e^{λ_{1}x} cos_{−}(λ_{2}x)]}{2|λ − μ|^{2}}+

+\frac{(λ_{2} − μ_{2})[e^{μ_{1}x} sin_{+}(μ_{2}x) + e^{λ_{1}x} sin_{−}(λ_{2}x)]}{2|λ − μ|^{2}}+

+\frac{(λ_{1} − μ_{1})[e^{μ_{1}x} cos_{+}(μ_{2}x) + e^{λ_{1}x} cos_{−}(λ_{2}x)]}{2|λ − μ̄|^{2}}−

−\frac{(λ_{2} + μ_{2})[e^{μ_{1}x} sin_{+}(μ_{2}x) − e^{λ_{1}x} sin_{−}(λ_{2}x)]}{2|λ − μ̄|^{2}}\}  (12)

for λ = λ_{1} + iλ_{2} ≠ μ = μ_{1} + iμ_{2}, λ ≠ μ̄ and r, s = 0, 1, 2, … and

[x^{r}e^{λ_{1}x} cos_{−}(λ_{2}x)] ◯ ∗ [x^{s}e^{λ_{1}x} cos_{+}(λ_{2}x)] =
= \frac{1}{2} B(r + 1, s + 1)x^{r+s+1}e^{λ_{1}x} cos_{−}(λ_{2}x) +

+ D_{λ_{1}}^{r}D_{μ_{1}}^{s} \{\frac{2(λ_{1} − μ_{1})[e^{μ_{1}x} cos_{+}(λ_{2}x) + e^{λ_{1}x} cos_{−}(λ_{2}x)]}{(λ_{1} − μ_{1})^{2} + 4λ_{2}^{2}}−

− \frac{2λ_{2}[e^{μ_{1}x} sin_{+}(λ_{2}x) − e^{λ_{1}x} sin_{−}(λ_{2}x)]}{(λ_{1} − μ_{1})^{2} + 4λ_{2}^{2}}\}_{μ=λ_{1}−iλ_{2}} , (13)

for all λ_{1}, λ_{2} ≠ 0 and r, s = 0, 1, 2, …. In particular

[e^{λ_{1}x} cos_{−}(λ_{2}x)] ◯ ∗ [e^{λ_{1}x} cos_{+}(λ_{2}x)] =

= \frac{λ_{2}xe^{λ_{1}x} cos_{−}(λ_{2}x) − e^{λ_{1}x} sin_{+}(λ_{2}x) + e^{λ_{1}x} sin_{−}(λ_{2}x)]}{2λ_{2}},

for λ_{2} ≠ 0. Proof. Using equation (4) with λ ≠ μ, μ̄, we have 4[e^{λ_{1}x} cos_{−}(λ_{2}x)] ◯ ∗ [e^{μ_{1}x} cos_{+}(μ_{2}x)] = = [e^{λ_{1}x}(e_{−}^{iλ_{2}x} + e_{−}^{−iλ_{2}x})] ◯ ∗ [e^{μ_{1}x}(e_{+}^{iμ_{2}x} + e_{+}^{−iμ_{2}x})] = = e_{−}^{λx} ◯ ∗ e_{+}^{μx} + e_{−}^{λ̄x} ◯ ∗ e_{+}^{μx} + e_{−}^{λx} ◯ ∗ e_{+}^{μ̄x} + e_{−}^{λ̄x} ◯ ∗ e_{+}^{μ̄x} =

= \frac{e_{+}^{μx} + e_{−}^{λx}}{λ − μ} + \frac{e_{+}^{μx} + e_{−}^{λ̄x}}{λ̄ − μ} + \frac{e_{+}^{μ̄x} + e_{−}^{λx}}{λ − μ̄} + \frac{e_{+}^{μ̄x} + e_{−}^{λ̄x}}{λ̄ − μ̄} =

= \frac{(λ_{1}−μ_{1})(e_{+}^{μx}+ e_{+}^{μ̄x}+ e_{−}^{λx}+ e_{−}^{λ̄x}) + i(λ_{2} − μ_{2})(e_{+}^{μ̄x} − e_{+}^{μx} + e_{−}^{λ̄x} − e_{−}^{λx})}{|λ − μ|^{2}} +

+ \frac{(λ_{1} − μ_{1})(e_{+}^{μx} + e_{+}^{μ̄x} + e_{−}^{λx} + e_{−}^{λ̄x}) + i(λ_{2} + μ_{2})(e_{+}^{μx} − e_{+}^{μ̄x} + e_{−}^{λ̄x} − e_{−}^{λx})}{|λ − μ̄|^{2}}

= \frac{2(λ_{1} − μ_{1})[e^{μ_{1}x} cos_{+}(μ_{2}x) + e^{λ_{1}x} cos_{−}(λ_{2}x)]}{|λ − μ|^{2}} +

+ \frac{2(λ_{2} − μ_{2})[e^{μ_{1}x} sin_{+}(μ_{2}x) + e^{λ_{1}x} sin_{−}(λ_{2}x)]}{|λ − μ|^{2}} +

+ \frac{2(λ_{1} − μ_{1})[e^{μ_{1}x} cos_{+}(μ_{2}x) + e^{λ_{1}x} cos_{−}(λ_{2}x)]}{|λ − μ̄|^{2}} −

− \frac{2(λ_{2} + μ_{2})[e^{μ_{1}x} sin_{+}(μ_{2}x) − e^{λ_{1}x} sin_{−}(λ_{2}x)]}{|λ − μ̄|^{2}}

and equation (12) follows.

Similarly, using equations (4) and (5) with λ ≠ λ̄ we have 4[x^{r}e^{λ_{1}x} cos_{−}(λ_{2}x)] ◯ ∗ [x^{s}e^{λ_{1}x} cos_{+}(λ_{2}x)] =

= [x^{r}e^{λ_{1}x}(e_{−}^{iλ_{2}x} + e_{−}^{−iλ_{2}x})] ◯ ∗ [x^{s}e^{λ_{1}x}(e_{+}^{iλ_{2}x} + e_{+}^{−iλ_{2}x})] = = (x^{r}e_{−}^{λx}) ◯ ∗ (x^{s}e_{+}^{λx}) + (x^{r}e_{−}^{λx}) ◯ ∗ (x^{s}e_{+}^{λ̄x}) + (x^{r}e_{−}^{λ̄x}) ◯ ∗ (x^{s}e_{+}^{λx}) +

+ (x^{r}e_{−}^{λ̄x}) ◯ ∗ (x^{s}e_{+}^{λ̄x}) =

= B(r + 1, +1)x^{r+s+1}(e_{−}^{λx} + e_{−}^{λ̄x}) + (x^{r}e_{−}^{λx}) ◯ ∗ (x^{s}e_{+}^{λ̄x}) +

+ (x^{r}e_{−}^{λ̄x}) ◯ ∗ (x^{s}e_{+}^{λx}) =

= 2B(r + 1, s + 1)x^{r+s+1}e^{λ_{1}x} cos_{−}(λ_{2}x) + (x^{r}e_{−}^{λx}) ◯ ∗ (x^{s}e_{+}^{λ̄x}) +

+ (x^{r}e_{−}^{λ̄x}) ◯ ∗ (x^{s}e_{+}^{λx}).

In order to evaluate (x^{r}e_{−}^{λx}) ◯ ∗ (x^{s}e_{+}^{λ̄x}) + (x^{r}e_{−}^{λ̄x}) ◯ ∗ (x^{s}e_{+}^{λx}) we put μ = μ_{1} − iλ_{2} and consider

e_{−}^{λx} ◯ ∗ e_{+}^{μx} + e_{−}^{λ̄x} ◯ ∗ e_{+}^{μ̄x} = \frac{e_{+}^{μx} + e_{−}^{λx}}{λ − μ} + \frac{e_{+}^{μ̄x} + e_{−}^{λ̄x}}{λ̄ − μ̄} =

= \frac{(λ_{1} − μ_{1})(e_{+}^{μx} + e_{+}^{μ̄x} + e_{−}^{λx} + e_{−}^{λ̄x}) + 2iλ_{2}(e_{+}^{μ̄x} − e_{+}^{μx} + e_{−}^{λ̄x} − e_{−}^{λx})}{|λ − μ|^{2}} =

= \frac{2(λ_{1} − μ_{1})[e^{μ_{1}x} cos_{+}(λ_{2}x) + e^{λ_{1}x} cos_{−}(λ_{2}x)]}{(λ_{1} − μ_{1})^{2} + 4λ_{2}^{2}} −

− \frac{2λ_{2}[e^{μ_{1}x} sin_{+}(λ_{2}x) − e^{λ_{1}x} sin_{−}(λ_{2}x)]}{(λ_{1} − μ_{1})^{2} + 4λ_{2}^{2}}.

Then

(x^{r}e_{−}^{λx}) ◯ ∗ (x^{s}e_{+}^{λ̄x}) + (x^{r}e_{−}^{λ̄x}) ◯ ∗ (x^{s}e_{+}^{λx}) =
= D_{λ_{1}}^{r}D_{μ_{1}}^{s}[e_{−}^{λx} ◯ ∗ e_{+}^{μx} + e_{−}^{λ̄x} ◯ ∗ e_{+}^{μ̄x}]_{μ=λ_{1}−iλ_{2}}

and equation (13) follows.

Note that by replacing x by −x in equation (12) gives an expression for

[x^{r}e^{λ_{1}x} cos_{+}(λ_{2}x)] ◯ ∗ [x^{s}e^{μ_{2}x} cos_{−}(μ_{2}x)].

Expressions for

[x^{r}e^{λ_{1}x} cos_{±}(λ_{2}x)] ◯ ∗ [x^{s}e^{μ_{2}x} sin_{∓}(μ_{2}x)],
[x^{r}e^{λ_{1}x} sin_{±}(λ_{2}x)] ◯ ∗ [x^{s}e^{μ_{2}x} cos_{∓}(μ_{2}x)],
[x^{r}e^{λ_{1}x} sin_{±}(λ_{2}x)] ◯ ∗ [x^{s}e^{μ_{2}x} sin_{∓}(μ_{2}x)]

can be obtained similarly.

References

1. I. M. Gel’fand and G. E. Shilov, Generalized functions, vol. I. Academic Press, 1964.

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