GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 2, 1996, 133-140

ON THE NON-COMMUTATIVE NEUTRIX PRODUCT (x_{+}^{r} ln x_{+}) ∘ x_{−}^{−s}

ADEM KILIÇMAN AND BRIAN FISHER

Abstract. The non-commutative neutrix product of the distributions x_{+}^{r} ln x_{+} and x_{−}^{−s} is evaluated for r = 0, 1, 2, … and s = 1, 2, . . .. Further neutrix products are then deduced.

In the following, we let N be the neutrix (see van der Corput [1]) having domain N′ = {1, 2, …, n, … } and range the real numbers, with negligible functions finite linear sums of the functions n^{λ} ln^{r−1} n, ln^{r} n, λ > 0, r = 1, 2, …, and all functions which converge to zero in the normal sense as n tends to infinity.

We now let ρ(x) be any infinitely differentiable function having the following properties:

(i) ρ(x) =0 for |x| ≥ 1,

(ii) ρ(x) ≥ 0,

(iii) ρ(x) = ρ(−x),

(iv) ∫_{−1}^{1} ρ(x) dx = 1.

Putting δ_{n}(x) = nρ(nx) for n = 1, 2, …, it follows that {δ_{n}(x)} is a regular sequence of infinitely differentiable functions converging to the Dirac deltafunction δ(x).

Now let D be the space of infinitely differentiable functions with compact support and let D′ be the space of distributions defined on D. Then if f is an arbitrary distribution in D′, we define f_{n}(x) = (f ∗δ_{n})(x) = ⟨f(t), δ_{n}(x−t)⟩ for n = 1, 2, . . .. It follows that {f_{n}(x)} is a regular sequence of infinitely differentiable functions converging to the distribution f(x).

A first extension of the product of a distribution and an infinitely differentiable function is the following (see for example [2] or [3]).

133

1991 Mathematics Subject Classification. 46F10.

Key words and phrases. Distribution, delta-function, neutrix, neutrix limit, neutrix product.

1072-947X/96/0300-0133$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

Definition 1. Let f and g be distributions in D′ for which on the interval (a, b), f is the kth derivative of a locally summable function F in L^{p}(a, b) and g^{(k)} is a locally summable function in L^{q}(a, b) with 1/p+1/q = 1. Then the product fg = gf of f and g is defined on the interval (a, b) by

fg = ∑_{i=0}^{k} \binom{k}{i}(−1)^{i}[F g^{(i)}]^{(k−i)}.

The following definition for the neutrix product of two distributions was given in [4] and generalizes Definition 1.

Definition 2. Let f and g be distributions in D′ and let g_{n}(x) = (g ∗ δ_{n})(x). We say that the neutrix product f ∘ g of f and g exists and is equal to the distribution h on the interval (a, b) if

N- lim_{n→∞}⟨f(x)g_{n}(x), ϕ(x)⟩ = ⟨h(x), ϕ(x)⟩^{1}

for all functions ϕ in D with support contained in the interval (a, b).

Note that if

lim_{n→∞}⟨f(x)g_{n}(x), ϕ(x)⟩ = ⟨h(x), ϕ(x)⟩,

we simply say that the product f.g exists and equals h (see [4]).

It is obvious that if the product f.g exists then the neutrix product f ∘ g exists and f.g = f ∘ g. Further, it was proved in [4] that if the product fg exists by Definition 1 then the product f.g exists by Definition 2 and fg = f.g. Note also that although the product defined in Definition 1 is always commutative, the product and neutrix product defined in Definition 2 is in general non-commutative.

The following theorem holds (see [5]).

Theorem 1. Let f and g be distributions in D′ and suppose that the neutrix products f ∘ g^{(i)} (or f^{(i)} ∘ g) exist on the interval (a, b) for i = 0, 1, 2, …, r. Then the neutrix products f^{(k)} ∘ g (or f ∘ g^{(k)}) exist on the interval (a, b) for k = 1, 2, …, r and

f^{(k)} ∘ g = ∑_{i=0}^{k} \binom{k}{i}(−1)^{i}[f ∘ g^{(i)}]^{(k−i)}  (1)

or

f ∘ g^{(k)} = ∑_{i=0}^{k} \binom{k}{i}(−1)^{i}[f^{(i)} ∘ g]^{(k−i)}  (2)

on the interval (a, b) for k = 1, 2, …, r.

1See [1] or [4] for the definition of N-lim.

In the next two theorems, which were proved in [5] and [6] respectively, the distributions x_{+}^{−r} and x_{−}^{−r} are defined by

x_{+}^{−r} = \frac{(−1)^{r−1}}{(r − 1)!} (ln x_{+})^{(r)}, x_{−}^{−r} = −\frac{1}{(r − 1)!} (ln x_{−})^{(r)},

for r = 1, 2, … and not as in the book of Gel’fand and Shilov [7].

Theorem 2. The neutrix products x_{+}^{r} ∘ x_{−}^{−s} and x_{−}^{−s} ∘ x_{+}^{r} exist and

x_{+}^{r} ∘ x_{−}^{−s} = x_{+}^{r}x_{−}^{−s} = 0,  (3)
x_{−}^{−s} ∘ x_{+}^{r} = x_{−}^{−s}x_{+}^{r} = 0  (4)

for r = s, s + 1, … and s = 1, 2, … and

x_{+}^{r} ∘ x_{−}^{−s} = ∑_{i=r+1}^{s} \binom{s}{i}\frac{(−1)^{i−1}r!}{(s − 1)!}c_{1}(ρ)δ^{(s−r−1)}(x),  (5)

x_{−}^{−s} ∘ x_{+}^{r} = ∑_{i=r+1}^{s} \binom{s}{i}\frac{(−1)^{i−1}r!}{(s − 1)!}[c_{1}(ρ)+ \frac{1}{2}ψ(i−r−1)]δ^{(s−r−1)}(x)  (6)

for r = 0, 1, …, s − 1 and s = 1, 2, …, where

c_{1}(ρ) = ∫_{0}^{1} ln tρ(t) dt, ψ(r) = \begin{cases}0, & r = 0, \\ ∑_{i=1}^{r} i^{−1}, & r ≥ 1. \end{cases}

Theorem 3. The neutrix products x_{+}^{−r} ∘ x_{−}^{−s} and x_{−}^{−s} ∘ x_{+}^{−r} exist and

x_{+}^{−r} ∘ x_{−}^{−s} = \frac{(−1)^{r}c_{1}(ρ)}{(r + s − 1)!}δ^{(r+s−1)}(x),  (7)
x_{−}^{−s} ∘ x_{+}^{−r} = \frac{(−1)^{r−1}c_{1}(ρ)}{(r + s − 1)!}δ^{(r+s−1)}(x) for r, s = 1, 2, ….  (8)

It was shown in [8] that with suitable choice of the function ρ, c_{1}(ρ) can take any negative value.

We now prove the following theorem.

Theorem 4. The neutrix products (x_{+}^{r} ln x_{+}) ∘ x_{−}^{−s} and x_{−}^{−s} ∘ (x_{+}^{r} ln x_{+}) exist and

(x_{+}^{r} ln x_{+}) ∘ x_{−}^{−s} = (x_{+}^{r} ln x_{+})x_{−}^{−s} = 0,  (9)
x_{−}^{−s} ∘ (x_{+}^{r} ln x_{+}) = x_{−}^{−s}(x_{+}^{r} ln x_{+}) = 0  (10)

136 ADEM KILIÇMAN AND BRIAN FISHER

for r = s, s + 1, s + 2 … and s = 1, 2, … and

(x_{+}^{r} ln x_{+}) ∘ x_{−}^{−s}=\frac{(−1)^{r}}{(s − r − 1)!}(c_{2} − \frac{π^{2}}{12} )δ^{(s−r−1)}(x)

− ∑_{i=r+1}^{s−1} \frac{(−1)^{i}r!c_{1}}{(s − i − 1)!i!(i − r)}δ^{(s−r−1)}(x)

− ψ(r) ∑_{i=r+1}^{s} \frac{(−1)^{i}sr!c_{1}}{i!(s − i)!} δ^{(s−r−1)}(x),  (11)
x_{−}^{−s} ∘ (x_{+}^{r} ln x_{+})=\frac{(−1)^{r}}{(s − r − 1)!}(c_{2} − \frac{π^{2}}{12} )δ^{(s−r−1)}(x)

− ∑_{i=r+1}^{s−1} \frac{(−1)^{i}r!c_{1}}{(s − i − 1)!i!(i − r)}δ^{(s−r−1)}(x)

− ψ(r) ∑_{i=r+1}^{s} \frac{(−1)^{i}sr!}{i!(s − i)!}[c_{1}+ \frac{1}{2}ψ(i−r−1)]δ^{(s−r−1)}(x) (12)

for r = 0, 1, 2, …, s − 1 and s = 1, 2 …, where

c_{2}(ρ) = ∫_{0}^{1} ln^{2} tρ(t) dt.

Proof. We first of all prove that

ln x_{+} ∘ x_{−}^{−1} = (c_{2} − \frac{π^{2}}{12} )δ(x).  (13)

We put (x_{−}^{−1})_{n} = x_{−}^{−1} ∗ δ_{n}(x) so that

(x_{−}^{−1})_{n} = − ∫_{x}^{1/n} ln(t − x)δ′_{n}(t) dt

on the interval [0, 1/n], the intersection of the supports of ln x_{+} and (x_{−}^{−1})_{n}. Then

⟨ln x_{+}, (x_{−}^{−1})_{n}⟩ = − ∫_{0}^{1/n} ln x ∫_{x}^{1/n} ln(t − x)δ′_{n}(t) dt dx
= − ∫_{0}^{1/n} δ′_{n}(t) ∫_{0}^{t} ln x ln(t − x) dx dt
= − ∫_{0}^{1} ρ′(u) ∫_{0}^{u}[ln v − ln n][ln(u − v) − ln n] dv du

ON THE NON-COMMUTATIVE NEUTRIX PRODUCT 137

on making the substitutions nt = u and nx = v. It follows that

N- lim_{n→∞}⟨ln x_{+}, (x_{−}^{−1})_{n}⟩ = − ∫_{0}^{1} ρ′(u) ∫_{0}^{u} ln v ln(u − v) dv du

= − ∫_{0}^{1} uρ′(u) ∫_{0}^{1}[ln u + ln y][ln u + ln(1 − y)] dy du  (14)

on making the substitution v = uy.

Now

∫_{0}^{1} ln y dy = ∫_{0}^{1} ln(1 − y) dy = −1,
∫_{0}^{1} ln y ln(1 − y) dy = − ∫_{0}^{1} ln(1 − y) dy + ∫_{0}^{1} \frac{y ln y}{1 − y} dy

= 1 + ∑_{i=1}^{∞} ∫_{0}^{1} y^{i} ln y dy = 1 − ∑_{i=1}^{∞}(i + 1)^{−2} = 2 − \frac{π^{2}}{6} ,

∫_{0}^{1} uρ′(u) du = − ∫_{0}^{1} ρ(u) du = −\frac{1}{2},
∫_{0}^{1} u ln uρ′(u) du = − ∫_{0}^{1}(1 + ln u)ρ(u) du = −\frac{1}{2} − c_{1},
∫_{0}^{1} u ln^{2} uρ′(u) du = − ∫ (2 ln u + ln^{2} u)ρ(u) du = −2c_{1} − c_{2}

and it follows from these equations and equation (14) that

N- lim_{n→∞}⟨ln x_{+}, (x_{−}^{−1})_{n}⟩ = c_{2} − \frac{π^{2}}{12} .  (15)

Further, it follows as above that ⟨ln x_{+}, x(x_{−}^{−1})_{n}⟩ = − ∫_{0}^{1/n} x ln x ∫_{x}^{1/n} ln(t − x)δ′_{n}(t) dt dx = −n^{−1} ∫_{0}^{1} ρ′(u) ∫_{0}^{u} v[ln v − ln n][ln(u − v) − ln u] dv du

= O(n^{−1} ln n).

Now let ϕ be an arbitrary function in D. Then ϕ(x) = ϕ(0) + xϕ′(ξx), where 0 < ξ < 1. It follows that

⟨ln x_{+}(x_{−}^{−1})_{n}, ϕ(x)⟩ − ϕ(0)⟨ln x_{+}, (x_{−}^{−1})_{n}⟩ = ⟨ln x_{+}, x(x_{−}^{−1})_{n}ϕ′(ξx)⟩
= O(n^{−1} ln n)  (16)

since ⟨ln x_{+}, x(x_{−}^{−1})_{n}⟩ = O(n^{−1} ln n). Thus

N- lim_{n→∞}⟨ln x_{+}(x_{−}^{−1})_{n}, ϕ(x)⟩ = N- lim_{n→∞}ϕ(0)⟨ln x_{+}, (x_{−}^{−1})_{n}⟩ = (c_{2} − \frac{π^{2}}{12} )ϕ(0)

on using equations (15) and (16). Equation (9) follows.

We now define the function f(x_{+}, r) by

f(x_{+}, r) = \frac{x_{+}^{r} ln x_{+} − ψ(r)x_{+}^{r}}{r!}

and it follows easily by induction that f^{(i)}(x_{+}, r) = f(x_{+}, r − i) , for i = 0, 1, …, r. In particular, f^{(r)}(x_{+}, r) = ln x_{+} , so that

f^{(i)}(x_{+}, r) = (−1)^{i−r−1}(i − r − 1)!x_{+}^{−i+r},

for i = r + 1, r + 2, …. Now the product of the functions x_{+}^{i} and x_{+}^{i} ln x_{+} and the distribution x_{−}^{−1} exists by Definition 1 and it is easily seen that

x_{+}^{i}x_{−}^{−1} = (x_{+}^{i} ln x_{+})x_{−}^{−1} = 0,  (17)

for i = 1, 2, …, r. Using equation (13) we have

f^{(r)}(x_{+}, r) ∘ x_{−}^{−1} = (c_{2} − \frac{π^{2}}{12} )δ(x)  (18)

and using equation (7) we have

f^{(i)}(x_{+}, r) ∘ x_{−}^{−1} = −\frac{c_{1}}{i − r} δ^{(i−r)}(x)  (19)

for i = r + 1, r + 2, . . ..

Using equations (2) and (17) we now have

(s − 1)!f(x_{+}, r)x_{−}^{−s} = ∑_{i=0}^{s−1} \binom{s − 1}{i}(−1)^{i}[f^{(i)}(x_{+}, r)x_{−}^{−1}]^{(s−i−1)}

= \frac{(s − 1)!}{r!}[x_{+}^{r} ln x_{+} − ψ(r)x_{+}^{r}]x_{−}^{−s} = 0,

for r = s, s + 1, s + 2, … and s = 1, 2, . . .. Equations (9) follow on using equations (3).

When r < s we have

(s − 1)!f(x_{+}, r) ∘ x_{−}^{−s} = ∑_{i=r}^{s−1} \binom{s − 1}{i}(−1)^{i}[f^{(i)}(x_{+}, r) ∘ x_{−}^{−1}]^{(s−i−1)}

= \binom{s − 1}{r}(−1)^{r}(c_{2} − 2 + \frac{π^{2}}{12} )δ^{(s−r−1)}(x) +

− ∑_{i=r+1}^{s−1} \binom{s − 1}{i}\frac{(−1)^{i}c_{1}}{i − r} δ^{(s−r−1)}(x)

on using equations (2), (17), (18) and (19). It now follows that

(x_{+}^{r} ln x_{+}) ∘ x_{−}^{−s} = r!f(x_{+}, r) ∘ x_{−}^{−s} + ψ(r)x_{+}^{r} ∘ x_{−}^{−s}

and equation (11) follows on using equation (5).

We now consider the product x_{−}^{−s} ∘ (x_{+}^{−r} ln x_{+}). The product ln x_{−} ln x_{+} exists by Definition 1 and ln x_{−} ln x_{+} = 0. Differentiating, we get

x_{−}^{−1} ∘ ln x_{+} = ln x_{−} ∘ x_{+}^{−1} = (c_{2} − \frac{π^{2}}{12} )δ(x)  (20)

on replacing x by −x in equation (13).

As above, we have

x_{−}^{−1}x_{+}^{i} = x_{−}^{−1}(x_{+}^{i} ln x_{+}) = 0,  (21)

for i = 0, 1, …, r − 1. Using equation (20) we have

x_{−}^{−1} ∘ f^{(r)}(x_{+}, r) = (c_{2} − \frac{π^{2}}{12} )δ(x)  (22)

and using equation (8) we have

x_{−}^{−1} ∘ f^{(i)}(x_{+}, r) = \frac{c_{1}}{i − r} δ^{(i−r)}(x),  (23)

for i = r + 1, r + 2, . . .. Equations (10) follow as above on using equations (1) and (21) and equations (12) follow on using equations (1), (6), (20), (21), (22), and (23).

Corollary. The neutrix products (x_{−}^{−r} ln x_{−}) ∘ x_{+}^{−s} and x_{+}^{−s} ∘ (x_{−}^{r} ln x_{−}) exist and

(x_{−}^{r} ln x_{−}) ∘ x_{+}^{−s} = (x_{−}^{−r} ln x_{−})x_{+}^{−s} = 0,
x_{+}^{−s} ∘ (x_{−}^{r} ln x_{−}) = x_{+}^{−s}(x_{−}^{r} ln x_{−}) = 0,

for r = s, s + 1, s + 2, … and s = 1, 2, … and

(x_{−}^{r} ln x_{−}) ∘ x_{+}^{−s} = \frac{(−1)^{s+1}}{(s − r − 1)!}(c_{2} − \frac{π^{2}}{12} )δ^{(s−r−1)}(x)

+ ∑_{i=r+1}^{s−1} \frac{(−1)^{s−r+i}r!c_{1}}{(s − i − 1)!i!(i − r)}δ^{(s−r−1)}(x)

+ ψ(r) ∑_{i=r+1}^{s} \frac{(−1)^{s−r+i}sr!c_{1}}{i!(s − i)!} δ^{(s−r−1)}(x),

x_{+}^{−s} ∘ (x_{−}^{r} ln x_{−}) = \frac{(−1)^{s+1}}{(s − r − 1)!}(c_{2} − \frac{π^{2}}{12} )δ^{(s−r−1)}(x)

+ ∑_{r+1}^{s−1} \frac{(−1)^{s−r+i}r!c_{1}}{(s − i − 1)!i!(i − r)}δ^{(s−r−1)}(x)

+ ψ(r) ∑_{i=r+1}^{s} \frac{(−1)^{s−r+i}sr!}{i!(s − i)!}[c_{1} + \frac{1}{2}ψ(i − r − 1)]δ^{(s−r−1)}(x)

for r = 0, 1, 2, …, s − 1 and s = 1, 2, . . ..

Proof. The results follow immediately on replacing x by −x in equations (9), (10), (11), and (12).

References

1. J. G. van der Corput, Introduction to the neutrix calculus. J. Analyse Math. 7(1959–60), 291–398.

2. B. Fisher, The product of distributions. Quart. J. Math. Oxford (2) 22(1971), 291–298.

3. B. Fisher, On defining the product of distributions. Math. Nachr. 99(1980), 239–249.

4. B. Fisher, A non-commutative neutrix product of distributions. Math. Nachr. 108(1982), 117–127.

5. B. Fisher, E. Savaş, S. Pehlivan, and E. Özçaḡ, Results on the noncommutative neutrix product of distributions. Math. Balkanica 7(1993), 347–356.

6. B. Fisher and A. Kiliçman, The non-commutative neutrix product of the distributions x_{+}^{−r} and x_{−}^{−s}. Math. Balkanica (to appear).

7. I. M. Gel’fand and G. E. Shilov, Generalized functions. Vol. I: Properties and operations. Academic Press, New York–London, 1964; Russian original: Fizmatgiz, Moscow, 1958.

8. B. Fisher, Some results on the non-commutative neutrix product of distributions. Trabajos de Matematica 44, Buenos Aires, 1983.

(Received 10.08.1994)

Authors’ address:

Department of Mathematics and Computer Science

University of Leicester, Leicester

LE1 7RH, England