Applicable Analysis and Discrete Mathematics
available online at http://pefmath.etf.bg.ac.yu
Appl. Anal. Discrete Math. 3 (2009), 212–223.
doi:10.2298/AADM0902212F
ON THE COMPOSITION OF THE DISTRIBUTIONS
µ
m
x−s
+ ln x+ AND x+
Brian Fisher
Let F be a distribution and let f be a locally summable function. The distribution F (f ) is defined as the neutrix limit of the sequence {Fn (f )}, where
Fn (x) = F (x) ∗ δn (x) and {δn (x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function δ(x). The composition
µ
m
of the distributions x−s
+ ln x+ and x+ is proved to exist and be equal to
−sµ
µm x+ lnm x+ for µ > 0 and s, m = 1, 2, . . . .
1. INTRODUCTION AND PRELIMINARIES
In the following we let ρ(x) be an infinitely differentiable function having the
following properties:
(i)
(iii)
ρ(x) = 0 for |x| ≥ 1,
ρ(x) = ρ(−x),
(ii) ρ(x) ≥ 0,
Z 1
(iv)
ρ(x) dx = 1.
−1
Putting δn (x) = nρ(nx) for n = 1, 2, . . . , it follows that {δn (x)} is a regular
sequence of infinitely differentiable functions converging to the Dirac delta-function
δ(x).
We let D be the space of infinitely differentiable functions with compact
support, let D(a, b) be the space of infinitely differentiable functions with support
contained in the interval [a, b], let D0 be the space of distributions defined on D
and let D0 (a, b) be the space of distributions defined on D(a, b). Then if f is an
arbitrary distribution in D0 , we define
fn (x) = (f ∗ δn )(x) = hf (t), δn (x − t)i
2000 Mathematics Subject Classification. 46F10.
Keywords and Phrases. Distribution, delta-function, composition of distributions, neutrix, neutrix
limit.
212
µ
m
On the composition of the distributions x−s
+ ln x+ and x+
213
for n = 1, 2, . . . . It follows that {fn (x)} is a regular sequence of infinitely differentiable functions converging to the distribution f (x).
m
We now define the distribution x−1
+ ln x+ by
m
x−1
+ ln x+ =
(lnm+1 x+ )0
m+1
for m = 1, 2, . . . and we define the distribution x−r−1
lnm x+ inductively by
+
m
0
mx−r−1 lnm−1 x+ − (x−r
+ ln x+ )
r
for r, m = 1, 2, . . . . This is not the same as Gel’fand and Shilov [8].
Next, we define the locally summable function xλ+ lnm x+ for λ > −1, and
m = 0, 1, 2, . . . by
λ m
x ln x, x > 0,
m
λ
x+ ln x+ =
.
0,
x<0
x−r−1
lnm x+ =
+
The distribution xλ+ lnm x+ is then defined inductively for λ < −1, λ 6=
−2, −3, . . . and m = 0, 1, 2, . . . , by the equation
λ−1
λ−1
(xλ+ lnm+1 x+ )0 = λ x+
lnm+1 x+ + (m + 1)x+
lnm x+ .
The distribution xλ− lnm x− is then defined for λ 6= −1, −2, . . . and m = 0, 1, 2, . . . ,
by
xλ− lnm x− = (−x)λ+ lnm (−x)+
and the distribution |x|λ lnm |x| is then defined for λ 6= −1, −2, . . . and m =
0, 1, 2, . . . , by
|x|λ lnm |x| = xλ+ lnm x+ + xλ− lnm |x|.
(1)
It follows that if r is a positive integer and −r − 1 < λ < −r, then
Z 1
r−1 (k)
h
X
ϕ (0) k i
m
λ
hx+ ln x+ , ϕ(x)i =
xλ lnm x ϕ(x) −
x dx
k!
0
k=0
+
r−1
X
(−1)m m!ϕ(k) (0)
,
k!(λ + k + 1)m+1
k=0
for an arbitrary ϕ in D[−1, 1].
If now f (x) is an infinitely differentiable function having a single simple
root at the point x = x0 , with f 0 (x) > 0, then putting t = f (x) and ψ(x) =
f 0 (x)ϕ(f (x)), we have
Z ∞
Z ∞
δn (t)ϕ(t) dt =
δn (f (x))f 0 (x)ϕ(f (x)) dx
−∞
−∞
Z ∞
=
δn (f (x))ψ(x) dx = hδn (f (x)), ψ(x)i.
−∞
214
Brian Fisher
Defining the distribution δ(f (x)) by
hδ(f (x)), ψ(x)i = lim
n→∞
Z
∞
δn (f (x))ψ(x) dx
−∞
we would get
hδ(f (x)), ψ(x)i =
1
hδ(x − x0 ), ψ(x)i
|f 0 (x0 )|
so that
δ(f (x)) =
1
hδ(x − x0 )
|f 0 (x0 )|
and more generally
δ (r) (f (x)) =
1 h 1 d ir
δ(x − x0 )
|f 0 (x0 )| f 0 (x) dx
for r = 0, 1, 2, . . . . This is of course in agreement with Gel’fand and Shilov’s definition of δ (r) (f (x)), see [8].
In order to generalize this definition of δ (r) (f (x)), the following definition was
given in [2].
Definition 1. Let f be an infinitely differentiable function. We say that the
distribution δ (r) (f (x)) exists and is equal to h on the open interval (a, b) if
Z ∞
N−lim
δn(r) (f (x))ϕ(x) dx = hh(x), ϕ(x)i
n→∞
−∞
for all functions ϕ(x) in D(a, b), where N is the neutrix , see [1], having domain
N 0 the positive integers and range the real numbers, with negligible functions which
are finite linear sums of the functions
nλ lnr−1 n, lnr n :
λ > 0, r = 1, 2, . . .
and all functions which converge to zero in the usual sense as n tends to infinity.
Note that taking a neutrix limit of a function is equivalent to picking out the
Hadamard finite part from the function and taking the usual limit of Hadamard’s
finite part.
The following definition generalizing Definition 1 was given in [3] and was
originally called the composition of distributions.
Definition 2. Let F be distribution in D0 and let f be a locally summable function.
We say the neutrix composition F (f (x)) exists and is equal to h on the open interval
(a, b) if
Z ∞
N−lim
Fn (f (x))ϕ(x) dx = hh(x), ϕ(x)i
n→∞
−∞
µ
m
On the composition of the distributions x−s
+ ln x+ and x+
215
for all functions ϕ(x) in D(a, b), where Fn (x) = F (x) ∗ δn (x) for n = 1, 2, . . . .
In particular, we say that the composition F (f (x)) exists and is equal to h
on the open interval (a, b) if
Z ∞
lim
Fn (f (x))ϕ(x) dx = hh(x), ϕ(x)i
n→∞
−∞
for all functions ϕ in D(a, b).
The following theorems were proved in [9], [6] and [5] respectively.
Theorem 1. The neutrix composition (xr )−s exists and
(xr )−s = x−rs
for r, s = 1, 2, . . . .
Theorem 2. If Fs (x) denotes the distribution x−s ln |x|, then the neutrix composition Fs (xr ) exists and
Fs (xr ) = rx−rs ln |x|
for r, s = 1, 2, . . . .
Theorem 3. If Fm (x) denotes the distribution x−1 lnm |x|, then the neutrix composition Fm (xr ) exists and
Fm (xr ) = rm x−r lnm |x|
for m, r = 1, 2, . . . .
The next theorem was proved in [7].
Theorem 4. If Fλ,m (x) denotes the distribution xλ+ lnm x+ , then the neutrix composition Fλ,m (xµ ) exists and
m
Fλ,m (xµ+ ) = µm xλµ
+ ln x+
for m = 1, 2, . . . , −1 < λ < 0, µ > 0 and λ, λµ 6= −1, −2, . . . .
Theorem 4 was then generalized in [4] with the following theorem.
Theorem 5. If Fλ,m (x) denotes the distribution xλ+ lnm x+ , then the neutrix composition Fλ,m (xµ ) exists and
m
Fλ,m (xµ+ ) = µm xλµ
+ ln x+
for m = 1, 2, . . . , λ < 0, µ > 0 and λ, λµ 6= −1, −2, . . . .
The following lemma can be proved easily by induction.
216
Brian Fisher
Lemma 1.
Z
1
v i ρ(r) (v) dv =
−1
0,
(−1)r r!,
0 ≤ i < r,
i=r
for r = 0, 1, 2, . . . .
The next two lemmas can be proved easily by induction.
Lemma 2. If ϕ is an arbitrary function in D with support contained in the interval
[−1, 1], then
−r
hx
m
ln |x|, ϕ(x)i =
r−1 (k)
h
X
ϕ (0) k i
x−r lnm |x| ϕ(x) −
x dx
k!
−1
Z
1
k=0
r−2
X
[1 + (−1)r+k ]m! (k)
ϕ (0),
−
(r − k − 1)m+1 k!
k=0
for m, r = 1, 2, . . . , where the second sum is empty when r = 1.
Lemma 3.
Z
1
n
uα lnr u du =
(−1)r+1 r!(1 − nα+1 )
+ E(ln n),
(α + 1)r+1
for α 6= −1 and r = 1, 2, . . . , where E(ln n) denotes a sum of negligible functions,
each containing a positive power of ln n.
2. MAIN RESULTS
We now prove the following extension of Theorem 5.
m
Theorem 6. If Fs,m (x) denotes the distribution x−s
+ ln x+ , then the neutrix comµ
position Fs,m (x+ ) exists and
(2)
lnm x+
Fs,m (xµ+ ) = µm x−sµ
+
for m = 0, 1, 2, . . . , s = 1, 2, . . . , µ > 0 and sµ 6= 1, 2, . . . .
Proof. We first of all put Gs,m (x) = (lnm+1 x+ )(s) and note that Gs,m (x) is of the
form
(3)
Gs,m (x) =
m
X
i
cs,m,i x−s
+ ln x+ ,
i=0
i
where cs,m,i = 0 if i ≤ m − s. Since x−s
+ ln x+ is an infinitely differentiable function
on any closed interval not containing the origin, it follows that
Fs,i (xµ+ ) = µi x−sµ
lni x+
+
µ
m
On the composition of the distributions x−s
+ ln x+ and x+
and then
Gs,m (xµ+ ) =
(4)
m
X
cs,m,i µi x−sµ
lni x+
+
i=0
on any closed interval not containing the origin.
Putting
Gs,m,n (x) = (lnm+1 x+ )(s) ∗ δn (x)
Z 1/n
=
lnm+1 (x − t)+ δn(s) (t) dt
−1/n
 R 1/n m+1
(s)
(x − t) δn (t) dt,

 −1/n ln
Rx
(s)
lnm+1 (x − t) δn (t) dt,

 −1/n
0,
=
we have
1/n < x,
−1/n ≤ x ≤ 1/n,
x < −1/n,
 R
1/n
(s)

lnm+1 (xµ − t) δn (t) dt,
1/n < xµ ,


 R−1/n
xµ
(s)
Gs,m,n (xµ+ ) =
lnm+1 (xµ − t) δn (t) dt, 0 ≤ xµ ≤ 1/n,
−1/n


R

0
(s)

lnm+1 (−t) δn (t) dt,
x < 0.
−1/n
Our problem now is to evaluate
(5)
Z
1
Gs,m,n (xµ+ )xk
dx =
+
1
xk
n−1/µ
+
Z
0
xk
−1
ns−(k+1)/µ
=
µ
n−1/µ
Z
1/n
k
x
0
−1
Z
Z
Z
xµ
lnm+1 (xµ − t) δn(s) (t) dt dx
−1/n
lnm+1 (xµ − t) δn(s) (t) dt dx
−1/n
Z
Z
0
lnm+1 (−t) δn(s) (t) dt dx
−1/n
1
(s)
ρ
(u)
−1
Z
1
u−(µ−k−1)/µ lnm+1 [(v − u)/n] du dv
0
Z
Z n
ns−(k+1)/µ 1 (s)
+
ρ (u)
v −(µ−k−1)/µ lnm+1 [(v − u)/n] dv du
µ
−1
1
Z 0
Z 0
+ n−s
xk
lnm+1 (−u/n)ρ(s) (u) du dx
−1
−1/n
= I1 + I2 + I3 ,
on using the substitutions u = nt and v = nxµ .
It is easily seen that
(6)
N−lim I1 = N−lim I3 = 0,
n→∞
n→∞
217
218
Brian Fisher
for k = 0, 1, 2 . . . .
Now,
ns−(k+1)/µ
I2 =
µ
(7)
=
1
ρ
(s)
m+1
X
(u)
n
Z
v −(µ−k−1)/µ [ln(1 − u/v)
1
−1
m+1
X
Z
m + 1 ns−(k+1)/µ 1
i
µ
−1
i=1
=
Z
+ ln v − ln n]m+1 dv du
Z n
ρ(s) (u)
v −(µ−k−1)/µ
1
i
m−i+1
× ln (1 − u/v) ln
u dv du + E(ln n)
Ji + E(ln n),
i=1
where E(ln n) denotes the terms containing powers of ln n and so are negligible and
R1
the term containing lnm+1 u is zero, since −1 ρ(s) (v) dv = 0 for s = 1, 2, . . . , by
Lemma 1.
We note that lni (1 − u/v) can be expanded in the form
lni (1 − u/v) =
∞
X
ai,p up
p=0
vp
,
where ai,p = 0 for p = 0, 1, . . . , i − 1 and then
Z 1
Z n
ns−(k+1)/µ
ρ(s) (u)
v −(µ−k−1)/µ lni (1 − u/v) lnm−i+1 u dv du
1
−1
=
=
∞
X
ns−(k+1)/µ ai,p
Z
1
up ρ(s) (u)
n
v −p−(µ−k−1)/µ lnm−i+1 v dv du
1
−1
p=0
∞
X
Z
m−i+1
s−p
ai,p (−1)
(m − i + 1)! [n
− ns−(k+1)/µ ]
[−p − (µ − k − 1)/µ + 1]m−i+2
p=0
Z
1
up ρ(s) (u) du + E(ln n)
−1
on using Lemma 3.
Using Lemma 1, it follows that
(8)
N−lim Ji = 0,
n→∞
if i > s and
(9)
N−lim Ji =
n→∞
m + 1 ai,s (−1)s+1 (m − i + 1)! s! µm−i+1
i
(sµ − k − 1)m−i+2
if i ≤ s.
It then follows from equations (7) to (9) that
(10)
N−lim I2 =
n→∞
m+1
X
i=1
m + 1 ai,s (−1)s+1 (m − i + 1)! s! µm−i+1
,
i
(sµ − k − 1)m−i+2
µ
m
On the composition of the distributions x−s
+ ln x+ and x+
219
for k = 0, 1, 2, . . . and it then follows from equations (5), (6) and (10) that
(11)
N−lim
n→∞
Z
1
Gs,m,n (xµ+ )xk dx
−1
=
m+1
X
m + 1 ai,s (−1)s+1 (m − i + 1)! s! µm−i+1
i
(sµ − k − 1)m−i+2
i=1
=
m X
m + 1 am−i+1,s (−1)s+1 i! s! µi
i
i=0
(sµ − k − 1)i+1
for k = 0, 1, 2, . . . .
We now consider the case k = r, where r is chosen so that sµ − r − 1 < 0,
and let ψ(x) be an arbitrary continuous function. Then
Z
n−1/µ
0
ns−(r+1)/µ
µ
xr ψ(x)Gs,m,n (xµ+ ) dx =
1
Z
v (r+1)/µ−1
0
Z
v
ψ[(v/n)1/µ ]
−1
× [ln(v − u) − ln n]m+1 du dv
and it follows that
(12)
lim
n→∞
Z
n−1/µ
0
xr ψ(x) dx Gs,m,n (xµ+ ) = 0.
When x ≤ 0, we have
Z
0
xr ψ(x)Gs,m,n (xµ+ ) dx =
−1
Z
0
xr ψ(x)
−1
=n
−s
Z
0
−1
r
Z
0
lnm+1 (−t)δn(s) (t) dt
−1/n
x ψ(x)
Z
0
[ln(−u) − ln n]m+1 ρ(s) (u) du dx
−1
and it follows that
(13)
N−lim
n→∞
Z
0
xr ψ(x)Gs,m,n (xµ+ ) dx = 0.
−1
Next, when xµ ≥ 1/n, we have
(14)
Gs,m,n (xµ+ ) =
Z
1/n
lnm+1 (xµ − t)δn(s) (t) dt
−1/n
1
u im+1 (s)
ln xµ + ln 1 −
ρ (u) du
nxµ
−1
#
Z 1"
m+1
X m + 1
u (s)
m+1 µ
m−i+1 µ
i
s
=n
ln
x +
ln
x ln 1 −
ρ (u) du
i
nxµ
−1
i=1
= ns
Z
h
220
Brian Fisher
= ns lnm+1 xµ
Z
+ns
1
ρ(s) (u) du
−1
m+1
X
i=1
=
m+1
X
∞ Z 1
X
ai,p up (s)
m+1
lnm−i+1 xµ
ρ (u) du
i
np xµp
p=i −1
∞
m + 1 m−i+1 µ X
ln
x
i
i=1
p=i
= (−1)s s!
m+1
X
i=1
Z
1
−1
ai,p up (s)
ρ (u) du
np−s xµp
m + 1 m−i+1
µ
ai,s x−sµ lnm−i+1 x + O(n−1 )
i
m X
m + 1 i
= (−1)s s!
µ am−i+1,s x−sµ lni x + O(n−1 ).
i
i=0
Letting n tend to infinity, it follows that
m X
m+1 i
(15)
lim Gs,m,n (xµ+ ) = Gs,m (xµ+ ) = (−1)s s!
µ am−i+1,s x−sµ lni x
i
n→∞
i=0
for x > 0.
Comparing equations (4) and (15) we see that
m + 1
(16)
cs,m,i = (−1)s s!
am−i+1,s ,
i
for i = 0, 1, 2, . . . , m.
We also see from equation (14) that
Z
m 1/n
X
m + 1 i µ am−i+1,s x−sµ lni x + O(n−1 ),
lnm+1 (xµ − t)δn(s) (t) dt ≤ s!
i
−1/n
i=0
µ
for x ≥ 1/n.
If now n−1/µ < η < 1, then
Z
Z η
1/n
m+1 µ
sµ (s)
ln
(x − t)δn (t) dt dx
x −1/n
n−1/µ
Z
m
η
X m+1
≤ s!
µi |am−i+1,s |
| lni x| dx + O(n−1 )
i
−1/µ
n
i=0
m X
m+1 i
= s!
µ |am−i+1,s |(η| lni η| + n−1/µ | lni n−1/µ | + . . .) + O(n−1 )
i
i=0
= O(η | lnm η|) + O(n−1/µ ).
It follows that if ψ is an arbitrary continuous function, then
Z 1/n
Z η
xr ψ(x)
lnm+1 (xµ − t) δn(s) (t) dt dx = O(η| lnm η|)
(17)
lim
n→∞
n−1/µ
−1/n
µ
m
On the composition of the distributions x−s
+ ln x+ and x+
221
for r = 1, 2, . . . .
Now let ϕ be an arbitrary function in D[−1, 1]. Then by Taylor’s Theorem
we have
r−1 k
X
x (k)
xr (r)
ϕ(x) =
ϕ (0) +
ϕ (ξx),
k!
r!
k=0
where 0 < ξ < 1. Then
hGs,m,n (xµ+ ), ϕ(x)i =
Z
1
ϕ(x)Gs,m,n (xµ+ ) dx
−1
=
Z
r−1 (k)
X
ϕ (0)
k!
k=0
+
Z
η
n−1/µ
1
xk Gs,m,n (xµ+ ) dx +
−1
Z
n−1/µ
0
xr ϕ(r) (ξx)
Gs,m,n (xµ+ ) dx +
r!
Z
η
1
xr ϕ(r) (ξx)
Gs,m,n (xµ+ ) dx
r!
xr Gs,m,n (xµ+ )ϕ(s) (ξx)
dx.
r!
Using equations (1), (11) to (13) and (15) to (17), it follows that
N−limhGs,m,n (xµ+ ), ϕ(x)i
n→∞
r−1 X
m X
m + 1 (−1)s+1 i! s! µi am−i+1,s
ϕ(k) (0) + O(η| lnm η|)
i+1 k!
(µs
−
k
−
1)
k=0 i=0
Z
m i
X
m + 1 1 µi am−i+1,s x−sµ ln xϕ(s) (ξx)
s
+(−1) s!
dx
i
r!
η
i=0
r−1 X
m
m Z 1
X
X
cs,m,i µi i!
cs,m,i µi lni xϕ(s) (ξx)
(k)
=−
ϕ
(0)
+
dx,
i+1
(µs − k − 1) k!
r!
i=0
i=0 0
=
i
k=0
since η can be made arbitrarily small. It now follows that
E
N−limhGs,m,n (xr+ ), ϕ(x)
n→∞
r−1 X
m
X
cs,m,i µi i!
ϕ(k) (0)
i+1 k!
(µs
−
k
−
1)
k=0 i=0
"
#
Z 1
m
r−1 k
X
X
x (k)
i
i
−sµ
+
cs,m,i µ
x
ln x ϕ(x) −
ϕ (0) dx
k!
−1
i=0
k=0
(Z
"
#
m
r−1 k
1
X
X
x (k)
i
i
−sµ
=
cs,m,i µ
x
ln x ϕ(x) −
ϕ (0) dx
k!
−1
i=0
k=0
)
r−1
X
(−1)i i!
(k)
+
ϕ (0)
(−sµ + k + 1)i+1 k!
=−
k=0
=
m
X
i=0
cs,m,i µi hx−sµ
lni x+ , ϕ(x)i
+
222
Brian Fisher
on using Lemmas 2 and 3. This proves that Gs,m (xµ+ ) and the sum
m
X
cs,m,i µi x−sµ
lni x+
+
i=0
exist and
(18)
Gs,m (xµ+ ) =
m
X
cs,m,i µi x−sµ
lni x+
+
i=0
on the interval [−1, 1] for m, r, s = 1, 2, . . . . However, equation (1) clearly holds on
any closed interval not containing the origin.
Now suppose that
lni x+
Fs,i (xµ+ ) = µi x−sµ
+
(19)
for i = 0, 1, . . . , m − 1 for some m and s = 1, 2, . . . . This is true when m = 1 since
we then have Fs,0 (xµ+ ) = Gs,0 (xµ+ ).
Note that equation (18) can be rewritten in the form
(20)
Gs,m (xµ+ ) =
m
X
cs,m,i µi Fs,i (xµ+ ).
i=0
Since Gs,m (xµ+ ) exists and Fs,i (xµ+ ) exists by our assumption for i = 0, 1, . . . , m − 1,
it follows that Fs,m (xµ+ ) exists and
m−1
X
Gs,m (xµ+ ) = cs,m,m µi Fs,m (xµ+ ) +
cs,m,i Fs,i (xµ+ )
i=0
= cs,m,m µi Fs,m (xµ+ ) +
m−1
X
cs,m,i µi x−sµ
lni x+
+
i=0
=
m
X
cs,m,i µi x−sµ
lni x+ ,
+
i=0
on using equations (19) and (20). It follows that
Fs,m (xµ+ ) = µm x−sµ
lnm x+
+
and so equation (19) holds for m. Equation (1) now follows by induction, completing
the proof of the theorem.
Replacing x by −x in Theorem 4, we get
m
Theorem 7. If Fs,m (x) denotes the distribution x−s
− ln x− , then the neutrix comµ
position Fs,m (x− ) exists and
(21)
Fs,m (xµ− ) = µm x−sµ
lnm x−
−
for m = 0, 1, 2, . . . , s = 1, 2, . . . , µ > 0 and sµ 6= 1, 2, . . . .
µ
m
On the composition of the distributions x−s
+ ln x+ and x+
223
Acknowledgement. The author would like to thank the referee for his help in
improving this paper and for pointing out some typing errors.
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Department of Mathematics,
University of Leicester,
Leicester, LE1 7RH,
England
E–mail: [email protected]
(Received February 23, 2009)
(Revised June 29, 2009)