I nternat. J. Math. & Math. Sci.
(1982)I-9
Vol. 5 No.
A DISTRIBUTIONAL REPRESENTATION OF
STRIP ANALYTIC FUNCTIONS
DRAGIA MITROVI(
Department of Mathematics
Faculty of Technology
University of Zagreb
Zagreb, Yugoslavia
(Received January 16, 1981)
ABSTRACT.
A strip analytic function converging in the
D’
topology to certain
boundary values (from the interior of the strip) is represented as the difference
of two generalized Cauchy integrals.
KEY WORDS AND PHRASES. Analytic function, dtbuion in D’, drb,n in ’,
convergence of tribtions Cauchy representation of a dstribution generazd
Cauchy integral), Plemelj dtribional formulas.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
I.
46F20, 46F10, 30E20.
INTRODUCTION.
In the theory of distributional behavior of analytic functions, two following
topics are central:
(i) the representation of distributions in terms of boundary
values of analytic functions; (2) the representation of analytic functions
terms of distributions.
The present paper, influenced by
[I, Theorem 97, p. 130]
via
[2, Theorem 3.6,
p. 68], continues the note [3] and contributes to the second topic.
In the cited
theorem of Beltrami and Wohlers, there is established a decomposition of strip
analytic functions into the difference of two Cauchy distributional representations
concerning the
S’ topology.
proved involving the
2.
Here, a version of this boundary value theorem is
D’ topology.
NOTATION AND PRELIMINARIES.
Throughout this paper the following symbols will be used:
MITROVI6
D.
+, "2-"
z,
4
the complex coordinates of points of
C ():
"
):
z
+
x
iy;
Im(z) > 0} and the open lower
Im(z) < 0} respectively;
the vector space of all infinitely differentiable complex valued
In;
functions defined on
D’
:
the open upper half-plane {z
half-plane {z E
C
,
In;
the real coordinate of a point of
t"
the vector space of all C -function with a compact support;
D’( In)"
.
the space of all continuous linear functionals (Schwartz distri-
butions) on
For the completeness we recall a few basic definitions and facts on the spaces
(IN).
(In)and
Let
non-negative integer p there exists a constant M
IR.
for all t
p
n
n
6
if
such that
IDp
for all t
IR.
n ;
(n)
exists a sequence
into
is dense in
in
sequence () that converges to
n
in
in
6
6
and
(i
P
+
It I)
The space
lim
6
6
p
It l)
M (i +
p
there
A linear functional T
n+
continuous linear functionals (distributions) on
inclusions
M
IR; (3) for each
IDp ,n(t)l
6).
T,
T,n
lim
n/
(t)
(that is, for each
6
which converges to
is continuous if
and for each
(2) for each p the sequence
of n, such that
p’ independent
The space
E C
is said to converge to zero in
e
(I) each
P
converges uniformly to zero on every compact subset of
there exists a constant M
on
(n) n
A sequence ()
if the following are satisfied"
(D
6
We say that a function
be a real number.
n
T,
for any
is the space of all
Finally, note the proper
6
In the following we shall use the same expression to denote a regular distribution and a function that generates it
3.
lwhen
no confusion is possible).
AUXILIARY RESULTS.
In order to establish the main result, we shall need the following three
simple lemmas.
LE 3.1"
zl
in
If
A+ and
h+(z) is
if h+(x +
a function analytic in A
i) converges to
+
h
x
+
in the
with
h+(z)
0(
topology as
i
]--I
as
+ 0,
DISTRIBUTIONAL REPRESENTATION OF ANALYTIC FUNCTIONS
3
that is,
+
<h X
\h+(x
z
lim
-++0
+
9, then h x E 6
for each
PROOF.
>
+ ic),
for all
+ i) (x)
dx
< O.
G
> 0 the function x + h + (x + iT) is continuous on
For each
> 0 the linear functional on
Therefore for each
h+(x
lim
g-++O
<h+(x
is a regular distribution in
I h+(x
+ ig),
?
defined by the integral
into
+ ig) (x) dx
By the hypothesis on the behavior of
exist the constants R > 0 and A > 0 such that for each
> 0 and all
h+(z)
xl > R
there
the
inequality
A
Ih
Then for all
holds.
Ixl >-
R}
r >
with a support contained in the set E
lim
X
the distribution h
+
X
I h+(x
x’
6G
+ i) (x)
has the asymptotic bound
[4, p. 54] it can be extended from t)
h
{x
IR"
it follows
< h+ @>I
us
Mx
# E
A
<
to
5
dx
-<
Ixl -I.
for all
A
I [xl-1
l(x)Idx
Hence, by Theorem
< 0.
In other words,
(<0).
Also, since
for each
> 0 and all
distribution i’
REMARK 3.1.
with Supp
E, we conclude that
h+(x
+ i)
is a regular
( < 0).
Perhaps it may be of interest to prove the above result directly.
Consider a linear functional on
<h+(x
For each
c
+ i), #
< 0) defined by means of
e(
i
h+(x
+ i) (x) dx,
> 0 the integral (3.1) exists because the integrand is equal to
4
D.
0(Ixl-l+).’
Let
(n)
MITROVI6
be any sequence which converges to zero in 6
as n
must show that
<h+(x +
lim
Let r denote a positive real number.
fh+(x
+ ig)
n(X)
We
0
Then we can write
-<
dx
x
lh+(x
n)
i),
-
I
h+(x
+
i)
n(X)
dx
+
(3.2)
-<r
+ +/-)
be an arbitrarily small positive real number, we may choose the number
Letting
r so large (r > R) that
]h+(x
+
dx -< A M
Cn(X)
ig)
0
I }xl-l+e
dx <
(3.3)
for all n.
The closed interval [-r, r] being now fixed, it follows from the con-
vergence of
(@n)
to zero in
lim
6
and the Lebesgue dominated convergence theorem that
I h+(x
+
ig)
n(X)
dx
(3.4)
0
The bound (3.3) and the limit (3.4) together show that the estimate (3.2) can be
made arbitrarily small for larg enough n.
(3.1) is a regular distribution in
’
Consequently, the linear functional
( < 0).
The previous results suggest the following lemma.
LENNA 3.2.
h+(x
+ +/-) converges
+,
<hx
6
for each
PROOF.
3.1.
h+(z)
if the function
to h
+
x
lim
+ O, that +/-s,
( < O) topology as
+/-n the
<h+(x
satisfies the conditions of Lemma 3.1, then
+ i),
+0
lim
/+0
I h+(x
+ i) (x) dx
( < 0).
Let e be a negative real number and let r be as in the proof of Lemma
To consider the limit we write
(x
+ ie) (x)
dx
(x + ie) (x) dx +
(x
+ i6) (x)
dx
DISTRIBUTIONAL REPRESENTATION OF ANALYTIC FUNCTIONS
where @ 6
6.
As each function
D
there exists a function in
is a C
-function, for any given compact of
that is identical to
So, by the hypothesis the first limit exists
{z 6
in the domain
Ixl
almost all
&+:
IRe(z)
over this compact [6, p. 4].
Since
e r > R} it follows
+ 0.
r as g
5
h+(z) is
that h+(x
lh+(x +
At the same time
ig)
analytic and bounded
+
ig) +
<
xA
h+(x)
for
> 0.
for all
Therefore, using the Lebesgue dominated convergence theorem,
I h+(x
lit
g/+0
+
I h+(x)
ig) (x) dx
6
Consequently, there exists a distribution H
H x’
D.
lim
h+(x
+ ig),
@(x) dx
( < 0) such that
for each
This implies h
+ 0
But
+
Hence, hx
is dense in
H
x
over
+
x
H
X
over
6
Obviously, the obtained results can be transposed bodily for a function h (z)
analytic in
A-
with
h-(z)
(--)
0
Iz{
as
and generating a regular distribution
by the integral
in
<h (x
LEMMA 3.3.
I
2zi
PROOF
on the space
hz)
If the function
< h+t
i
t
I
)
i),
z
i) (x) dx.
h (x
satisfies the condition of Lemma 3.1, then
>
h+(z)
for
z 6
0
for
Z 6
+
(3 5)
From Lemma 3.1 we kDow, in particular, that the distribution h+ acts
t
e
with
-i.
1
Since the function t
t
z
belongs to this space
(Im(z) z 0), the Cauchy representation of h+ is well defined.
t
To prove the lemma,
we shall first evaluate the limit of the integral
i
2i
+ 0 (observe that
as g
in
+.
A
< h+ (t +
ig),
i
t
z
>
this integral exists for each e > 0).
Let z be any point
By the Cauchy integral formular applied to the function
h+(
+ iE)
along the closed path consisting of a sufficiently large semicircle in
&+ of
D. MITROVlC
6
[-r, r], we get
radius r and the segment
I h+(t
i
2i
For z E A
+ is)
lim
< h+(t
+ is)
Thus, letting s
this integral vanishes.
1
2zi
+(z
h
dt
z
t
i
+ is)
+
+(z)
>
z
t
+
Z EA
for
+0, we have
A+
h
for
z6
0
for
Z6 A
Now by Lemma 3.2 the representation (3.5) follows.
For a function h (z) analytic in
to ones of
h+(z),
and satisfying here the conditions similar
we infer by the same procedure that
1
1
h
2i
)
z
t
h-(z)
for
z,
A-
0
for
z.
A+
THE MAIN RESULT.
4.
We are now prepared to prove the main result of this paper.
Yl
< Im(z) <
that
fl
logy.
y2
with f(z)
lim
f(x + i(y
s
+0
Then for
Yl
2--
< fl
t
YI’
Ira(z) <
Y2"
PROOF.
in
4,
f2
and
s
+
i
iy
i
2i
z
I
fl
Yl
< a < b <
an application of
decomposition f(z)
+ i(y 2
+
in
A.
Suppose
s)) in the 9’ topo-
< f2
t
+
i
iy
2
f2
is analytic in the lower half-plane
Since f(z) tends uniformly to zero as
Y2"
(4.1)
z
is analytic in the upper half-plane
and the Cauchy representation of
Let
lim f(x
+0
+
zl
Y2
where the Cauchy representation of
Ira(z) >
for some % > 0 as
z11+%
+ s))
I
< Im(z) <
i
f(z)
1
0(
:
{z,
Let f(z) be a function analytic in the strip A
THEOREM 4.1.
Izl
Cauchy’s integral formula [7, Lemma I, p.293] leads to the
f+(z)
+ f-(z),
where
+ia
f+(z)
1
2vi
I
f()
z
-oo+ia
oo+ib
f (z)
i
2z---
J[ f()%
Z
-oo+ib
DISTRIBUTIONAL REPRESENTATION OF ANALYTIC FUNCTIONS
f+(z)
We recall that the function
7
Im(z) > a,
is analytic in the upper half-plane
and f (z) is analytic in the lower half-plane Im(z) < b.
trarily closeness of the points a and b to the points
strip A is the common domain of analyticity for
Yl
f+(z)
By virtue of the arbiand
Y2
respectively, the
f-(z).
and
In order to
investigate the behavior of these functions at the point at infinity consider the
equality
+.ia
f+(z)
z
2i
f
d
oo+ia
f()
-oo+ia
z f ()
z
j
-ooia
-
oo+ia
i
2i
[
I
z
I
i
2i
d
f()
-oo+i a
Iz
The integral of the Cauchy type in (4.2) vanishes as
plane Im(z) >
other
while
YI’
From this we conclude that
gral representation for z
half-plane Im(z) <
f+(z)
f-(z)
D’
0(
--T
we infer
as
zl
f-(z)
oo.
0(
Let z
+ i(a + )
x
Also, from a similar inte-
Iz
in the lower
f-(z)
really converge in
as
f+(z)
and
Yl
and Im(z)
Y2
respectively
be a point in the half-plane
Then in the distributional setting
f+(x
+ i(a + ))
1
2i
f+(x
+ i(y I + g))
lim
a/Y
(f(t + ia)
f+(x
1
-(x + i)
t
+ i(a + ))
I
1
lim
< f(t
0(
By Lemma 3.1 the analyticity of f(z)
to
fl
as a
/
Yl
in
together imply
fie 6’
t-(x + ig)
A(Iz
oo) and the convergence
(-i _< e < 0).
hand, according to Lemma 3.2 we have
f+(x
+ i(y + g))
I
I
2i
<f i
i
t
1
+ ia),
a/Y I
+ ia)
% > 0.
Y2"
(from the interior of A).
of f(t
i
0(
1
T
z
topology to certain boundary values on Im(z)
Im(z) > a.
in the upper half-
one converges since f()
Further, we must verify that the functions
the
(4 2)
d
-(x + i)
On the other
MITROVI6
D.
8
Now, in view of the distributional Plemelj formulas [5, Theorem 2] we get
f+x
in the
f+(x
lim
+
+ g))
i
f
g ++0
1
x
I
2i
l
9’ topology.
Let z
) be a point in the half-plane Ira(z) <
+ i(b
x
+ i(b
f (x
i
2i
g))
Y2"
i
t -(x- ig)
(f(t + ib)
Starting from
>
and proceeding along the same lines as before, we find
f (x
+ i(y 2
I
))
i
(f2
2i
-(x- i.g)
t
)
e ++0
in the
9’ topology.
So we have proved that the function
half-plane Im(z) >
Izl
/
,
Im(z)
Yl
[resp. Im(z) < y2
f+(z)
is analytic in the
0(
with the order relation
f+x
and that it converges in the 9’ topology to
y2 ].
f-(z)]
[resp.
on Im(z)
as
Yl
[resp. f x on
In view of Lemma 3.3, it follows that
I
< f+t
i
2zi
t
+
iy
z
I
>
f+(z)
for
Im(z) >
Yl
0
for
Im(z) <
Yl
Analogously,
i
2i
<f
t
i
iy 2
+
z
>
f-(z)
for
Im(z) <
0
for
Im(z) >
Y2
Y2
Now we shall cmpute the value of the integral
i
2i
for Im(z) <
Y2"
< f+ (t +
iY2)
t
+
i
iy
2
(4.3)
z
For such z the function
f+ ()
z
[-r + iy 2,
is analytic inside the closed path which consists of the segment
r
+
iy
2]
and the semicircle L
r
of radius r lying in Im(z) >
Cauchy integral theorem, we may write
i
2i
f+ (t
+
+
iy
t
-r
iY2)2
z
dt
+
--i
2i
!
-
f+ ()
r
z
d
According to
Y2"
0
9
DISTRIBUTIONAL REPRESENTATION OF ANALYTIC FUNCTIONS
The integral along L r tends to zero as r
zero for Im(z) <
Y2"
I
jv
f (z)
(f-(t +
iy
iy
2)
+
t
i
iy
i
2i
f+t +
f-(t + iYl),
t
+
i
iy
i
2i
f-t +
f+ (t
t
+
i
iy
+
iY2),
f+(z)
+
the boundary value of f(z) on Im(z)
+
I)
(4.4)
0
z
I
f+t
f-t with
and
(4.4) and
we have
From the decomposition f(z)
f+(t
Thus the integral (4.3) is equal to
Combining the Cauchy representation of
I.
(4.3) respectively,
f+(z)
.
Also, as an immediate consequence of the derivation above,
2i
for Im(z) >
/
z
2
f (z) we see that
Yl
is the boundary value of
>
z
I
in the
for
Im(z) >
Yl
for
Im(z) <
Y2
f+t
fl
D’ topology
f(z) on Im(z)
Y2
+ f-(t +
and
f2
f
t
iy
I)
is
+
in the same topology.
Consequently,
f+(z)
f-(z)
i
2i
< fl’
t
+
i
iy
1
2i
< f 2’
t
+
1
iy
I
2
z
z
for
Im(z) >
Yl
for
Im(z) <
Y2
Again returning to the decomposition of the function
f(z),
the representation (4.1)
follows at once.
REFERENCE S
Introduction to the theory of Fourier integrals, 2nd.,
Oxford University Press (1948).
i.
TITCHMARSH, E.C.
2.
BELTRAMI, E.J.
3.
MITROVI6,
4.
BREMERMANN, H.J.
5.
MITROVI6,
and WOHLERS, M.R. Distributions and the boundary values of
analytic functions, Academic Press, New York (1966).
D. A distributional representation of analytic function, Math.
Balkanica 4, (1974), pp. 437-440.
Distributions, complex variables, and Fourier transforms,
Addison-Wesley, New York (1965).
D.
The Plemelj distributional formulas, Bull. Amer. Math. Soc. 77
(1971) pp. 562-563.
6.
ZEMANIAN, A.H. Distribution theory and transform analysis, Mc Graw-Hill,
New York (1965).
7.
SVESHNIKOV, A. and TIKHONOV, A. The theory of functions of a complex variable,
Mir Publisher, Moscow (19 78).