Internat. J. Math. & Math. Sci.
Vol. 9 No. 2 (1986) 301-312
301
FOURIER TRANSFORMS OF DINI-LIPSCHITZ FUNCTIONS
M.S. YOUNIS
Department of Mathematics
Yarmouk University, Irbid, Jordan
(Received October 29, 1984
It
ABSTRACT.
well
is
known
that
if
Lipschitz
conditions
of a certain order are
f(x), then these conditions affect considerably the absolute
convergence of the Fourier series and Fourier transforms of f. In general, if f(x)
imposed on a function
belongs to a certain function class, then the Lipschitz conditions have bearing as to
the dual space to which the Fourier coefficients and transforms of
f(x)
belong.
In
the present work we do study the same phenomena for the wider Dini-Lipschitz class as
well as for some other allied classes of functions.
Dini-Lipschitz functions, Fourier series, Fourier transforms.
KEY WORDS AND PHRASES.
1980 AMS SUBJECT CLASSIFICATION CODE.
42A, 44.
INTRODUCTION.
f(x) belongs to the Lipschitz
TITCHMARSH ([I] Theorems 84,85) proved that if
p
(a, p) in the L norm on the real Line R, then its Fourier transform f
class Lip
be longs to
L
B (R)
for
P
p+a p-1
o
<
<
1,
In [2]
<
p
<
< B < p"
P
p-I
2.
and [3]
we extended Titchmarsh’s Theorems to heigher differences and to
n
n
n
is the n dimensional torus
and T
where T
R
functions of several variables on
group.
In
this
paper we
try,
among other
thlngs,
to
explore
the validity of
those
theorems in case of functions of the wider Dini-Lipschitz class on various groups.
2.
DEFINITIONS AND NOTATIONS.
In the sequel,
Euclidian space,
T
torus respectively.
n
stands for the n-dimenslonal
will denote the real llne, R
n
and T
denote the circle group [o, 2hi and the n-dimensional
p
consists of all equivalent classes of functions such that
L
R
302
M. S. YOUNIS
DEFINITION 2.1.
LP(R).
f(x)
Let
The Fourier Transform
(u)
f
is defined by
ixud x
f(x)
R
LP(T)
f(x)
If
of
however, its Fourier series is given by
Z
f(x)
e
c
-iux
With the usual modifications of these two definitions for functions of several
variables in
LP(R n)
LP(T n)
and
respectively.
DEFINITION 2o2o
f(x) e
Let
LP(R)
LP(T).
or
w (h,f)
Then the integral modulus of continuity
For
P
is defined by
we write
p
DEFINITION 2.3.
Let
0 (h a)
w (h,f)
P
as
h
o.
o (h
f(x)
Then we say that
a)
belongs to the Lipschitz class Lip(a,p) or to the
Little Lipschitz class lip(a,p) respectively.
DEFINITION 2.4.
Krovokin ([4], p.
65) defines the Dini-Lipschitz class as those functions such
that
()
w(h,f) Log
Lira
o.
h+ o
Equivalently one could write
Lim w(h,f)
o [Log
(--Ih)]-I
h+ o
For functions in
L
p
spaces.
We can define the Dini-Lipschltz classes as those
for which
Lira w (h f)
p
h+O
o
[Log
(_)]-I
FOURIER TRANSFORMS OF DINI-LIPSCHITZ FUNCTIONS
303
A still further extension is possible if we write
h/0
.
for some
()]-Y"
o [Log
Lim w (h f)
p
FOURIER TRANSFORMS OF DINI-LIPSCHITZ FUNCTIONS.
3.
Our aim is to show that the conclusion of Titchmarsh’s Theorem 84
about their Fourier transforms
f
115]
and that all what we can say
p
I.
belongs to L (R) where --+--.
f
is that
p.
[|,
LP(R)
does not hold for the Dini-Lipschitz functions in
P
P
Thus we prove the following.
THEOREM 3.1.
Let f(x) belong to the Dini-Lipschitz class in
<
p
<
LP(R).
Then
^f
belongs to
L
p’,
2.
PROOF.
Notice that
!
o [Log()]
w (h f)
p
-I
n
is equivalent to
:o
f(x+h)-f(x)l,
Thus by taking the Fourier Transform of
Young inequality, and following Titchmarsh’s proof we arrive at
o[Log(--) ]-P"
du
and hence
Then for
8
< p"
Log(--)]-P"
o [h
du
and by Holder inequality we obtain
@(X)
o
o [Log
[X
-I
(X)]
Log(X)]S[X] I- + --p
-8
X
+
p
which yields
X
: lISdu
o[Log
x]-Sx -s
+
-p
applying the Hausdorff-
304
M. S. YOUNIS
For the right hand side of the last estimate to be bounded as
X
we must
have
B+:<
0
p-and
IB
<
i.e.
which is always the case in our situation.
The first condition, however, gives
p-B p+#<O
p’<
p_
p"
The case
indicates
that,
Lipschitz
conditions
in
is
rejected of course and we are left with
contrast
on our
to
the
Lipschitz
functions,
the
p"
B
imposition
of
which
Dinl-
functions does not improve upon the conclusion of
the
Hausdorff-Young theorem and the proof is complete.
We cemark at this point that if we employ the condition
w (h f)
p
!
o[Log(-)]
-y
in theorem 3.1 we obtain the two conditions.
-B+-<o
p-and
-YB <The first yields the previous conclusion
B
p’,
and the second gives
--. <
P
y
or
For
p
p*
2
we get
--<y
REMARK 3.2.
In
situated
this
in
paragraph we would
between the
like
Lipschitz
inspired from Weiss and Zygmund [5].
and
to
the
employ some
conditions
Dini-Lipschltz
which
conditions.
Thus we prove the following theorem.
are
rather
These were
FOURIER TRANSFORMS OF DINI-LIPSCHITZ FUNCTIONS
305
THEOREM 3.3.
f(x)
Let
LP(R)
belong to
<
p
<
such that
2
o[
0
<
a
<
as h
0.
Then
Lg(R)
f
P
p +
p-I
<
g
h
for
< p- =____
p-I
and
PROOF.
The proof goes exactly as that of theorem (3.1) and yields
X
II
f
du
O[Log X]
-Yg
X
I-
d +
and for the right hand of this estimate to be bounded as
a
X
--p
one must have
+ p-< 0
Y <
and
The first restriction gives the original conclusion of Titchmarsh’s theorem 84
<
and the second gives
y.
The choice
gives
O[ -,h
Log hY
and we get
P,
2p-I
and for
p
p"
must be greater than
2, Y
2"
NOTICE 3.4.
In [2]
Tltchmarsh’s Theorems were proved for higher differences or equivalently
for higher derivatives of
f(x).
This indicates that if we use the conditions
o[
Log()
or
Wp(h)
f
r)
x)
O[
Log hY
-I
306
M. S. YOUNIS
where
r
g
A r f(x)
h
(-1)r-i (ri)
f(x+ih)
Then we will arrive at the same results proved in the previous theorems.
Another valid point here is that if we turn to the realm of Fourier Series of
functions on
LP(T)
LS(R)
we will get the same conclusions except
is replaced now
and the summation is taken over the integers
with the sequence space
Z.
Appart
from that, the definitions and the proofs are exactly the same.
4.
FUNCTIONS OF SEVERAL VARIABLES.
functions
Titchmarsh’s Theorems were generalized also for
in
LP(Rn)
and
LP(T n)
(see [2] and [3]), without any change in the results.
In contrast, we expect that for
LP(Tn)
the foregoing conclusions hold
LP(Rn)
the Dini-Lipschltz functions in
and
verbally.
To
see
this
we
would
llke
to
there
that
out
point
are
two
definitions
Llpschltz functions of several variables. We confine ourselves to functions in
2
T, for simplicity. Thus we introduce the following definitions.
for
2
R
and
DEFINITION 4.1.
f(x,y)
Let
X
LP(R2).
belong to
LiP(2, P)
and to
y
in
lf(x+h,
Then we say that
f
belongs to
LiP(el, p)
in
if
f(x+h, y) + f(x,y)
f(x,y +k)
y +k)
ll p
I 2
O[h
<
0
k
at, a2
<
I.
Another definition states that
c
I[f(x+h,y
We
indicates
functions in
theorems.
that
LP(Rn)
+ k)
[2]
in
and
f(x,y)
and
LP(Tn)
ll p
[3]
we
a2
O[h
+ k
have
employed the two definitions
for
and obtained the same conclusions of Titchmarsh’s
However the steps of the proofs when the first definition was employed were
straight forward, where as the arguments in case of the second definition needed to be
handled with special care.
In view of the previous considerations we introduce the following:
DEFINITION 4.2.
The function
f(x,y)
in
lf(x+h,y+k)
LP(R2)
f(x,y+k)
o
as
h,
k/
O.
belongs the Dini-Lipschltz class if
f(x+h,y) + f(x,y)
tLog()Log ()]-I
ll p
FOURIER TRANSFORMS OF DINI-LIPSCHITZ FUNCTIONS
307
Other classes of functions can be obtained by replacing the right hand side of
this estimate by
o[
Log()
-Y
-Y 2
[Log()
a
a2
h’lyl]
o[
k
Y2
Log h
Log k
and
h
0[
k
Log h
Logk k
re s pe c t ive ly.
DEFINITION 4.3.
The function
f(x,y)
belongs to the Dini-Lipschitz class in
LP(R 2)
o[(Log(’))-I + (LoE())-l],
other function classes are defined as
0[Log()
-YI
+
Log() -Y2
1
llf<x+h,y+k)-
f(x,y)
ll p
a2
h
0[(
Log h
h
+
"
Log k
We now state and prove the following theorem
THEOREM 4.4.
Let
f(x,y)
belong to
lf(x+h,y+k)
LP(R2)
<
f(x,y+k)
<
p
Then its Fourier transform
h
and let
f(x*h,y) +
u2
u
0[
2,
k
,]
2
],
Log h
Log k
;(u,v)
belongs to
P,.
p+ al p-
p+a2 p-
<
< p"
<
< p"
o
L
<
,,
where
z_<
_.)
if
308
M. S. YOUNIS
and
PROOF.
As in the proof of theorem 84 of Titchmarsh, we obtain
I__ I
0
2-I
Oil-I
fk[uvf^[P"
0
h
O:
dudv
k
Log h
p
Log k
Now let
X
Y
f
Then for
p
du dv
and by the Holder inequality we arrive at
+
l-a 1B
(X,Y)=0[X
P Log X -YIB ][Y l-aZB
+
P Log Y -Y2B ],
so that
s
X
Y
S I}l
O[X
du dv
1-B-e IB +
B +
l-B-ol
[Y
P (LogX
-’
P (Log Y)
-a
B a2
+--p--<
-Y2B<-I
which give
p + x p-I
where
and
-
<
min
: < O.
p-
(Y1
0
)] x
-y22B
For the last quantity to be bounded as
i-ail +
IB
x, Y
’iI < -I
we get the required conclusions
FOURIER TRANSFORMS OF DINI-LIPSCHITZ FUNCTIONS
309
We indicate that if we use the previous definition with higher differences or if
we employ the
LP(T 2)
Fourier Series in
we still get the same results of theorem 4.4
The proofs are direct and we omit them.
We hint also that the conclusion of theorem 4.4 could have been stated equivalently
f(u,v)
in the following manner that
E
n L
L
where
P
< 8 i < p"
P+I p-I
i= 1,2.
We conclude this section by adding that we could have used definition 4.3, and
here we could have arrived at similar results in case of Fourier transform and Fourier
Series, however, the proofs in this case are not so direct as in the previous cases.
2
(R) AND
L2(T).
5. FUNCTIONS IN L
The special case
ation in this work.
0
<
a
<
and
2 [I, Theorem 85]
p
Titchmarsh proved that if
as
f(x)
e
L2(R)
deserves some consider-
then the conditions
h/ 0
and
-X
=r
j_ 2
-2a
X
as
X
are equivalent.
In
L2(R n)
[2]
and
L2(T n)
and
[3]
we extended this theorem to higher differences of functions in
respectively.
Here we examine the analogus situation for the
Dini-Lipschitz class and start with the following.
THEOREM 5.1.
Let f(x)
as
h/
L2(R).
Then the conditions
0,
-X
2
du=o[Lo x]
as
X
-1
are equivalent.
PROOF.
Applying the Parseval’s Identity and following Titchmarsh’s proof we get in this
case
du
o
[Log(--lh)]-2
310
M.S. YOUNIS
so that
x
i12
But
l+
4x
2x
1+(1+
nLog 2
)Log x
[I
2
du
+ (Log 2X) -2 + (Log 4X) -2 + ...]
-2
o[(Log X)
+ (1+
"’’]
+
+
x
o[ (Log X -2,
YX
4X
2X
[
du
Log 2
-I
+ (1+
x
Log
2Log
Log x
2
-1
...]
nLOgx 2)
.Log
X
which tends to zero as
and
n
+ (I +
(1 +
Log X
go to infinity, we also notice that
2Log 2 -1
Lo=..gA
"
nLOgx2
+ (I +
Log
Log X
[Log
+
X + Log2
Log X + 2Log2
and the series in the brackets reduces to
[Lo--’O’ +
as
X
+ 3Log
2"’’
(R).
Which is convergent by the comparison with the power series.
Hence we arrive at
last to the estimate.
S
x]
o
-I
x
which proves the first assertion.
us examine the estimate
f(x)
g
0
<
<
I.
We state the following theorem.
h
THEOREM 5.2.
Let
0[-]
Log
The converse can be delt with in a similar fashion.
L2(R).
Then the conditions
B
Log h
-)
FOURIER TRANSFORMS OF DINI-LIPSCHITZ FUNCTIONS
as
and
0
h
-X
f
as
311
X
X
-28
-2
2
I1
f
+
<Log h)
O[X
du
>
and in fact if
-2
are equivalent
O[X
estimate can be replaced with
the right hand side of
the last
].
We shall not prove this in detail since the main trend of the proof is quite
clear.
In comparision with the previous theorem, thus
-2
-2
117
x
f
-2
-2
0[X
-2
+ (2 n)
+ (2 n)
-2
][I + (2)
[Log X]
nLog
[I + Log X 2
-2
[I +
2Lgx2
Log
-28
-2
+ (X
(Log X)
0[X
du
(Log 2X
)...l
Log 2
[I + Log X
-2B
"’’]
-2
...l
Now the terms in the brackets
(1 +
k
kLOgx2
Log
1,2...n
[1 +
are all bounded by
=[
Log 2
Log X
Log 2 + Log X
which tends to
X
as
-2
=]
Log
2
.
So that we are left with
-2
-2a
Y
x
du
0[X
(Log X)
-2
][I + 2
which proves the first part of the theorem.
-4ct
+ 2
+ ...]
The converse can be carried in exactly
the same manner as in Titchmarsh’s theorem 85 and the proof is complete.
Here
again
the
8
choice
>
reduces
original case. i.e.
-X
[
+
X
I1
-2a
2
du
O[X
the conclusions of
the
theorem
to
the
M. S. YOUNIS
312
6.
CONCLUDING REMARKS.
The treatments in the previous section convince us that for the various types of
2
Dini-Lipschitz functions in
L (T).
2
L
(Rn),
2
and
L (T n)
the analysis can be carried
almost without much difficulty, however, even the statements of the results in case of
2
2
2
L (R)
and
2
L (T)
would be fairly complicated.
We conclude finally that Titchmarsh’s theorems [especlal[y Theorem 85] were extended in [3] as well as in various papers in the Literature to other groups such as
the
0-dimensional, the finite dimensional and compact Lie groups.
We indicate that
In a forth-coming paper we shall be dealing with the present subject along those
direct [os.
REFERENCES
[.
TITCHMARSH, E.C.
Theory of Fourier Integral, 2nd Ed., Oxford Univ. Press, 1948.
p
Fourier Transforms In L
Spaces, M. Phi1. Thesis, Chelsea College,
2.
YOUNIS, M.S.
3.
London, 1970
YOUNIS, M.S. Fourier Transforms of Lipschltz Functions on Compact Groups.
Thesis McMaster University, Hamilton, Ontario, Canada, 1974.
4.
KROVOKIN, P.P. Linear Operators and Approximation
graphs on Advanced Mathematics and Physics, 1960.
5.
WEISS, M. and ZYGMUND, A.
Theory.
Ph.d.
International Mono-
A Note on Smooth Functions, Inda._Math. 2 (1959), 52-58.