lnternat. J. Math. & Math. Sci.
VOL. 16, NO. 4 (1993) 737-748
737
AN ORTHONORMAL SYSTEM ON THE CONSTRUCTION
OF THE GENERALIZED CANTOR SET
RAAFAT RIAD RIZKALLA
Mathematics Department, Faculty of Education
Atn Shams University, Heltopolts, Cairo, Egypt
(Received January 16, 1992 and in revised form June 22, 1992)
ABSTRACT.
This paper presents a new complete orthonormal system of functions
[0,1] and whose supports shrink to nothlng. Thls
system related to the construction of the Cantor ternary set. We defined the
canonical map
and proved the equivalence between this system and the Walsh
defined on the interval
ystem.
The generalized Cantor set with any dissection ratio is established
and the constructed system Is defined in the general case.
KEY WORD AND PIStASES. Dyadic expansion, Walsh functlons, Cantor set.
1992 Al SU CLASSIFICATION CODE. 42C10
I.
INTRODUCTION.
We shall denote the set of positlve integers by P, the set of non-negatlve
integers by N, the set of real numbers by R, and the set of dyadlc rationals
in the unit interval [0,1} by q. In partlcular, each element of Q has the form
n
p/Z for some p,n e N, 0
Each n e N has a unique dyadic expansion
p < Z
n.
nk 2k
nke {0,1}
k=O
are called the dyadic coefficients of n. Likewise
n
E
where n
k
a {unique} dyadic expansion
x
{1.1}
ech
x
[0,1} has
z-(k+1)
Z
(1.2)
xk
Xke {0,1}
k=O
the finite expansion being chosen in case x belongs to the dyadic rationals q.
In terms of the dyadic expansions, the dyadic sum of two numbers x,y
[0,1}
is defined by
x $ y
(R)
Yk
Integers
The dyadic sum of a pair of
n
E
k=O
m
2
-(k+l]
[1.3)
n,m e N is defined by
nk
mk
2-k
(1.4)
k=O
where (n ,k e N) and (m ,k e N) are the dyadic coefficients of n and m.
k
k
Let r be the function deflned on [0, I) by
x e
r(x)
-1
x e
[0,)
[,1}
extended to R by periodicity of period 1. The Rademacher system (r ,n
n
N) is
R.R. RIZKALLA
738
rn(X)
defined by,
r(Z
n
N. The first three Rademacher functions
R,n
x), x
are shown In Flgure I.
r
o
1/4
112
-1
z
314
Fig. 1.
The Rademcher functions form an incomplete set of orthonormal functlons on
[0,I).
The Walsh system
w
was introduced by Paley [11] in 193Z
N
n
as products of Rademacher functlons In the followlng way.
n
coefficients
k
N
Wn
Walsh
functlons
If n
N has dyadic
then
to
belong
(1.5)
rk
-kTO
the class
of
plecewlse
constant
basis
functions that have been developed In the twentleth century and have played an
Important role in sclentlflc
appllcatlons.
The foundations of the Walsh
functlons field were made by Rademacher [IZ], Walsh [16], Flne [10], and Paley
[11]. Owlng to their sallent propertles, Walsh functions proved to be very
powerful In solvlng varlous problems such as the
analysis of
dynamic systems
[4-8], the deslsn of optlmal controllers [3], and the Identlflcatlon problem
systems [Z],[14]. The flrst elght Walsh functlons are shown in
Flgure Z.
of dynamlc
w0
w
w
4
5
w
6
w?
+1
or
+1
ill
1/2
or
3/4
-1
114
112
314
-1
Fig. 2.
Notlce by deflnltlon that
Wzn
Wo=
r
n
for n e N
and each Walsh
dlscontlnultles on
function Is piecewlse constant wlth flnltely many Jump
on
[0,1) and takes only the values +I or -I. The Walsh system Is orthonormal
[0,1) and possess the properties (see [13]),
w
w
w
(1.6)
Wn(X
y)
Wn(X Wn(Y
(1.7)
n
m
AN ORTHONORMAL SYSTEM ON THE CONSTRUCTION OF THE GENERALIZED CANTOR SET 739
By a dyadic interval in
[0,1)
p
we shall mean an interval of the form
p+l
[--n
I(p,n)
n
p < 2
0
n
2
[0,1) and n
N
n p
2
For each x
length 2
-n
N we shall denote the dyadlc Interval of
n
I (x)
I(p,n) where 0 -< p < 2 Is uniquely
whlch contalns x by
n
x
determined by the relatlonshlp
I(p,n).
The Walsh functions,
belng a
complete orthonormal system (see [I],[16]) allows a representatlon of every
absolutely Integrable function f(x) on [0, I) In Walsh serles In the form
where
a w (x)
n
n--O n
being the Walsh-Fourler coefflclents of f.
f(x)
J
n
f(x)
Wn(X)
dx
sum of Walsh
The n-th partial
DIrlchlet kernel are denoted respectively by [15]
n-I
n-1
and
f
series and the
Sn
D n (x)
2
Dn
k=O ak wk
Flne[10], has shown that
satlsfles
a
2
n
Snf(X)
Dn(X
f(t)
kO wk
$ y) dt
The Dlrlchlet kernel
[0,2 -n) and zero elsewhere.
for x e I (0)
n
THE CANTOR SET AND THE CANTOR f3JNCTION
The Cantor ternary set C formed by removing middle thirds from the
interval
[0,1] can be defined as follows
for each n e N, construct the
2.
closed intervals
2n/k
where k
2k +I
2kn
j
3
T.
k
I=0
i
(2 I)
is defined by
k
k
n
k < 2
0
n
3
I
31
(2.2)
belng the dyadic coefficients of k In {I. I}.
of ratio I/3
Then the Cantor ternary set
or more briefly the Cantor set C Is
(R)
C=
0
n=O
2n-1
U
J
2n/k
k=O
The middle open Intervals removed In the above construction are defined by
E
U
2n-1
U
n= 0 k=O
E
where
2n+k
6k’+
3n+1
6k +I
E
3n+l
2n+k
Arlthmetlcally, the Cantor set consists exactly of those points which can be
expanded in the ternary system without using the digit
x
x
k=O
k
3
-(k+1)
Xke
We deflne a class of closed intervals
n
:0 k < 2
J
,i.e. for every x e C
{0,2)
(2.3)
n e N
-n.
n
The Cantor
N, n contains 2 closed Interval each of length 3
For each n
set can be associated with a monotone non-decreaslng continuous function
Lebesgue function and defined by the followlng process.
called the Cantor
For each closed interval I, let
IRI(X
and
I
Xl(t)
dt
where
xi(t)
{I
0
tI
I
the length of the
and
function
characteristic
represents the
O
740
R.R. RIZKALLA
interval I, respectively.
N, let
For each n
Fn(X)
2
-n
2n-I
{2.4)
Rj ,i(x)
k=O
k
Since
X (t)
Rj .(x)
/k
J2n*k
Jz".k
0
2
x
3n[
OJ
dt
X (t)
dt
Jzn/k
x < 2k .3
0
3nx
xJn
2k
2 /k
x > {2k +I}. 3-n
Hence, from (2.4),
{
Fn(X)
F0
Fo(x)
x
2k’+
2-nl3nx
x e
x e
k)
(2k+I).2 -m
x
[0,1]
J1
Osk<2
Jzn+k
n
m-I
0 s k < 2
E2m-l/&
P
and for n
s m s n
{x),,J
F (x)
/
F2 (x)
_.__/-
f
/
F3 (x)
t’-/----/0
0
Flg.3.
We observe that,
the function F {x} is continuous, nonn
decreasing on [0,1],and satisfies Fn{O}
O, Fn{1}
1 {see figure 3}. Indei,
2k+1
Fn(X
lira
N,
for each n
2
It is easy to see that
m
F(x)
x
E2m-l/k
0
m-I
k < 2
1
<
=
[0, I] exists, is continuous,
llm Fn (x), x
nm
I.
O, F{1}
F{O}
monotone non-decreaslng and satisfies
This function is
called the Cantor-Lebesgue function associated with the Cantor set C.
(x)
=r. Xk.2
k=O
where
Xk
xk
[0,1)
C
Now, we introduce the mapping
-(k+l)
{o
1
x
if
c
(2.s)
5= o
Xk= 2,
being the coefficients in the ternary expansion of x in {2.3}. The mapping
satisfies the following properties:
1-
is one-to-one mapping from
contalns all
x
C/C
onto [0, I) where
is a countable set
C
whlch has ternary expansion termlnates in 2’s. In fact,
C
={
2k+1
n
3
Oak<2
n
neN
}.
-
AN ORTHONORMAL SYSTEM ON THE CONSTRUCTION OF THE GENERALIZED CANTOR SET
n
N, 0 -< k < 2
2- For each n
x
f]
k
Define the Lebesgue-stleltjes measure
2
n
n
n
which contains y.
F(a)
F()
N
n
where F Is
(k__
n
2
AN ORTHONORMAL SYSTEM OF FUNCTIONS
3.
We construct a function system
for each x
In the followlng way
n
k
I
-n
[a,)
F
2-n
F Jn+k
2
0 -< k < 2
the Cantor function. Indeed, for each
k
In
(x)
Iff
J2n/k
and
2n
Is the dyadic interval of length
In(Y)
where
C/C
2k"
(
741
21
k
I=0 I
@
|0,1)
nand nne NN,
on the Interval [0,I)
n+1
e
0 s k < 2
define
k
(-1)
{
n(X
o
x e Ja
0
/k
(3.1)
otherwise
has the constant value +1 on half of the closed
nwhlle
has constant value -1.
on the other half
n
n+l
(see figure 4). Consequently,
is odd function about t
It Is obvlous that
l,ntervals
Indeed,
in
n
1/2
[ n dF
O
N
n
O
o
nu
13
FI U
u
+1
o|
-1L
F!
u
l,,
’/,
2/9
7/9
6/9
3/9
e/9
FIg. 4.
THEOREM 1.
measure
F
The system
n:
n e N
Is orthonormal
with.
respect to the
l.e.
n m dF
nm
(0
(3.21
n
m.
PR(X)F. Let n > m z 0 and J e
Slnce
Pm has constant value +1 or
-I on J, and J contains an even number {namely 2n-m
of closed intervals in
on half of these intervals
Pn has the constant value +1 whlle on the
other half
has constant value -I, consequently,
m+l"
m+l
n
Summing over all J
&+l
we get
J
n dF
nmd
Jean+
Since
{J}
interval, then
2
-(n+l)
V J e
+1
1 n d@=l
z
0
O.
O.
If n
m, then
Je+
n e N
Mn+1
d
n
contains 2
n+l
c 1 osed
R.R. RIZKALLA
742
For each x
COROLLARY 1.
(-11 n
n(X)
where
r
N
Let n
PROOF.
k
(-1)
0
then
J2n+l
X
(-I)
n
rn,
r
g
2-
n
x
)(x)
n k=O k
n
C/C
On
Now, we define the product system
N
n
Since
ko"
(-1)
r ((x))
n
)(x)
From the
N J2"/l/k
n
k
In+l(
and
/k
(C/C)
x
and
(x)
we have
x
n(X)
n+l
k < 2
0
properties of the mapping
&0n(X)
(3.3)
)(x)
rnO
be the Rademacher system.
N
n
n
N
n
C/C
xn
N.
by
N
n
N) are the dyadic coefficients of n.
where (n ,k
k
lf3
l-I
M
U
I-i
U
i-I
U
U
+1
o|
|,l
-1 [
6/9
3/9
2/9
/9
?/9
8/9
Fig. 5.
The function system
I
satisfies the following properties:
For each n,m e N,
2. For each n,m
to
and
@n @m @n
2n+k,
N, m
@2 n
n
closed intervals in
0 < k < 2
n, @m
an+1
2n+I’ @m
is even function with respect
{see figure S).
respect to
is odd function with
3. For each n,m N, 2n s m <
is the dyadic sum.
where
m
has constant value +! on half of the
while on the other half
@m
has constant value -1,
consequently,
@m dF
n
m <
2
0
2n+l
n
N.
In general,
j
4.
@n dF
P
n
0
Let n e N with dyadic coefficients
n
k
k
(3.5)
N
equations (3.3),{3.4), we have
Cn(X)
{-!) nkXk
{-1) <n,Cx)>
k=O
where
{n,(x})
Z nk
k=O
x
k
(rood 2}
and x
C/C
From
AN ORTHONORMAL SYSTEM ON THE CONSTRUCTION OF THE GENERALIZED CANTOR SET
For each x
COROLLARY 2.
@n(X)
w
where
(3.6)
)(x)
(WnO
be the Walsh system.
N
n
n
N
n
C/C
From equations (1.5),(3.3) and (3.4), we have
PROOF.
@n(X)
(k(X))
k=O
((rko
k=O
--
)(x))
Wn((x)
(rk({(x)))
k=O
)(x).
w
n
According to corollary 2 and the properties of the mapping
obvious that
2k"
n
TIREM 2.
measure
fF
J
let
First, let
O,
n,m e g
2
n
aEOREM 3.
it
{3.7}
n e N
0
(
(3.8)
Since
it suffices to show that
i’J i(R)J
Second, in case i
which is clear from {3.5}.
n+1
is
is orthonormal with respect to the
N
n
J,
then
dg
Jean+
F{J)
F
2
n+1.
2
-(n+l)
Jean+
The system
Is complete, with respect to
n e N
@n
on the Cantor set C.
F
PROOF.
iJ.
m < 2
Om dF
the measure
n’
,
ijd
N,
m
n
0 s k < 2
k
Wn(
)=
The system
i.e.
PROOF.
J Cm
743
Let
n,m e N
n
m < 2
0
and suppose that
S
is Integrable
functlon for which
Slnce each function
,In
2 /k
Let
O r @m dF
0 s m < 2
@
n
0 s m < 2
0
n
(3.9)
has only one flxed value on each Interval
-(2k +1).3 -n
r(k)
fnnJH
J
Then {3.9} glves,
Y
J
dF
k .3
/k
2n-1
#m dPF
k=O
r(k)
n
-n
dF’
m
0 -< k <
2n"
n
0 s m < 2
0
From {3.7), we get
@m(
then
2k
k
w)m
0 -< k < 2
2n
2n_l
r.
k=O
Now, the determlnant
(k)n win(
r
m(
w
k
k
n,
0
)l
n
0 -< k < 2
and does not vanlsh. Thus the numbers
w
k
m
m < 2
0
n,
n
0 s m < 2
0
n
m < 2
n e N,
{3.10}
is of order 2
n
are llnearly independent and
R.R. RIZKALLA
744
it follows from (3.10) that
rn(k}
0 -< k < Z
0
FSd
continuous function G(x)
G(
Uslng (3.11), we get
F
0
_2k +I_
2k--=
G(
n
(3.11)
N and let G(x) denote the
m
Suppose that {3.9) is satisfied for a11
n
0 -< k < 2
n
n
3
3
is equivalent to zero on the
and so G(x} is a constant on C. It follows that
Cantor set C. Thus the proof is completed.
[0,1]. The
Let f be an absolutely lntegrable function on the interval
is defined
N
n
Fourier series of f with respect to the system
#n
by
S
a
f
n=O
_n
n
where
a
[
n
f(x)
0
The n-th partial sums of
@n(X) fF
n-1
a,. #k
_’.
_.Qnf
this series will be denoted by
k=O
P
Notice for n
N.
n
[0,1]
and x
n-1
{Snf)(x)
(j f(t),k(t)dF(t)),k
k=O
whe r e
(x)
=J f(t)Dn(x,t)dF(t)
n-
(x)
k=O
denote to the n-th Dirichlet kernel.
k
Dn(x,t)=
COROLLARY 3.
N
Let n
(3.12}
k (t)
C/C"
x,t
n-1
(I)
Dzn(X,t) I]
Dzn(X, t) ..n
+
i=0
II
(t)
J
x
Jn /k
x,t
N and
(I)
CC
2n-1
I=0
.
Fix
,2n-1
0,1
I n (k.2-n).
Dlrlchlet kernel
},
+
I=0
Wl((x)
l(X) l(t)
the
the properties of
2k +1
n
3
and
2kn
J
2n+k
Using
).
3
mapping
the
in the Walsh case, we get
D2n
D2n(X,t)
x
and let
2n-1
THEORE 4.
I=0
n-1
k
(x)
then
I=0
Wl((x) wi((t) )
rl((x)) rl((t))
+
i=0
(II)
2n-1
2n_1
@i(x) Oi(t)
n-I
[l
k <
Using (1.7), (3.6), and (3.12),
we have
Dzn(X,t)
0
for some k
2 /k
2
Let n
PROOF.
llx)i(t)
i=O
2n
(t))
wi((x)
2n
((t))
I n (k.2-")
Let f be a continuous function on CC
(t)
J
z
then
2
/k
n
f converges
uniformly to f
PROOF.
n
n
where k
(x)
For every
CC
x
and
n
N
we define
2k +1
<- x <
n
n
3
3
is defined in (2.2). By the canonical map
and
n
n(X)
by
maps onto the dyadic interval
2k
n
In(k.2-n)
and
F([=n
n
for some k
the set
n ))
(CC)N
2
-n
n
k < -2
0
[an,
n
Then the
AN ORTHONORMAL SYSTEM ON THE CONSTRUCTION OF THE GENERALIZED
CANTOR SET
function
f
745
can be written as
f(x)
2
n
[
f{x)
dF{t)
(3.13)
n
Using corollary 3
(S2n f)(x) j
n
2nj
From
2no
O2n(X,t} dF(t)
f(t)
f(t}
< e whenever
f(x}
(S2n f)(x}
(t}
[e
n
,gn dF(t)
dF
n
{3.13} and since f is continuous, i.e
f{t}
f(t)
t
x
f(x}
V
< 8
> 0
8 > 0
3
,n }.
[an
x,t e
n
2n [ (f(t}
n
n If(t)
2nj
f(x))
f(x}
Therefore,
dF{t)
< e
dF{tl
n
and the proof is completed.
M CANTOR SET ITH CONSTANT DISSECTION RATIO
4.
in (3.1} can be descrlbe on any Cantor set
The constructed system
with constant dlssectlon ratlo. The generatlng intervals of the Cantor set of
general ratio
0 < p < q
p/q
2 /k
are deflned by
2n.k
]
I
n
O<k<2
n
2q
2q
are determlned by the recurslve formula
2n/k
where t
(q-p)n
[kn
P/q
p,q e P
1-k
A
t
and
{q_p)
n-1
+
)t
2n-l/k
2n+k
n
(q+p)(2q) n-1
k
being the dyadic coefficients of k in {I.I). It is easy to see that
k
p/q
e N. Then the Cantor set C
for each
C
C/q
P/q
m
intervals
m
(l)
J.+
2
(ili)
p/q
fl
O
n=O
k=O
of
each
J
2n+k
closed
a
a
2 +k
2
n+l
The
closed
2n+k
P/q
P/q
.=o,
J.+
(li)
2n+k
+2k+s
p/q
interval
satisfies the following four properties:
J
c
n
Is
of ratlo p/q
P/q
P/q
=2
2 -1
end-polnts
contains the
R2n
2
2
2
+2k
0 s k < 2
for
n+l
2r+k
/2k
J.+
fl
2k*!
n
=
N, and
n
/2k+1
P/q
(iv)
llm
J2n+k
ma
Ok<2
-)
0
.
The open Intervals removed In this construction are defined by
2
p/q
E
U
p/q
p/q
U
k= 0
E
where
2n+k
2 +k
P/q
For each
N, the class J
I
2
n+
2q
0
2
n+l
n+
P/q
{ J2n/k
k
.2k
k < 2
n
(q_p)n+l
{2q)n+1
.2k.!
contains 2
n
closed
2 -1
interval each of length
q-P
-q
and
Fn (x)
2
-n
R
k=O
p/q(x)
’J2n/ k
where
R.R. RIZKALLA
746
(t) dt
;[
J2n+
0
P/q
J2n, k
k
t
x <
0
"{q-pa
/k
n
n
(2q)
p/q
2 +k
+
(q_p)n
/
>
X
Hence,
I
Fn(X)
"
x
2n+k
(2q) n
p/q
2 +k
(q_p)n
2 +k
p/q
xEEm_
2
2k +___I
m
2
0 -< k < 2
non-decreaslng on [0,1]. In fact
assoclated with the Cantor set
each F
n
/q
n
-
is linear on the lhervals
n
U
x e
J
2n+k
0
k <
E
2
and
2 .k
k =0
is
Jn
p/q
P/q
2k+l
m
2
F (x)
n
m
+k
constant on each component of the set [0,1]/(
llm
m-1
(II), and (III) above that F is continuous and
n
p/q
Observe by properties (I),
F(x)
O-<k<2n
xe,J
+l+k
). The Cantor function
2m-I
m <
+k
P <
then there is a continuous strictly increasing
TIOIIE S. If
C.
function L on [0,1] such that L(/q)
<
we have
k
2
lOOF. For each n e N, 0
n,
P/q
q-P
-q
where
n
2q
p <
1,then 0<1. Let L be the monotone-lncreaslng plecewlse llnear
n
Since
3
p/q
function, L :[0, I]--[0 I]
0 s k < 2
n.
mapping the end-polnts of
Then for n>m, L
n
and L
m
differ only on
I So, for each x
m"
3
defines a function x) on o,}. As
Ln
Lm
II/I
>
[P/q}
2n+k
-
L(/q)
J2n+k
k < 2
0
m,
m
Xlm
{nX)-mX){
So
for each n and k,
0
-
In fact,
the sequence ( L (x)> converges
L Is continuous
[0, I] and
Lm
converges uniformly to L [9].
monotone increasing. We have L([O,I])
L j
J
onto those of J
2m+k
{tmx)-x){=
<
J
and Is clearly
n
k < 2
We need only to show that L is one-to-one. Suppose
c
either
[0,1].
If
x or y lles in removed Interval it is easy to see
(x,y)
then as /q is
So suppose
that
L(y) > L(x).
x and y are in /q
SO
C.
p/q
nowhere dense there is an Interval
L(y)
L(x)
J
so
n
.k
{2q} n
for some n,k.
But then
is strictly increasing and the proof Is completed.
L
According to theorem 5, for each n
L
(x,y)
2n+k
(q_p)n
}
k < 2
N, 0
,
2k
3
n
and
L
n,
1
and ] s P < 1,
{2q} n
}
2k +I
n
3
AN ORTfIONORMAL SYSTEM ON THE CONSTRUCTION OF THE GENERALIZED CANTOR SET
set
0
747
Now, the constructed system @ in (3.1) can be generallzed for any Cantor
[0,1), p,q P and n N,
for each x
Cp/q In the following way
k < 2
n
n+l,
k
l.e. k
2
define
i=O
(-1)
n(X)
k0
P/q
x
2
otherwise
0
Is defined as in (3.4), it is orthonormal and complete
on the general Cantor set Cp/. Ust
system with respect to the measure
for the
theorem 5 and the mappt L, the results obtained in sect ton 3
The product system
@
F
ternary Cantor set C c be easily generalized for the case C
REFERENCES
1.
Convergence Problems of
ALEXITS, G.
New York
Oxford
Press,
Pergamon
Orthoonal Serf,s,
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Time
CHEN, C.F., and HSIAO, C.H.
Pro@. IEE 122 {1975}, 565-570.
3.
CHEN, C.F. ,and HSIAO, C.H.
Control Via Walsh Functions,
596-602.
Design
Domain Synthesis Vla Walsh Functions,
Of Plecewlse Constant
Gains for Optimal
IEEE Trans. Autom. Controll..
(1975),
AC-.O
4.
CHEN, C.F.,and HSIAO, C.H. A State Space Approach to Walsh Serles Solution
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Analysls and Optimal Control of Time Varying
CHEN, W.L., and SHIH, Y.P.
Linear Systems Vla Walsh Functlons, Int. J. Control, 27. No.6 {1978},
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6.
CHEN,W.L.,
Equatlons,
and SHIH,Y.P.
Shlft
Walsh Matrlx and Delay dlfferentlal
IEEE Trans. Autom. ControlL AC-23 (1978), 1023-1028.
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Walsh Series Analysis of Multi
Cl, W.L.
Inst. 313 (1982}, 207-217.
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CORRINGTON, M.S.
Walsh Functions,
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Delay Systems,
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Fr’artk.l.l.n
Solution of Dlfferentlal and Integral Euatlons with
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DEBARRA, G. Introductlon t_9o Measure Theory, Van Nostrand Relnhold press,
New York London, 1975.
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Math. Soc. 34 {1932}, 241-279.
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Elnlge Stze Ober Relhen yon Allgemelnen Orthogonalen
12. Rademacher, H.
Funktlonen, Math. Ann. 87 {1922}, 122-138.
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DYadic Harmonlc Analvsls, Adam Hllger England, 1990.
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14. TZAFESTAS, S.G.
Identification.
Walsh Series Approach to Lumped and Distributed System
J. Franklin Inst. 305 (1978), 199-220.
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J. Hath. &_ 4ath.
5. No.4 (1982), 62S-673.
15. WADE, W. R.
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