Generalized Wavelet Transform Associated with
Legendre Polynomials
C.P. Pandey
Ajay Kumar Garg Engineering College,
Ghaziabad –201001, India
M.M. Dixit
North Eastern Regional Institute of Science and
Technology, Nirjuli-791109, India
Rajesh Kumar
Noida Institute of Engineering and Technology, Greater Noida, India
ABSTRACT
The convolution structure for the Legendre transform developed by Gegenbauer is exploited to define Legendre translation by means of which a new wavelet and wavelet transform involving Legendre Polynomials is defined. A general reconstruction formula is derived.
MSC
33A40; 42C10
Keywords
Legendre function, Legendre transforms, Legendre convolution, Wavelet transforms.
1.
INTRODUCTION
Special functions play an important role in the construction of wavelets. Pathak and Dixit [5] have constructed Bessel wavelets using Bessel functions. But the above construction of wavelets is on semi-infinite interval (0,
). Wavelets on finite intervals involving solution of certain Sturm-Liouville system have been studied by U. Depczynski [2]. In this paper we describe a new construction of wavelet on the bounded interval (-1, 1) R using Legendre function. We follow the notation and terminology used in [7].Let X denote the space
[-1,1]
C
or
,
p
1
),
1
,
1
(
L
p
endowed withthe norms
,
p
1
,
dx
|
)
x
(
f
|
2
1
||
f
||
p / 1 1
1 p
p
(1.1)
|
)
x
(
f
|
sup
||
f
||
1 x 1 C
. (1.2) An inner product on X is given by
dx
g(x)
)
x
(
f
2
1
g
,
f
1
1
. (1.3) As usual we denote the Legendre polynomial of degree n N0 by Pn(x), i.e.
(x
1
)
;
x
[-1,1].
dx
d
n!
2
)
x
(
P
2 nn 1 n
n
For these polynomials one has
(i)
|
P
n(
x
)
|
P
n(
1
)
1
;
x
[-1,1]
( 1.4)(ii)
(
1
x
)
P
(
x
)
2
x
P
(
x
)
n(n
1)
P
(
x
)
0
n '
n "
n
2
; (1.5)(iii)
2 ) 1 n ( n ) 1 ( P'
n
(1.6)
The Legendre transform of a function f
X
is defined by
1
1
; ) ( ) ( 2 1 ) ( ) ](
[f k f k f x P x dx
L k
(1.7)
The operator L associates to each
f
X
sequence of real(complex) numbers
0 k
) k ( f
, called the Fourier Legendre coefficients.
The inverse Legendre transform is given by
0 k
k v
)
x
(
P
)
k
(
f
)
1
k
2
(
)
x
(
f
)
x
(
]
f
[
L
. (1.8)
Lemma 1.1. Assume f, g
X
, k N0 and c R, then(i)
L
[
f
]
(k)
f
X;(ii)
L
[
f
g
]
(k)
L[f]
(k)
L[g]
(k)
,
cL[f](k);
)
k
](
cf
[
L
(iii)
L
[
f
]
(k)
0
for
all
k
N0iff f(x) = 0 a.e ;(iv)
0 k
N j) (k, j, k , 0
j k , 1 2k
1 (j)
] P [ L
Let us recall the function K(x,y,z) which plays role in our investigation
otherwise,
0
z z z , xyz 2 z y x 1 ) z , y , x ( K
2 1 2
2 2
(1.9)
where z1 = xy – [(1-x2) (1-y2)]1/2 and z2 = xy + [(1-x2)
Then the function K(x,y,z) possesses the following properties; (i) K(x,y,z) is symmetric in all the three variables
(ii)
11
dz
)
z
,
y
,
x
(
K
.Also it has been shown in [8] that
11 k k
k
P
(
z
)
K
(
x
,
y
,
z
)
dz
1
)
y
(
P
)
x
(
P
(1.10)Applying (1.8) to (1.10), we have
0
k k k
(
)
P
(
)
P
(
)
P
)
1
2
(
2
)
,
,
(
k
z
y
x
k
z
y
x
K
. (1.11)The generalized Legendre translation y for
y
[
1
,
1
]
ofa function
f
X
is defined by
11
y
f
(
z
)
K(x,
y,
z)dz
1
)
y
,
x
(
f
)
x
)(
f
(
(1.12)Using Hölder’s inequality it can be shown that
X X
y
f
||
||
f
||
||
(1.13)and the map
y
yf
is a positive linear operator from X into itself.As in [7], for functions f,g defined on [-1,1] thegeneralized Legendre convolution is given by
11
y
f
)
(x)
g(y)
dy
(
2
1
(x)
)
g
*
f
(
11 1
1
dz
dy
z)
y,
K(x,
g(y)
)
(
2
1
z
f
(1.14)Lemma 1.2. If
f
X
,
g
L
1(
1
,
1
),
then the convolution (f*g) (x) exists (a.e.) and belongs to X. Moreover,l X X
||
f
||
||
g
||
||
g
*
f
||
, (1.15))
k
(
g
)
k
(
f
)
k
(
)
g
*
f
(
(1.16)
For any
f
L
2(
1
,
1
)
the following Parseval identity holds for Legendre transform,
k
2 2 2
||
f
||
|
)
k
(
f
|
)
1
k
2
(
. (1.17)In this paper, motivated from the work on classical wavelet transforms (cf. [1], [3]) we define the generalized wavelet transform and study its properties. A general reconstruction formula is derived. A reconstruction formula, under a suitable stability condition is obtained. Furthermore, discrete LWT is investigated.
Using Legendre Wavelet, frame and Riesz basis are also studied. A few examples of LWT are given. Similar constructions of wavelets and wavelet transforms on semi-infinite interval can be found in [4] and [5].
2.
GENARALAISED WAVELET
TRANSFORM
For a function
X
,
define the dilation Da by1
a
0
),
(
)
(
t
at
D
a
. (2.1)
Using the Legendre translation and the above dilation, the
wavelet
b,a(
t
)
is defined as follows:)
at
(
)
t
(
D
)
t
(
b a ba ,
b
(2.2)
11
X,
(az)dz,
)
z
,
t
,
b
(
K
1
(2.3)
where
1
b
1
and
0
a
1.
The integral is convergent by virtue of (1.13).Now, using the wavelet b,a the Legendre wavelet transform (LWT) is defined as follows:
f
)
(b,
a)
f(t),
(
t
)
L
(
b,a(2.4)
dt
)
t
(
)
t
(
f
2
1
11
a , b
(2.5)
11 1
1
dt
z)dz
t,
K(b,
)
az
(
)
t
(
f
2
1
(2.6) provided the integral is convergent.
Since by (1.13) and (2.2)
b,a
X
whenever
X
,
by Lemma 1.2, the integral (2.6) is convergent for).
1
,
1
(
L
f
1
The admissibility condition for the Legendre wavelet is given by
0 k
2
k
|
)
k
(
|
A
. (2.7)
From (2.7) it follows that
(
0
)
0
.
But
11
k
(
t
)
dt
P
)
t
(
2
1
)
k
(
Yields
11
1
1
-0
(
t
)
dt
(
t
)
dt
0
P
)
t
(
)
0
(
2
Theorem 2.1. If
X
defines a Legendre wavelet and),
1
,
1
(
L
1
then the convolution (
) defines a Legendre wavelet.Proof. Let
X
andL
(
1
,
1
),
1
so that
is abounded function on (-1,1). By Lemma 1.2, (
)
X
.
We have
k
2
k
)
k
(
)
(
A
k2 2
k
|
)
k
(
|
|
)
k
(
|
k
2
1
k
|
)
k
(
|
||
||
.
Therefore,
(
)
represents a Legendre wavelet. Theorem2.2. Let)
f
(L
and
X
and
)
1
,
1
(
L
f
1
(b,a) be thecontinuous Legendre wavelet transform. Then, we have the following inequality
X l X
||
f
||
||
||
||
a)
(b,
)
f
L
(
||
. Proof. The above inequality follows from (1.15).
3.
A GENERAL RECONSTRUCTION
FORMULA
In this section we derive a general reconstruction formula and show that the function f can be recovered from its Legendre wavelet transform. Using representation (2.6), we have
1
1 1
1
dt
dz
z)
t,
K(b,
(az)
)
t
(
f
2
1
a)
(b,
)
f
L
(
k
k k
k
b
P
t
P
z
P
k
t
f
2
1
(
)
(
)
(
)
2
dt
dz
(az)
)
(
2
1
11 1
1
k
1
1
k 1
1 k
k
(
az
)
P
(
z
)
dz
2
1
dt
)
t
(
P
)
t
(
f
2
1
)
b
(
P
)
1
k
2
(
kk
(
b
)
f
(
k
)
(
a
,
k
)
P
)
1
k
2
(
where
.
dz
)
z
(
P
)
az
(
2
1
)
k
,
a
(
k1
1
(3.1)
Therefore,
)
k
,
a
(
)
k
(
f
)
a
,
k
(
)
f
L
(
;so that
.
)
k
,
a
(
)
k
(
f
db
)
b
(
P
a)
(b,
)
f
L
(
2
1
11
k
Multiplying both sides by
(
a
,
k
)
and a weight function q(a) and integrating both sides with respect to a from 0 to 1,we have
). k ( f da | ) k , a ( | q(a) da db ) b ( P ) a , b )( f L ( ) k , a ( ) a ( q 2
1 2
1
0 1
0
1
1
k
(3.2) Assume that
1 0
2
da
|
k
,
a
(
|
)
a
(
q
)
k
(
Q
0. (3.3) Using (3.3), (3.2) can be written as
da
db
)
b
(
a)P
(b,
)
f
L
(
)
k
,
a
(
)
a
(
q
)
k
(
Q
2
1
)
k
(
f
1
0
1
1
k
da
db
)
b
(
P
)
k
,
a
(
a)
(b,
)
f
L
(
)
a
(
q
)
k
(
Q
2
1
10
1
1
k
(3.4) We have from (2.3)
11 a
,
b
K
(
b
,
t
,
z
)
(
az
)
dz
1
)
t
(
11 k
k k
k
(
b
)
P
(
t
)
P
(
z
)
dz
P
)
1
k
2
(
)
az
(
2
1
k1
1
-k k
k
(
b
)
P
(
t
)
(az)
P
(
z
)
dz
P
)
1
k
2
(
2
1
kk
k
(
b
)
P
(
t
)
(
a
,
k
)
P
)
1
k
2
(
=
)
t
(
)
b
(
P
)
k
,
a
(
v
k
. (3.5)
Using (3.5) in (3.4) we have
10 1
1
a ,
b
(
k
)
dadb
a)
(b,
)
f
L
(
)
a
(
q
)
k
(
Q
2
1
)
k
(
f
Set
(
k
)
b,a(
k
)
/
Q
(
k
)
a,
b
Then
1 1 ,
0 1
1
( )
( ) (
) (b,a)
( )
2
b a
f k
q a
L f
k dadb
. (3.7) Putting (3.7) in (1.8), we have
k
1
0 1
1
a , b
k
(
t
)
q
(
a
)(
L
f
)(
b
,
a
)
(
k
)
dadb
P
)
1
k
2
(
2
1
)
t
(
f
10 1
1 k
k a , b
dadb
)
t
(
P
)
k
(
)
1
k
2
(
)
a
,
b
)(
f
(L
)
a
(
q
2
1
Therefore
f(t)
10 1
1
a , b
dadb
)
t
(
)
a
,
b
)(
f
(L
)
a
(
q
2
1
(3.8)
4.
THE DISCRETE TRANSFORM
The continuous Legendre wavelet transform of the function f in terms of two continuous parameters a and b can be converted into a semi-discrete Legendre wavelet transform by assuming that a = 2-m; m
Z and
1
b
1
.
Now, we assume that
L
(
1
,
1
)
2
satisfies the so called “stability condition”
m2 m
B
|
)
k
2
(
|
A
(4.1)
for certain positive constants A and B, 0 < A
B
.
The function
L
(
1
,
1
)
2
satisfying (4.1) is called dyadic wavelet.
Using the definition (2.4), we define the semi-discrete
Legendre wavelet transform of any
f
L
(
1
,
1
)
2
by
)
(
),
(
)
2
1
(b,
)
(L
(b)
)
(
L
mf
f
mf
t
b,2mt
(4.2)
11
m b
(
2
t
)
dt
)
t
(
f
2
1
(4.3)
)
f
(
m
, (4.4)
where
m
),
z
2
(
)
z
(
mm Z.
Now, using Parseval identity (1.17), (4.1) yields the following
)
1
,
1
(
L
f
,
||
f
||
B
||
f
L
||
||
f
||
A
22 2m
2 2 m 2
2
(4.5)
Theorem 4.1. Assume that the semi-discrete LWT of any
)
1
,
1
(
L
f
2
is defined by (5.2). Let us consider another wavelet * defined by means of its Legendre transform.
j
2 j *
|
)
k
2
(
|
2
)
k
(
)
k
(
. (4.6) Then
m
v 1
1
k m *
m
f
(
b
)
(
2
k
)
P
(
t
)
(
b
)
db
L
)
t
(
f
(4.7)
Proof: In view of (4.4), for any
f
L
(
1
,
1
),
2
we have
m 1
1
v
k m *
m
f
(
b
)
(
2
k
)
P
(
t
)
(
b
)
db
L
=
m 1
1 k
k k m *
m
f
(
b
)
(2k
1)
(
2
k
)
P
(
t
)
P
(
b
)
db
L
=
m k1
1
k m k
m *
db
)
b
(
P
)
b
(
f
L
)
t
(
P
)
k
2
(
)
1
k
2
(
=
m km k m *
)
k
(
f
L
)
t
(
P
)
k
2
(
)
1
k
2
(
2
=
m
m *
k
m
k
(t)
f
(
k
)
(
2
k
)
(
2
k
)
P
)
1
k
2
(
2
=
mj
2 j m
m m
k
k
k
2
2
(
2
)
k
2
(
)
k
2
(
)
k
(
f
(t)
P
)
1
k
2
(
2
kk
(
t
)
P
)
k
(
f
)
1
k
2
(
=f(t) .
The above theorem leads to the following definition of dyadic dual.
Definition 4.2. A function
L
(
1
,
1
)
2 ~
is called adyadic dual of a dyadic wavelet , if every
f
L
(
1
,
1
)
2
can be expressed as
m 1
1
k m
m
(
2
k
)
P
(
t
)
(
b
)
db
.
~
)
b
(
f
L
)
t
(
f
(4.8) So far we have considered semi-discrete Legendre wavelet
transform of any
f
L
(
1
,
1
)
2
a. Now, we discretize the translation parameter b also by restricting it to the discrete set of points
b
,
m
2
n
b
m,n m 0Z, n
N0 , (4.9)where
b
0
[
1
,
1
]
is a fixed constant. We write)
b
n
2
,t
2
(
)
t
(
)
t
(
m m 0a ; b n , m ;
b0 m,n m
(4.10) Then the discrete Legendre wavelet transform of any
)
1
,
1
(
L
f
2
can be expressed as
f
)
(b
m,n;a
m)
f
,
b0;m,nL
(
m
Z, n
N0.(4.11)The “stability” condition for this reconstruction takes the form
)
1
,
1
(
L
f
,
||
||
,
||
||
22 ; , 2 22 20
0
N n
Z m
n m
b
B
f
f
f
A
, (4.12)
where A and B are positive constants such that
.
B
A
0
Theorem 4.3. Assume that the discrete LWT of any
)
1
,
1
(
L
f
2
is defined by (4.11) and stability condition (4.12) holds. Let T be a linear operator on L2(-1,1) defined by
n , m ; b
N n
Z m
n , m ;
b 0
0
0
,
f
Tf
. (4.13) Then
m,nb n , m ;
b0 0
,
f
f
(4.14) where
T
;
m
n , m ; b 1 n , m
b0 0
Z, n
N0Proof. From the stability condition (4.12), it follows that the operator defined by (4.13) is a one-one bounded linear operator.
Set
)
1
,
1
(
L
f
,
Tf
g
2
Then, we have
0
0
N n
Z m
2 n m, ; b
f,
f
,
Tf
. Therefore
2 1 2 1
-2 2 2
2 1
||
||
||
||
T
g,
f
Tf,
||
f
||
A
||
||
g
T
g
g
g
T
A
so that
2 2
1
||
g
||
A
1
||
g
T
||
.
Hence, every
f
L
(
1
,
1
)
2
can be reconstructed from its discrete LWT given by (4.11). Thus
Tf
T
f
1
0
0 0
N n
Z m
n , m ; b 1 n m, ;
b
T
f,
(4.15) Finally, set
m
;
T
1 b ;m,nn , m
b0 0
Z, n
N0 Then, the reconstruction (4.15) can be expressed as
0
0 0
N n
Z m
n , m b n m, ; b
f,
f
which completes the proof of theorem 4.3.
5.
FRAMES AND RIESZ BASIS IN L2(-1,
1)
In this section, using
b0;m,n a frame is defined and Rieszbasis of
L2(-1,1) is studied.
Definition 5.1. A function
L
(
1
,
1
)
2
is said togenerate a frame
of
L
(
1
,
1
)
2 n, m ;
b0
with sampling rate b0 if (5.12) holds for some positive constants A and B. If A = B, then the frame is called a tight frame.
Definition 5.2. A function
L
(
1
,
1
)
2
is said to
generate a Riesz basis of
b0;m,n
with sampling rate b0 ifthe following two properties are satisfied.
The linear span
b0;m,n:
m
Z > is dense in L2(-1,1)(5.1)
There exist positive constants A and B with 0<A
B
such that2 , 2
2 , ;
N n
Z m
n m, 2
2
, 2
0
}
{
c
||
}
{
||
n m n
m bo n
m
B
c
c
A
(5.2)
for all
2 n ,
m
}
c
{
(N 2
0). Here A and B are called the
Riesz bounds of
{
b0;m,n}
.Theorem 5.3. Let
L
(
1
,
1
),
2
}
{
b ;m,n0
is a Riesz basis of L2(-1,1);
}
{
b ;m,n0
is a frame of L2(-1,1) and is also an 2
-linearly independent family in the sense that if
0
c
m,n n , m ;b0
and{c
m,n}
2,
then cm,n = 0.
Furthermore, the Riesz bounds and frame bounds agree.
Proof. It follows from (5.2) that any Riesz basis is 2
-linearly independent.
Let
{
b0;m,n}
be a Riesz basis with Riesz bounds A and B,and consider the matrix operator
0 0 N
N ) n , m ( ), s , r ( n , m , s , r
M
where the entries are defined by
n , m ; b s , r ; b n , m , s ,
r
0,
0
. (5.3)
Then from (5.2), we have
2 2 n , m n
, m n
m, s, r,
n , m , s , r s r, 2
n ,
m
}
||
c
c
B
||
{
c
}
||
c
{
||
A
2
so that M is positive definite. We denote the inverse of M by
20
N ) n , m ( ), s , r ( n , m , s , r 1
M
(5.4) which means that both
t,u;m,n r,m s,m;r,
s,
m,
n
u , t
u , t ; s , r
N0 (5.5)
and
2 , 1 -n m, n m, s, r, , , ,
, 2 , 1
2
2
c
A
{
}
}
{
mn
n m s r
s r n
m
c
c
c
B
(5.6) are satisfied. This allows us to introduce
)
x
(
)
x
(
b;m,nn , m
n , m ; s , r s
, r
0
. (5.7)
Clearly,
L
(
1
,
1
)
2 s ,r
and it follows from (5.3) and (5.5) that
;
b ;m,n r,m s,n;
r,
s,
m,
n
s , r
0
N which means that {r,s} is the basis of L2(-1,1), which is
dual to
{
b0;m,n}
.Furthermore, from (6.5) and (6.6); we conclude that
n , m , s , r n , m s , r
,
and the Riesz bounds of {r,s} are B-1 and A-1.
In particular, for any
f
L
(
1
,
1
)
2
we may write
n , mn m, n , m ;
bo
(
x
)
,
f
)
x
(
f
and
n , m
2 n , m ; bo 1
-2 2 2
n , m
n , m ; bo 1
,
f
A
||
f
||
,
f
B
(5.8) Since, (5.8) is equivalent to (4.12), therefore, statement (i) implies statement (ii). To prove the converse part, we recall
Theorem 4.3 and we have for any
g
L
(
1
,
1
)
2
and f = T-1g,
0
N n
Z m
n , m ; bo n , m ; bo
,
f
)
x
(
g
.
Also, by the
2
linear independence of{
b0;m,n}
, this representation is unique. From the Banach-Steinhaus and openmapping theorem it follows that
{
b0;m,n}
is a Riesz basisof L2(-1,1).
6.
REFERENCES
[1] C.K. Chui, An Introdcution to Wavelets, Acadmic Press, New York (1992).
[2] U. Depczynski, Sturm-Liouville wavelets, Applied and Computational Harmonic Analysis, 5 (1998), 216-247. [3] G. Kaiser, A Friendly Guide to Wavelets, Birkhauser
Verlag, Boston (1994).
[4] R.S. Pathak, Fourier-Jacobi wavelet transform, Vijnana Parishad Anushandhan Patrika 47 (2004), 7-15.
[5] R.S. Pathak and M.M. Dixit, Continuous and discrete Bessel Wavelet transforms, J. Computational and Applied Mathematics, 160 (2003) 241-250.
[6] E.D. Rainville, Special Functions, Macmillan Co., New York (1963).
