### Generalized Wavelet Transform Associated with Legendre

```International Journal of Computer Applications (0975 – 8887)
Volume 108 – No 12, December 2014
Generalized Wavelet Transform Associated with
Legendre Polynomials
C.P. Pandey
M.M. Dixit
Rajesh Kumar
Ajay Kumar Garg Engineering
College,
North Eastern Regional
Institute of Science and
Technology,
Nirjuli-791109, India
Noida Institute of Engineering
and Technology,
Greater Noida, India
ABSTRACT
(i) |
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.
33A40; 42C10
(ii) (1  x 2 ) Pn" (x)  2x Pn' ( x)  n(n  1) Pn (x)  0 ; (1.5)
(iii)
Pn' (1) 
n ( n  1)
2

L[ f ](k )  f (k ) 
Keywords
Legendre function, Legendre
convolution, Wavelet transforms.
transforms,
Legendre
(1.6)
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].
X
denote
the
Lp (1,1), 1  p  , or C [-1,1]
space
endowed with
f ( k ) 


 k 0 ,
(complex) numbers 
Legendre coefficients.
(1.1)
|| f || C  sup | f ( x ) |
.
An inner product on X is given by
11
 f , g    f ( x ) g(x) dx
2 1
.
(1.3)
As usual we denote the Legendre polynomial of degree n N0
by Pn(x), i.e.
n
d
Pn (x )  2 n!   (x 2  1) n ; x  [-1,1].
 dx 
1
For these polynomials one has
called
the
Fourier


L[f ]v (x)  f (x)   (2k  1) f (k) Pk (x)
k 0
Lemma 1.1. Assume f, g
(i)
(iii)
(iv)
(1.2)
sequence of real
The inverse Legendre transform is given by
(ii)
1/ p
1 1

| f | p    | f (x) |p dx  , 1  p  ,
 2 1

f X
(1.7)


the norms
1 x 1
1
1
f ( x) Pk ( x)dx;
2 1
The operator L associates to each
1. INTRODUCTION
n
( 1.4)
The Legendre transform of a function f  X is defined by
MSC
Let
Pn ( x) |  Pn (1)  1 ; x  [-1,1]
. (1.8)
 X , k N0 and c R, then
L[f ] (k)  f
X;
L[f  g] (k)  L[f] (k)  L[g] (k) ,
L[cf ](k)  cL[f](k);
L[f ] (k)  0 for all k N0iff f(x) = 0 a.e ;
1

 2k  1 , k  j

L[ Pk ] (j)  
0
, k  j, (k, j)  N 0


Let us recall the function K(x,y,z) which plays role in our
investigation
1  x 2  y 2  z 2  2xyz ,
z1  z  z 2
(1.9)

K ( x, y, z)  
0
otherwise,

where z1 = xy – [(1-x2) (1-y2)]1/2 and z2 = xy + [(1-x2) (1y2)]1/2.
35
International Journal of Computer Applications (0975 – 8887)
Volume 108 – No 12, December 2014
Then the function K(x,y,z) possesses the following properties;
(i)
K(x,y,z) is symmetric in all the three variables
2. GENARALAISED WAVELET
TRANSFORM
1
(ii)
 K(x, y, z)dz   .
1
1
Pk ( x ) Pk ( y)   Pk (z)K( x, y, z)dz
 1
wavelet
(1.11)
y [1,1]
11
( y f )( x )  f ( x, y)   f (z) K(x, y, z)dz
 1
of
(1.12)
(1.13)
where  1  b  1 and
convergent by virtue of (1.13).
is a positive linear operator from X
As in [7], for functions f,g defined on [-1,1] thegeneralized
Legendre convolution is given by
1
f ( t ) b ,a ( t ) dt
2 1
(1.14)
Lemma 1.2.
If f  X, g  L (1,1), then the
convolution (f*g) (x) exists (a.e.) and belongs to X. Moreover,
1
1
f ( t )(az) K(b, t, z)dz dt
2 1 1
(2.6)
X
  X,
(1.15)
f  L1 (1,1).
The admissibility condition for the Legendre wavelet is given
by

| (k ) | 2
A  

k
k 0
.


(f * g )  ( k )  f ( k ) g ( k )
(2.5)
1 1

1 1


(2.4)
1
b ,a
Since by (1.13) and (2.2)
whenever
by Lemma 1.2, the integral (2.6) is convergent for
1 1
|| f * g || X  || f || X || g || l ,
The integral is
provided the integral is convergent.
11
(f * g) (x)   ( y f ) (x) g(y) dy
2 1
  f ( z) g(y) K(x, y,z) dy dz
0  a  1.
Now, using the wavelet b,a the Legendre wavelet transform
(LWT) is defined as follows:

into itself.
(2.2)
1
K(b, t, z) (az)dz,   X,
 1
(2.3)
(L  f ) (b, a)   f(t),  b,a (t ) 
Using Hölder’s inequality it can be shown that
1

2
is defined as follows:
1

f  X is defined by
||  y f || X  || f || X
 b ,a ( t )
 b,a (t )   b D a (t )   b (at)
 
K ( x, y, z)   (2k  1) Pk ( x) Pk ( y) Pk ( z) .
2 k 0
The generalized Legendre translation y for
(2.1)
Using the Legendre translation and the above dilation, the
(1.10)
Applying (1.8) to (1.10), we have
y  yf
define the dilation Da by
Da (t )   (at ), 0  a  1 .
1
and the map
  X,
For a function
Also it has been shown in [8] that
a function
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 semiinfinite interval can be found in [4] and [5].

(1.16)
For any f  L (1,1) the following Parseval identity
holds for Legendre transform,
(2.7)
From (2.7) it follows that
(0)  0. But
2
11
(k )   ( t ) Pk ( t )dt
2 1


 (2k  1) | f (k) |2 || f || 22 .
(1.17)
Yields
k
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.

1
2 (0)   ( t ) P0 ( t )dt 
1
1
 (t )dt  0
-1
.
wavelet.
36
International Journal of Computer Applications (0975 – 8887)
Volume 108 – No 12, December 2014
Theorem 2.1. If
  X defines a
  L1 (1,1),
then the convolution ( ) defines a
Therefore,
Legendre wavelet and

Legendre wavelet.


11
(L  f ) (b, a) Pk (b)db  f (k ) (a , k ).

2 1
We have
A   
(  )  (k )

2
 (a , k )
Multiplying both sides by
and a weight function
q(a) and integrating both sides with respect to a from 0 to 1,
we
have



1

1

11
q(a ) (a, k )  (L  f )( b, a ) Pk (b)db da   q(a) | (a, k ) |2 da f (k ).
2 0
 1

0

(3.2)
k
k


| (k ) |2 | (k ) | 2

k
k

 ||  ||1 
k
Assume that
| (k ) |

k
.
2
1
0

f  L1 (1,1) and   X and (L  f )
(b,a) be the
continuous Legendre wavelet transform. Then, we have the
following inequality
|| (L  f ) (b, a) || X  || f || l ||  || X
f (k ) 

1

1 1
  (L  f ) (b, a)Pk (b)db da
q
(
a
)

(
a
,
k
)
2Q(k ) 0
 1


1

1 1

q
(
a
)
(
L
f
)
(b,
a)

(a , k )Pk (b)db da




2Q(k ) 0
 1


.
Proof. The above inequality follows from (1.15).
(3.4)
We have from (2.3)
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 11
f ( t )  (az) K(b, t, z) dz dt
2 1 1
1


  f (t ) (az) dz dt  2k  1Pk (b) Pk (t ) Pk ( z) 
2 11
2 k

11
1 1
1 1

  (2k  1)Pk (b)  f (t )Pk (t )dt     (az)Pk (z)dz 
k
 2 1
  2 1

11
 b ,a ( t )   K (b, t , z)(az)dz
 1

  (2k  1) Pk (b) f (k )(a , k )
k
11
(az) (2k  1) Pk (b) Pk ( t ) Pk (z)dz
2 1
k

1
1
  (2k  1) Pk (b) Pk ( t )  (az) Pk (z)dz
2 k
-1

  (2k  1) Pk (b) Pk ( t ) (a , k )
k
v
=



 (a , k )Pk (b)  ( t )


.
Using

f (k ) 
where

11
 (az) Pk (z)dz.
(a, k) 2 1
0. (3.3)
Using (3.3), (3.2) can be written as
Theorem2.2. Let


Q(k )   q(a ) | (a , k |2 da 
(  ) represents a Legendre wavelet.
Therefore,
(L  f ) (b, a) 
;
so that

 is a
Proof. Let
and
so that
bounded function on (-1,1). By Lemma 1.2, ( )  X.
  L1 (1,1),
X

( L  f )  ( k , a )  f ( k )  (a , k )
 b ,a
Set
(3.5)
in
1
(3.5)
(3.4)
1
we
have

1
q(a )  (L  f ) (b, a)  b ,a (k )dadb
2Q(k ) 0
1

 (k )   b ,a (k ) / Q(k )
. (3.6)
(3.1)
37
International Journal of Computer Applications (0975 – 8887)
Volume 108 – No 12, December 2014
1
Then
Putting

1
 b,a
1
f (k )   q(a)  ( L f ) (b,a)  (k )dadb
2 0 1

(3.7)
in
(1.8),
(k )
 *
 (k ) 
. (3.7)
we
1 1
have


2  | (2  j k ) |2
j 
.
 b ,a
1
f ( t )   (2k  1)Pk ( t )   q(a )( L  f )( b, a )  (k )dadb
2 k
0 1
1 1
 b ,a
1
   q(a ) (L  f )( b, a ) (2k  1)  (k )Pk ( t )dadb
2 0 1
k
Then
v
  * m

f (t )    L f (b)  (2 k)Pk (t )  (b)db
m   1



1

m
Therefore
1 1
1
   q(a ) (L  f )( b, a ) b ,a ( t )dadb
2 0 1
f(t)
Now, we assume that
“stability condition”

  L2 (1,1)
m
v

m
  L f (b)  (2k  1)  (2
*

m
m 1
m
satisfies the so called
*
 (2k  1)  (2
k
k) |  B
2
m
k ) Pk (t )  Lm f (b)Pk (b)db
1
1
2 (2k  1)  (2 m k )Pk ( t )Lm f  (k )
*
(4.1)
for certain positive constants A and B, 0 < A
  L2 (1,1) satisfying
 B  .
m
=

k

(4.1) is called
m
k
Using the definition (2.4), we define the semi-discrete
f  L2 (1,1)
11
  f ( t ) b (2 m t )dt
2 1
|| L f ||
m  
k

2  ( 2 2 k
2
m  j
j
=

(4.3)
m  Z.
Now, using Parseval identity (1.17), (4.1) yields the following
A || f || 22 
m
  (2k  1) f (k )Pk ( t )
(4.4)
 m (z)  (2  m z),

2  (2k  1)Pk (t) f (k )
by

(2 m k ) (2 m k )
k
=f(t) .
 (f   m ) ,
where


1
( Lm f ) (b)  (L f ) (b, m )   f (t ), b,2  m (t ) 
2
(4.2)
*

2 (2k  1)Pk (t) f (k ) (2 m k )  (2 m k )
=
Legendre wavelet transform of any
k)Pk (t )Pk (b)db
k
=
m  
The function
we have
=
1
m

 | (2
A
 1  b  1.
(4.7)
  * m
L f (b)   (2 k)Pk (t )  (b)db


m 1


1
4. THE DISCRETE TRANSFORM
assuming that a = 2-m; m  Z and
f  L2 (1,1),
Proof: In view of (4.4), for any
(3.8)
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
(4.6)

m
2
2
 B || f || 22 , f  L2 (1,1)
(4.5)
Theorem 4.1. Assume that the semi-discrete LWT of any
f  L (1,1) is defined by (5.2). Let us consider another
2
wavelet * defined by means of its Legendre transform.
dual.
~
Definition 4.2. A function
  L2 (1,1)
is called a
f  L2 (1,1)
can
be
expressed
as


f (t )    Lm f (b) ~ (2 m k)Pk (t )  (b)db.


m 1
1
(4.8)
So far we have considered semi-discrete Legendre wavelet
transform of any
f  L2 (1,1)
discretizing only variable
38
International Journal of Computer Applications (0975 – 8887)
Volume 108 – No 12, December 2014
a. Now, we discretize the translation parameter b also by
restricting it to the discrete set of points
b 0  [1,1]
where
 b ;m , n ( t )   b
0
so that
n
b0 , m 
2m
Z, n  N0 ,
b m ,n 
m , n ;a m
f  L2 (1,1)
(4.9)
Hence, every
can be reconstructed from its
discrete LWT given by (4.11). Thus
( t )  ( 2 t ,2 n b 0 )
m
f  T Tf 
(4.10)
0
Z,
m
n
N0.(4.11)
mZ
nN 0
2
A
and
0  A  B  .
Theorem 4.3.
are
positive
constants
such
0
f
  f, 
mZ
nN 0
  mb ,n
b 0 ;m, n
0
that
In this section, using
basis
L2(-1,1) is studied.
b
0 ;m , n
Assume that the discrete LWT of any
f  L (1,1)
is defined by (4.11) and stability condition
(4.12) holds. Let T be a linear operator on L2(-1,1) defined by
 f ,
mZ
nN 0
 b
b 0 ;m , n
Definition 5.1. A function

b ;m , n
of
0 ;m , n
(4.13)

Then
f    f ,  b ;m,n   mb,n
0
(4.14)

 T b
0 ;m , n
; m
The linear span
Z, n  N0
Proof. From the stability condition (4.12), it follows that the
operator defined by (4.13) is a one-one bounded linear
operator.
Set
g  Tf , f  L (1,1)
0
There exist positive constants A and B with 0<A 
such that
  f, 
b 0 ; m, n

2
2
for all
.
Therefore
A || T g ||  A || f ||  Tf, f 
2
2
B
c
mZ
nN 0
2
 bo;m,n  B {c m,n }  2
m, n
2
(5.2)
mZ
nN 0
1
Z > is dense in L2(-1,1)
(5.1)
Then, we have
 Tf , f  

  b ;m , n : m 
A || {c m,n } || 22 
2
b ;m , n
is said to
0
generate a Riesz basis of
with sampling rate b0 if
the following two properties are satisfied.
where
1
L (1,1)
  L2 (1,1)
Definition 5.2. A function
m ,n
b0
is said to
2
0
generate a frame
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.
.
0
a frame is defined and Riesz
of
  L2 (1,1)
2
Tf 
Z, n  N0
5. FRAMES AND RIESZ BASIS IN L2(-1,
1)
,
B
(4.15)
which completes the proof of theorem 4.3.
  B || f || 22 , f  L2 (1,1)
(4.12)
where
0
Then, the reconstruction (4.15) can be expressed as
The “stability” condition for this reconstruction takes the form
b0 ;m, n
 T 1 b ;m,n
 mb,n  T 1 b ;m,n ; m 
0
  f ,
b 0 ;m, n
Finally, set
can be expressed as
(L  f ) (b m, n;a m )  f ,  b ;m,n 
A || f || 22 
mZ
nN 0
1
Then the discrete Legendre wavelet transform of any
f  L2 (1,1)
  f, 
is a fixed constant. We write
m
1
|| g || 2
A
.
|| T 1g || 2 
2
2
 g, T -1 g  || g || 2 || T 1 g || 2
{c m,n }   2
Riesz bounds of
2
(N 0 ). Here A and B are called the
{ b ;m,n }
0
.
  L2 (1,1),
Theorem 5.3. Let
statements are equivalent.
then the following
39
International Journal of Computer Applications (0975 – 8887)
Volume 108 – No 12, December 2014
{ b ;m,n }
Furthermore, from (6.5) and (6.6); we conclude that
0
 r ,s ,  m , n   r ,s , m , n
is a Riesz basis of L2(-1,1);
{ b ;m,n }
2
is a frame of L2(-1,1) and is also an  independent family in the sense that if
0
linearly
  b ;m,n c m, n  0
0
{c m, n }   2 ,
and
and the Riesz bounds of {r,s} are B-1 and A-1.
In particular, for any
Furthermore, the Riesz bounds and frame bounds agree.
m ,n
2 -
{ b ;m,n }
1
  f ,
  || f || 22  A -1   f ,  bo;m,n 
2
bo;m , n
m ,n
M   r ,s,m,n ( r ,s ), ( m,n )N N
0
0
from
A || {c m,n } || 2 
2
c
0
.
(5.2),
r, s
we
so that M is positive definite. We denote the inverse of M by
(5.4)
t ,u
  bo;m ,n
.
{ b ;m,n }
2
{ b ;m,n }
0
is a Riesz basis
[1] C.K. Chui, An Introdcution to Wavelets, Acadmic Press,
New York (1992).
N0 (5.5)
and
bo;m , n
and f =
6. REFERENCES
  r ,s;t ,u  t ,u;m,n   r ,m s,m; r, s,m, n 
2
g  L2 (1,1)
0
Also, by the   linear independence of
, this
representation is unique. From the Banach-Steinhaus and open
mapping theorem it follows that
of L2(-1,1).
which means that both
B 1 {cm,n } 2 
 f ,
mZ
nN 0
have
r, s,m, n
2
0
Since, (5.8) is equivalent to (4.12), therefore, statement
(i) implies statement (ii). To prove the converse part, we recall
(5.3)
 r ,s ,m,n c m,n  B || {c m,n } || 22
M 1   r ,s,m,n ( r ,s ),( m,n )N
(5.8)
g( x ) 
 r ,s ,m,n   b ;r ,s , b ;m,n
0
m ,n
Theorem 4.3 and we have for any
T-1g,
where the entries are defined by
c
2
r ,s
r , s ,m,n
 r,s,m, n c m, n  A -1 {cm,n } 2
are satisfied. This allows us to introduce
 r ,s ( x )    r , s ; m , n  b ; m , n ( x )
0
m,n
 r ,s  L2 (1,1)
.
(5.7)
and it follows from (5.3) and
 r ,s ;  b ;m,n   r ,m  s ,n ; r, s, m, n 
0
[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).
(5.6)
Clearly,
(5.5) that
and
B
0
Let
be a Riesz basis with Riesz bounds A and B,
and consider the matrix operator
Then
we may write
f ( x )    f ,  bo;m,n   m, n ( x )
then cm,n = 0.
Proof. It follows from (5.2) that any Riesz basis is
linearly independent.
f  L2 (1,1)
N
[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).
[7] R.L. Stens and M. Wehrens, Legendre Transform
Methods and Best Algebraic Approximation, Comment.
Math.
Prace
Mat
21(2)
(1980),
351-380.
which means that {r,s} is the basis of L2(-1,1), which is
dual to
{ b ;m,n }
0
.
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