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,
Ghaziabad –201001, India
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
.
Hence, (t) changes sign in (-1,1) therefore it represents a
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
dyadic wavelet.
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.
The above theorem leads to the following definition of dyadic
dual.
~
Definition 4.2. A function
  L2 (1,1)
is called a
f  L2 (1,1)
dyadic dual of a dyadic wavelet , if every
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
.
IJCATM : www.ijcaonline.org
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