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 1Pk (b) Pk (t ) Pk ( z) 2 11 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 ) Lm f (b)Pk (b)db 1 1 2 (2k 1) (2 m k )Pk ( t )Lm 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 ( Lm 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 ) Lm 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) mZ nN 0 2 A and 0 A B . Theorem 4.3. are positive constants such 0 f f, mZ nN 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 , mZ nN 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 mZ nN 0 2 bo;m,n B {c m,n } 2 m, n 2 (5.2) mZ nN 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 mZ nN 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 , mZ nN 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 40 2

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