The Bessel Wavelet Transform

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1 Int. Journal of Math. Analysis, Vol. 5, 211, no. 2, The Bessel Wavelet Transform Akhilesh Prasad* and Ashutosh Mahato Department of Applied Mathematics Indian School of Mines Dhanbad-8264, Jharkhand, India apr (Corresponding Author); M. M. Dixit Department of Mathematics NERIST, Itanagar, India Abstract In this paper continuity of the Bessel wavelet transform B ψ of function φ in terms of a mother wavelet ψ is investigated on certain distribution spaces when the Hankel transform of ψ defined by ψ(x, y) C (R 2 +). A sobolev space boundedness result is obtained. Mathematics Subject Classification: 44A35, 33A4, 42C1 Keywords: Bessel wavelet transform, Hankel transform, Bessel operator, Sobolev space 1 Introduction The Hankel transformation is usually defined by φ(x) (h μ φ)(x) (xy) 1/2 J μ (xy)φ(y)dy, x R + (, ) (1.1) wherej μ represents the Bessel function of the first kind and order μ. We shall assume throughout this paper that μ 1/2 and φ L 1 (R + ), R + (, ). The inversion formula for (1.1) [2, p.239 is given by φ(y) (xy) 1/2 J μ (xy)(h μ φ)(x)dx, y R + (1.2)

2 88 A. Prasad, A. Mahato and M. M. Dixit The above transformation has been extended to distributions by Zemanian [6. For every μ R +, he introduced the space H μ (R + ) consisting of all infinitely differentiable function φ defined on R + (, ), such that m, k N the quantities γ μ m,k sup x m (x 1 d/dx) k x μ 1/2 φ(x) <. (1.3) x R + Using theory of H μ space of Zemanian [6, Pathak and Dixit [3 investigated the Bessel wavelet transform B ψ defined as follows: (B ψ φ)(b, a) : where φ(u) (h μ φ)(u). Let us assume that for any real number ρ, ψ satisfies (bu) 1/2 J μ (bu) φ(u) ψ(au)du, (1.4) (1 + x) l (x 1 d/dx) α (y 1 d/dy) β ψ(xy) Cα,β,l (1 + y) ρ β, α, β, l N (1.5) where C α,β,l > is a constant and ψ denotes the Hankel transform of the basic wavelet ψ. The class of all such wavelet ψ is denoted by H ρ. This permits us to define the Hankel transform with respect to x of ψ(ax) h μ [(h μ (ψ)) (aξ) (xξ) 1/2 J μ (xξ)(h μ ψ)(ax)dx. (1.6) We shall use some notation and terminology of [2,4,5,7. The differential operator S μ,x (or μ,x ) is defined by S μ,x d2 dx + (1 4μ2 ). (1.7) 2 4x 2 From [2,4 we know that for any φ H μ, h μ (S μ φ) y 2 h μ φ, (1.8) (x 1 d/dx) k (ψφ) and from [2 also we have k ν ( ) k (x 1 d/dx) ν φ(x 1 d/dx) k ν ψ (1.9) ν r Sμ,x r φ(x) b j x 2j+μ+1/2 (x 1 d/dx) r+j (x μ 1/2 φ(x)), (1.1) j where b j are constants depending only on μ.

3 The Bessel wavelet transform 89 Definition 1.1 A tempered distribution φ H μ (R +) is said to belong to the Sobolev space G μ,s,μ R, 1 p <, if its Hankel transform h μφ corresponds to a locally integrable function over R + (, ) such that ( 1/p φ G μ (R + ) (1 + ξ 2 ) s (h μ φ)(ξ) dξ) p. (1.11) 2 The Bessel Wavelet Transform Lee [1 has defined the space B μ,b and Υ 2q μ,b as follows: Definition 2.1 φ B μ,b if φ is smooth function φ(x) for x>band α μ b,k (φ) sup (x 1 d/dx) k x μ 1/2 φ(x) <,k, 1, 2,... (2.1) x R + where b> is a constant and μ is real number. Definition 2.2 For each q1,2,3,..., Φ Υ 2q μ,b,ifz μ 1/2 Φ is an even entire function and β μ,2q b,k (Φ) sup zx+iy e by2q z 2k μ 1/2 Φ(z) where Φ(h μ φ),b> is constant and μ is real number. <,k, 1, 2,... (2.2) The topology of the spaces B μ,b and Υ 2q μ,b are generated by the seminorms { α μ } b,k and { β μ,2q } k b,k respectively. It follows from Definition 2.1 and 2.2 k that B μ,b and Υ 2q μ,b are Frechet spaces. We define σ μ b,k (φ) max νk αμ b,ν (φ); ρμ,2q b,k (φ) max νk βμ,2q b,ν (φ). (2.3) Then σ μ b,k and ρμ,2q b,k define a norm on the space B μ,b and Υ 2q μ,b respectively. Following technique of Zemanian [7, p.141, we can write x ν (x 1 d/dx) n x μ 1/2 h μ φ(x) y 2μ+2n+ν+1 (y 1 d/dy) ν (y μ 1/2 φ(y)) (xy) (μ+n) J μ+ν+n (xy)dy. (2.4) Theorem 2.3 The Bessel wavelet transform B ψ is a continuous linear mapping of B μ,b into Υ 2q μ,b.

4 9 A. Prasad, A. Mahato and M. M. Dixit Proof: Let z x + iy and μ 1/2, the Bessel wavelet transform B ψ has the representation (B ψ φ)(z, a) (zu) 1/2 J μ (zu)(h μ φ)(u)(h μ ψ)(au)du, μ 1/2 with b> and (h μ φ)(u)(h μ ψ)(au) L 2 (,b) if and only if (B ψ φ)(z, a) L 2 (, ), z μ 1/2 (B ψ φ)(z, a) is an even entire function of z and there exists a constant C such that Let φ B μ,b, then (B ψ φ)(z, a) C exp(b y ), z. (B ψ φ)(z, a) (zu) 1/2 J μ (zu)(h μ φ)(u)(h μ ψ)(au)du, μ 1/2 h μ [(h μ φ)(u)(h μ ψ)(au) (z). Applying the technique of the Zemanian for fixed a, from (2.4), z 2k μ 1/2 (B ψ φ)(z, a) u [(u 2k+2μ+1 1 D) 2k u μ 1/2 (h μ ψ)(au)(h μ φ)(u) So that e by2q z 2k μ 1/2 (B ψ φ)(z, a) [ (zu) μ J μ+2k (zu) du. u [(u 2k+2μ+1 1 D) 2k u μ 1/2 (h μ ψ)(au)(h μ φ)(u) [ (zu) μ J μ+2k (zu) e by2q du u2k+2μ+1 2k s u μ 1/2 (h μ φ)(u) sup z,u sup u 2k s ( ) 2k s sup u ( ) 2k (u 1 D) s (h μ ψ)(au)(u 1 D) 2k s s [ (zu) μ J μ+2k (zu) e by2q du (1 + u) 2k+2μ+1 (u 1 D) s ψ(au) (u 1 D) 2k s u μ 1/2 (h μ φ)(u) sup z,u [ (zu) μ J μ+2k (zu) e by2q du.

5 The Bessel wavelet transform 91 Applying inequalities (1.5) and (2.1), then from the above, we have 2k ( ) 2k (1 + u) 2k+2μ+1 C s (1 + u) ρ α μ b,2k s s (h μφ) s 2k s ( ) 2k s sup z,u [ (zu) μ J μ+2k (zu) e by2q du C s α μ b,2k s (h μφ) sup z,u (zu) μ J μ+2k (zu)e by2q (1 + u) 2k+2μ+ρ+1 du. (2.5) We note that for all z such that z 1 z μ J μ+2k (z) 2 μ e μ! and for z > 1 z μ J μ+2k (z) C(π/2) 1 z μ 1/2 exp( Imz ). Since y y 2q,if y 1 and y y 2q,if y < 1, then (zu) μ J μ+2k (zu)e by2q 2 μ e e by2q, y 1 μ! C(π/2) 1 z μ 1/2 e b( y 2q y ), y >1. Thus (zu) μ J μ+2k (zu)e by2q L1 for μ +1/2. Therefore e by2q z 2k μ 1/2 (B ψ φ)(z, a) L 1 This completes the proof of the theorem. 2k s ( ) 2k C s s α μ b,2k s (h μφ). 3 The Sobolev Type Space We have already defined the Sobolev space G μ (R + ) by (1.11). In the following, we shall make use of the following norm on G μ (R + R + ) in the proof of the boundedness result ( 1/p φ G μ (R + R + ) (1 + ξ 2 ) s (1 + η 2 ) s (h μ φ)(ξ,η) dξdη) p, φ H μ (R + R + ).

6 92 A. Prasad, A. Mahato and M. M. Dixit Lemma 3.1 Let us assume that for any positive real number ρ, ψ(x) satisfies ( x 1 d ) l x μ 1/2 ψ(x) dx C l,ρ(1 + x) ρ l, l N (3.1) then there exists a positive constant C such that h μ [(h μ ψ) (aξ) C (1 + a) ρ+2l+μ+1/2 (1 + ξ 2 ) l. (3.2) Proof: From the definition (1.6), we know that h μ [(h μ ψ) (aξ) (xξ) 1/2 J μ (xξ)(h μ ψ)(ax)dx. So that from [4, (1 + ξ 2 ) l h μ [(h μ ψ) (aξ) ξ,a R +,l N and Δ μ,x defined as (1.7). Now, (1 Δ μ,x ) l (h μ ψ)(ax) (xξ) 1/2 J μ (xξ)(1 Δ μ,x ) l (h μ ψ)(ax)dx, (3.3) l r ( ) l ( 1) r Δ r μ,x r (h μψ)(ax) l r ( ) l ( 1) r r r j ( b j x 2j+μ+1/2 x 1 d ) r+j ( x μ 1/2 (h μ ψ)(ax) ). (3.4) dx Hence by (3.3) and (3.4), and inequality (3.1), we have h μ [(h μ ψ) l (aξ) Q μ (1 + a) ρ+2l+μ+1/2 (1 + ξ 2 ) l (1 + x) 2j+2μ+ρ+1 2l dx. r j Choosing l j>μ+ ρ/2 + 1, we conclude that h μ [(h μ ψ) (aξ) C (1 + a) ρ+2l+μ+1/2 (1 + ξ 2 ) l. r ( ) l b j C r+j,ρ r

7 The Bessel wavelet transform 93 Definition 3.2 Let (h μ ψ)(aξ) be a wavelet in H ρ defined by (1.5), Then the Bessel wavelet transform B ψ defined by (B ψ φ)(y, x) (yη) 1/2 J μ (yη)(h μ ψ)(xη)(h μ φ)(η)dη (3.5) exists for φ H μ (R + ). Theorem 3.3 For any wavelet h μ ψ H ρ, the Bessel wavelet transform (B ψ φ)(y, x) admits the representation (B ψ φ)(y, x) where φ H μ (R + ). Proof: From definition (3.5) (B ψ φ)(y, x) (xξ) 1/2 J μ (xξ)(yη) 1/2 J μ (yη)(h μ ψ)(ξη)(h μ φ)(η)dξdη, (3.6) (yη) 1/2 J μ (yη)(h μ ψ)(xη)(h μ φ)(η)dη [ (yη) 1/2 J μ (yη) (xξ) 1/2 J μ (xξ)(h μ ψ)(ξη)dξ (h μ φ)(η)dη (xξ) 1/2 J μ (xξ)(yη) 1/2 J μ (yη)(h μ ψ)(ξη)(h μ φ)(η)dξdη. The last integral exists because (h μ φ)(η) H μ (R + ) and (h μ ψ)(ξη) satisfies the inequality (3.2). Corollary 3.4 For any basic wavelet h μ ψ H ρ, the Bessel wavelet transform h μ [B ψ φ(ξ,η) admits the representation h μ [B ψ φ(ξ,η)(h μ ψ)(ξη)(h μ φ)(η), (3.7) where φ H μ (R + ). Proof: The right hand side of (3.7) (h μ ψ)(ξη)(h μ φ)(η) h μ [(h μ ψ)(ξη) h μ [(h μ φ)(η) [ (xξ) 1/2 J μ (xξ)(h μ ψ)(xη)dx [ (yη) 1/2 J μ (yη)(h μ φ)(y)dy (h μ φ)(y)dxdy h μ [B ψ φ(ξ,η). (xξ) 1/2 J μ (xξ)(yη) 1/2 J μ (yη)(h μ ψ)(xη)

8 94 A. Prasad, A. Mahato and M. M. Dixit Theorem 3.5 Let h μ ψ H ρ and (B ψ φ)(x, y) be the Bessel wavelet transform then there exists D> such that for ρ R + and l N, (B ψ φ) G μ (R + R + ) D h μ φ s+(ρ+2l+μ+1/2)/2,p G (R μ + ), φ H μ (R + ). Proof: Using Lemma 3.1, we have (B ψ φ) G μ (R + R + ) ( (1 + ξ 2 ) s (1 + η 2 ) s (h μ [B ψ φ(ξ,η)) ) 1/p p dξdη ( ) (1 + ξ 2 ) s (1 + η 2 ) s (h μ ψ)(ξη)(h μ φ)(η) p 1/p dξdη ( (1 + ξ 2 ) s (1 + η 2 ) s ) p hμ (h μ ψ)(ξη) p 1/p (h μ φ)(η) p dξdη ( (1 + ξ 2 ) s (1 + η 2 ) s C (1 + η) ρ+2l+μ+1/2 (1 + ξ 2 ) l (h μ φ)(η) p dξdη ) 1/p. We note that and Therefore (1 + η) ρ+2l+μ+1/2 2 (ρ+2l+μ+1/2)/2 ( 1+η 2) (ρ+2l+μ+1/2)/2,ρ (1 + η) ρ+2l+μ+1/2 ( 1+η 2) (ρ+2l+μ+1/2)/2,ρ<. (1 + η) ρ+2l+μ+1/2 max ( 1, 2 (ρ+2l+μ+1/2)/2)( 1+η 2) (ρ+2l+μ+1/2)/2. Hence (B ψ φ) G μ (R + R + ) ( C (1 + ξ 2 ) s l max(1, 2 (ρ+2l+μ+1/2)/2 )(1 + η 2 ) s+(ρ+2l+μ+1/2)/2 (h μ φ)(η) p dξdη) 1/p C ( ( (1 + η 2 ) s+(ρ+2l+μ+1/2)/2 (h μ φ)(η) p dη (1 + ξ 2 ) s l p dξ) 1/p, ) 1/p where C is certain constant. The ξ integral is convergent as l can be chosen large enough so that (B ψ φ) G μ (R + R + ) D h μ φ G s+(ρ+2l+μ+1/2)/2,p μ (R + ).

9 The Bessel wavelet transform 95 4 Product of two Bessel Wavelet Transforms Let B ψ1 and B ψ2 be two Bessel Wavelet Transforms of φ H μ (R + ) defined as follows: (B ψ1 φ)(b, a) :B 1 (b, a) : and (B ψ2 φ)(d, c) :B 2 (d, c) : Then, their product B ψ1 ob ψ2 (orb 1 ob 2 ) is defined by B(b, a, c) (B 1 ob 2 )(b, a, c) (bu) 1/2 J μ (bu) ψ 1 (au) φ(u)du, (4.1) (du) 1/2 J μ (du) ψ 2 (cu) φ(u)du, (4.2) (bu) 1/2 J μ (bu) ψ 1 (au)h μ [B ψ2 φ(u, c)du (4.3) (bu) 1/2 J μ (bu) ψ 1 (au) ψ 2 (cu) φ(u)du (4.4) (bu) 1/2 J μ (bu)χ(a, c, u) φ(u)du, where φ denotes the Hankel transformation of φ and χ(a, c, u) ψ 1 (au) ψ 2 (cu), provided the integral is convergent. and ψ 2 (cu) H ρ 2, then for certain con- Theorem 4.1 Let ψ 1 (au) H ρ 1 stant D> and ρ 1,ρ 2 R +, (B ψ1 B ψ2 φ)(b, a, c) G μ (R + R + ) D φ. s+(ρ G 1 +ρ 2 )/2,p μ (R + ) Proof: By definition (4.3), we have (B ψ1 B ψ2 φ)(b, a, c) (bu) 1/2 J μ (bu) ψ 1 (au) h μ [B ψ2 φ(u, c)du. From (4.4), it follows that (B ψ1 B ψ2 φ) has Hankel transform equal to [ ψ1 (au) ψ 2 (au) φ(u)du. Therefore

10 96 A. Prasad, A. Mahato and M. M. Dixit (B ψ1 B ψ2 φ)(b, a, c) G μ (R + R + ) ( 1/p (1 + u 2 ) s (1 + a 2 ) s ψ1 (au) ψ 2 (au) φ(u)du dadu) p. Since from (1.5) ψ 1 (au) C ρ1,l(1 + a) l (1 + u) ρ 1 C ρ1,l max ( 1, 2 l/2)( 1+a 2) l/2 max ( 1, 2 ρ 1 /2 )( 1+u 2) ρ 1 /2 and ψ 2 (au) C ρ2,l(1 + a) l (1 + u) ρ 2 C ρ2,l max ( 1, 2 l/2)( 1+a 2) l/2 max ( 1, 2 )( ρ 2/2 1+u 2) ρ 2 /2. Therefore (B ψ1 B ψ2 φ) G μ (R + R + ) ( (1 + u 2 ) s (1 + a 2 ) s C ρ1,lc ρ 2,l (1 + u2 ) (ρ 1+ρ 2 )/2 (1 + a 2 ) l φ(u) p dadu ) 1/p ( C ρ1,ρ 2,l (1 + a 2 ) s l ) 1/p ( ) p da (1 + u 2 ) s+(ρ 1+ρ 2 )/2 p 1/p φ(u) du, where C ρ1,ρ 2,l is certain positive constant. The a-integral can be made convergent by choosing l sufficiently large, so that (B ψ1 B ψ2 φ) G μ (R + R + ) D φ, s+(ρ G 1 +ρ 2 )/2,p μ (R + ) where D is positive constant. Acknowledgement: This work is supported by Indian School of Mines, Dhanbad, under grant number 6132 /ISM JRF/Acad/29. References [1 W. Y. Lee, On the Cauchy Problem of the Differential Operator S μ., Proc. Amer. Math. Soc., 51(1) (1975),

11 The Bessel wavelet transform 97 [2 R. S. Pathak, Integral Transforms of Generalized Functions and Their Applications, Gordon and Breach Publishers, Amsterdam. (1997). [3 R. S. Pathak, and M. M. Dixit, Bessel Wavelet Transform on Certain Function and Distribution Spaces., Journal of Analysis and Applications., 1 (23), [4 A. Prasad, The Generalized Pseudo-Differential Operator on Some Gevrey Spaces., International Journal of Mathematical Sciences., 5(1) (26), [5 K. Trimèche, Generalized Wavelets and Hypergroups, Gordon and Breach Science Publishers, Amsterdam. (1997). [6 A. H. Zemanian, The Hankel Transform of Certain Distributions of Rapid Growth, SIAM. J. Appl. Math., 14 (1966), [7 A. H. Zemanian, Generalized Integral Transformations, Interscience Publishers, New York (1968). Received: August, 21