Characterization of W p -type of spaces involving fractional Fourier transform

Journal of Inequalities and Applications, Jan 2015

The characterizations of W p -type of spaces and mapping relations between W- and W p -type of spaces are discussed by using the fractional Fourier transform. The uniqueness of the Cauchy problems is also investigated by using the same transform. MSC: 46F12, 46E15.

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Characterization of W p -type of spaces involving fractional Fourier transform

Upadhyay and Kumar Journal of Inequalities and Applications Characterization of W p-type of spaces involving fractional Fourier transform Santosh Kumar Upadhyay 0 1 2 Anuj Kumar 1 2 0 Department of Mathematical Sciences, Indian Institute of Technology (BHU) , Varanasi, 221005 , India 1 Using the theory of the Hankel transform, Betancor and Rodriguez-Mesa [] gave a new characterization of the space of We 2 DST-Centre for Interdisciplinary Mathematical Sciences, Banaras Hindu University , Varanasi, 221005 , India The characterizations of Wp-type of spaces and mapping relations between W- and Wp-type of spaces are discussed by using the fractional Fourier transform. The uniqueness of the Cauchy problems is also investigated by using the same transform. MSC: 46F12; 46E15 fractional Fourier transform; convex functions; Gel'fand and Shilov spaces of type W; Lp-space - x, y = x · y = and the norm of x is defined by |x| = = x + · · · + xn  . The n-dimensional fractional Fourier transform (FrFT) with parameter α of f (x) on x ∈ Rn is denoted by (Fαf )(ξ ) [, ] and defined as f p = Kα(x, ξ )f (x) dx, ξ ∈ Rn, ∀n ∈ Z Cα = (π i sin α) –n e inα = The corresponding inversion formula is given by f (x) = (π) n Rn Kα(x, ξ )fˆα(ξ ) dξ , x ∈ Rn, where the kernel Kα(x, ξ ) = Cαe– i(|x|+|ξ|)cotα +i x,ξ csc α, and Cα is defined by (.). Now from the technique of [, p.], (.) can be written as Fα e– i|x|cotα φ(x) (ξ ) = (π ) n Cαe i|ξ|cotα ψ (w sin α) = (π ) n Cαe i|wsin α| cotα Now we recall the definitions of W - and W p-type of spaces from [–], which are given below. Let μj and wj, j = , . . . , n, be continuous and increasing functions on [, ∞) with μj() = wj() =  and μj(∞) = wj(∞) = ∞. We define Mj(xj) = j(yj) = wj(ηj) dηj (yj ≥ ), Zn, (b + ρ)y , k ∈ + zk = zk · · · znkn , Mj(–xj) = Mj(xj), Mj(xj) + Mj xj ≤ Mj xj + xj , j(–yj) = j(yj), j(yj) + j yj ≤ j yj + yj . We define μ(ξ ) = μ(ξ) , . . . , μn(ξn) , w(η) = w(η) , . . . , wn(ηn) . The space WM,a(Rn) consists of all C∞-complex valued functions φ(x) on x ∈ Rn, which for any δ ∈ Rn+ satisfy the inequality Dkxφ(x) ≤ Ck,δ exp –M (a – δ)x , and the space W Mp,a(Rn) consists of all infinitely differentiable functions φ(x) on x ∈ Rn, which for any δ ∈ Rn+ satisfy the inequality The space WM,,ab(Cn) consists of all entire analytic functions φ(z) such that there exist constants ρ, δ ∈ Rn+ and Cδ,ρ >  such that φ(z) ≤ Cδ,ρ exp –M (a – δ)x + (y) be the pair of functions which are dual in the sense of xjyj ≤ Mj(xj) + j(yj), holds for any xj ≥, yj ≥ . Theorem . Let M(x) and Young. Then 2 Characterization of Wp-type of spaces Fα WM,a ⊂ W , a ,r for any  ≤ p, r < ∞. p Proof Let e– i|x|cotα φ(x) ∈ W Mp,a(Rn) and σ = w + iτ . Then for any p and r, using the technique of [, pp.-] and (.), we have (σ sin α)kψ (σ sin α) r = Now using the inequality |σ ||k| ≤ |σ ||k+|+|σ ||k| , we have |w|+ Using (.), we get In the above expression, we set γ = ( a + ρ), since γ = a – δ and ρ is arbitrarily small together with δ. Therefore, we have ≤ Ck,r,η,s,ρ exp –yw +  (b + ρ)y From the arguments of [, p.] we have Dkwψ (w sin α) ≤ Ck,ρ,r,η,s exp –M exp M This implies that ψ (w sin α) ∈ W Mr, b . Theorem . Let M(x) and (y) be the functions which are dual in the sense of Young to the functions M(x) and (y), respectively. Then for any  ≤ p, r < ∞. ψ (σ sin α) ≤ (π i sin α) –n e inα ≤ (π i sin α) –n e inα = Cρ,δ,α exp – w, y + Now using the arguments of [, p.], we have ψ (σ sin α) ≤ Cρ,δ,α exp –M exp M 3 Relation between W- and Wp-types of spaces p WM,a = WM,a,  ≤ p < ∞. p  Fα WM,a ⊂ W , a . p  WM,a ⊂ Fα– W , a . p Lemma . Let  ≤ p < ∞. Then WM,a ⊂ WM,a. r≤k r≤k r≤k e– x,τ csc α |σ ||k–r–m+λ|+ + |σ ||k–r–m+λ| (|w| + ) ≤ Cα,k exp – x, τ csc α + Now using the arguments of [, p.], we get Fα– W , a ⊂ WM,a. Thus (.) and (.) imply that Rn φ(x) ∈ WM,a( ) and σ = w + iτ ∈ . Then from [, Theorem .], it Proof Let e follows that a . Then from (.) we have φk(x) ≤ Cα,k exp – x, τ csc α + Using (.) and [, p.] we get This implies that ⊂ WM,a. From (.) and (.) we find Now from (.) and (.) we get the result WM,a ⊂ WM,a. M,a = WM,a. = W ,  ≤ p < ∞. ⊂ WM, a . Proof Let e (σ csc α)kφ(s) ≤ |Cα| r≤k r≤k r≤k r≤k Now using the arguments of [, p.], we get (σ csc α)kφ(σ ) ≤ Cα,k exp Therefore (.) and (.) yield ⊂ W Again we take e ⊂ WM, b . p (iσ csc α)kφ(σ ) dw |iσ csc α||k|+ + |iσ csc α||k| p (|w| + ) ≤ Cα,k,δCp Rn exp – x, τ csc α – M (iσ csc α)kφ(σ ) p ≤ Cα,k,δ,p exp This implies that Fα–(WM, b ) ⊂ W ,b,p. Now (.) and (.) give W ,b ⊂ W ,b,p. Finally, (.) and (.) give W ,b = W ,b,p. Theorem . Let (y) and M(x) be the functions which are dual in the sense of Young to the functions M(x) and (y), respectively. Then WM,,ab,p = WM,,ab,  ≤ p < ∞. By the inverse property of the fractional Fourier transform we get – i(|z|+|σ|)cotα +i σ ,z csc αφˆα(z) dx.  × exp –M exp –M Now using (.), we have φ(σ + iτ ) ≤ Cδ,ρ,α exp –M ,b, ,b Fα– WM,a ⊂ WM,a. Thus from (.) and (.), we get ,b,p ,b WM,a ⊂ WM,a. ,b ,b,p WM,a ⊂ WM,a . Similarly it is easy to show that Finally, (.) and (.) imply that u(x, ) = u(x), = P(i x)u(x, t), ∀(x, t) ∈ Rn × [, T ], kx = φ(x, t) = φ(x) ∈ , 4 Uniqueness class of a Cauchy problem in the space  for  ≤ t ≤ t, where t is any point in the interval  ≤ t ≤ T , P˜ is the adjoint of P and x∗ is the conjugate of x. Applying the fractional Fourier transform to (.) and (.), we get Let us write of the system (.) and (.), using the inequality and the arguments of [, p.] in (.) we obtain If we set M(w csc α) = |w csc α|p /p, (τ csc α) = |τ csc α|p /p, Now, let us assume that = W ,b ψα,(σ ) = (Fαφ)(x) ∈ WM,a. Q(σ csc α, t, t) will be a multiplier in the space WM,,ab which maps this space into the space ,b+θ WM,a–θ taking T sufficiently small. Thus the Cauchy problem (.) and (.) has a unique ,b+θ Fα– WM,a–θ = Competing interests The authors declare that they have no competing interests. Authors’ contributions The main idea of this paper was proposed by SKU. AK prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript. Acknowledgements The first author is thankful to DST-CIMS, Banaras Hindu University, Varanasi, India for providing the research facilities and the second author is also thankful to DST-CIMS, Banaras Hindu University, Varanasi, India for awarding the Junior Research Fellowship since December 2012. 1. Gurevich , BL: New types of test function spaces and spaces of generalized functions and the Cauchy problem for operator equations . Dissertation, Kharkov ( 1956 ) 2. Gel 'fand, IM, Shilov, GE: Generalized Functions , vol. 3. Academic Press, New York ( 1967 ) 3. Friedman , A: Generalized Functions and Partial Differential Equations. Prentice Hall , New York ( 1963 ) 4. Pathak , RS, Upadhyay, SK: Wp-Space and Fourier transformation. Proc. Am. Math. Soc . 121 ( 3 ), 733 - 738 ( 1994 ) 5. Betancor , JJ, Rodriguez-Mesa , L: Characterization of W-type spaces . Proc. Am. Math. Soc . 126 ( 5 ), 1371 - 1379 ( 1988 ) 6. Upadhyay , SK: W-Spaces and pseudo-differential operators . Appl. Anal . 82 , 381 - 397 ( 2003 ) 7. De Bie , H, De Schepper , N: Fractional Fourier transforms of hyper complex signals . Signal Image Video Process . 6 , 381 - 388 ( 2012 ) 8. Upadhyay , SK, Kumar , A, Dubey, JK : Characterization of spaces of type W and pseudo-differential operators of infinite order involving fractional Fourier transform . J. Pseud.-Differ. Oper. Appl . 5 ( 2 ), 215 - 230 ( 2014 ) 9. Prasad , A, Mahato, A: The fractional wavelet transform on spaces of type W. Integral Transforms Spec . Funct. 24 ( 3 ), 239 - 250 ( 2012 ) 10. Pathak , RS: On Hankel transformable spaces and a Cauchy problem . Can . J. Math. 34 ( 1 ), 84 - 106 ( 1985 )


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Santosh Upadhyay, Anuj Kumar. Characterization of W p -type of spaces involving fractional Fourier transform, Journal of Inequalities and Applications, 2015, 31, DOI: 10.1186/s13660-014-0544-9