Characterization of W p type of spaces involving fractional Fourier transform
Upadhyay and Kumar Journal of Inequalities and Applications
Characterization of W ptype 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 RodriguezMesa [] gave a new characterization of the space of We
2 DSTCentre for Interdisciplinary Mathematical Sciences, Banaras Hindu University , Varanasi, 221005 , India
The characterizations of Wptype of spaces and mapping relations between W and Wptype 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; Lpspace

x, y = x · y =
and the norm of x is defined by
x =
= x + · · · + xn .
The ndimensional 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 inα =
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– ixcotα φ(x) (ξ ) = (π ) n Cαe iξcotα
ψ (w sin α) = (π ) n Cαe iwsin α cotα
Now we recall the definitions of W  and W ptype 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 = zk · · · 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 Wptype of spaces
Fα WM,a ⊂ W , a ,r for any ≤ p, r < ∞.
p
Proof Let e– ixcotα 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 –yw + (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 inα
≤ (π i sin α) –n e inα
= 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 Wptypes 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 αk+ + iσ csc αk 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 αp /p,
(τ csc α) = τ csc αp /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 DSTCIMS, Banaras Hindu University, Varanasi, India for providing the research facilities and
the second author is also thankful to DSTCIMS, 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: WpSpace and Fourier transformation. Proc. Am. Math. Soc . 121 ( 3 ), 733  738 ( 1994 )
5. Betancor , JJ, RodriguezMesa , L: Characterization of Wtype spaces . Proc. Am. Math. Soc . 126 ( 5 ), 1371  1379 ( 1988 )
6. Upadhyay , SK: WSpaces and pseudodifferential 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 pseudodifferential 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 )