Post on 12-Dec-2016
J. Pseudo-Differ. Oper. Appl. (2013) 4:251–265DOI 10.1007/s11868-013-0067-z
The fractional S-transform on spaces of type W
Sunil Kumar Singh
Received: 7 October 2012 / Revised: 11 February 2013 / Accepted: 6 March 2013 /Published online: 23 March 2013© Springer Basel 2013
Abstract In this paper, we generalize the results of Singh (Integr Transforms SpecialFunct 23:891–899, 2012). Following the techniques of Singh (Integr TransformsSpecial Funct 23:891–899, 2012), the continuity results for the fractional S-transformare obtained on some suitably designed spaces of type W .
Keywords Fractional Fourier transform · Fractional S-transform · S-transform ·Wavelet transform
Mathematics Subject Classification (2010) 42C40 · 46F12
1 Introduction
The fractional S-transform based on the S-transform [7] and fractional Fourier trans-form [1], generalizes the S-transform time-frequency representation to a time fractionalfrequency representation [9]. Let us recall the definition of the S-transform:
The one-dimensional continuous S-transform of u(t) is defined as [8]
(Su)(τ, f ) = S(u(t))(τ, f ) =∫
R
u(t)ω(τ − t, f )e−i2π f t dt, (1.1)
S. K. Singh (B)Department of Mathematics, Rajiv Gandhi University,Doimukh 791112, Arunachal Pradesh, Indiae-mail: sks_math@yahoo.com
252 S. K. Singh
where the window ω is assumed to satisfy the following:
∫
R
ω(t, f )dt = 1 for all f ∈ R\{0} . (1.2)
The most usual window ω is the Gaussian one
ω(t, f ) = | f |k√
2πe− f 2 t2
2k2 , k > 0, (1.3)
where f is the frequency, t is the time variable, and k is a scaling factor that controlsthe number of oscillations in the window.
Then, Eq. (1.1) can be rewritten as a convolution
(Su)(τ, f ) =(
u(·)e−i2π f · ∗ ω(·, f ))
(τ ). (1.4)
Applying the convolution property for the Fourier transform, we obtain
(Su)(τ, f ) = F−1{u(· + f )ω(·, f )}(τ ), (1.5)
where F−1 is the inverse Fourier transform. For the Gaussian window case (1.3),
F {ω(t, f )}(α, f ) = ω(α, f ) = e−2(πkα/ f )2. (1.6)
Thus we can write the S-transform in the following form:
(Su)(τ, f ) =∫
R
u(α + f )e−2(πkα/ f )2ei2πατ dα. (1.7)
Also, if u( f ) and (Su)(τ, f ) are the Fourier transform and S-transform of u respec-tively, then
u( f ) =∫
R
(Su)(τ, f )dτ ; (1.8)
so that
u(t) = F−1
⎛⎝
∫
R
(Su)(τ, ·)dτ
⎞⎠ (t). (1.9)
Some basic properties of S-transform can be found in [5,6].
The fractional S-transform 253
2 The fractional Fourier transform
The fractional Fourier transform(FRFT) has played an important role in signal process-ing. The ath order FRFT of a signal u(t) is defined as [1]:
Fau ( f ) =
∫
R
u(t)Ka(t, f )dt, (2.1)
where the transform kernel Ka(t, f ) is given by
Ka(t, f ) =⎧⎨⎩
Aθeiπ( f 2 cot θ−2 f t csc θ+t2 cot θ), if θ �= nπ
δ(t − f ), if θ = 2nπ
δ(t + f ), if θ + π = 2nπ,
(2.2)
where Aθ = √1 − i cot θ , θ = aπ/2, a ∈ [0, 4), i is the complex unit, n is an integer,
and f is the fractional Fourier frequency(FRFfr) (Fig. 1).The inverse FRFT of Eq. (2.1) is:
u(t) =∫
R
Fau ( f )Ka(t, f )d f. (2.3)
We can write (2.1) as
Fau ( f ) =
∫
R
u(t)Aθeiπ( f 2 cot θ−2 f t csc θ+t2 cot θ)dt
= Aθ eiπ f 2 cot θ∫
R
u(t)e−i2π f t csc θeiπ t2 cot θ dt (2.4)
= Aθ eiπ f 2 cot θ [eiπ t2 cot θ u(t)]( f csc θ).
Fig. 1 The time-FRFfrdomain plan
254 S. K. Singh
3 The fractional S-transform
The fractional S-transform(FRST) is a generalization of the S-transform. The ath ordercontinuous fractional S-transform of u(t) is defined as [9]:
F RST au (τ, f ) =
∫
R
u(t)g(τ − t, f )Ka(t, f )dt, (3.1)
where the window g is given by
g(t, f ) = | f csc θ |p
k√
2πe−t2( f csc θ)2p/2k2; k, p > 0, (3.2)
which satisfies the condition:
∫
R
g(t, f )dt = 1 for all f ∈ R\{0} . (3.3)
Inverse fractional S-transform is defined by
u(t) =∫
R
⎡⎣
∫
R
F RST au (τ, f )dτ
⎤⎦ Ka(t, f )d f. (3.4)
Note that the fractional S-transform depends on a parameter θ and can be interpreted asa rotation by an angle θ in the time-frequency plane. An FRST with θ = π
2 correspondsto the S-transform, and an FRST with θ = 0 corresponds to the zero operator. Theparameters p and k can be used to adjust the window function space.
Let
h(t, τ, f ) = g(τ − t, f )Ka(t, f ), (3.5)
and
Ha(τ, f, f1) =∫
R
h(t, τ, f )Ka(t, f1) =∫
R
g(τ − t, f )Ka(t, f )Ka(t, f1)dt.
(3.6)
Since
Ka(t, f )Ka(t, f1) = Aθ Aθeiπ [( f 2− f 21 ) cot θ−2( f − f1)t csc θ], (3.7)
The fractional S-transform 255
using (3.2) and (3.7) in (3.6) we obtain
Ha(τ, f, f1) =∫
R
| f csc θ |p
k√
2πe−(τ−t)2( f csc θ)2p/2k2
×Aθ Aθ eiπ [( f 2− f 21 ) cot θ−2( f − f1)t csc θ]dt
= | f csc θ |p
k√
2πAθ Aθ eiπ [( f 2− f 2
1 ) cot θ]
×∫
R
e−(τ−t)2( f csc θ)2p/2k2e−i2π( f − f1)t csc θ dt. (3.8)
By using the technique of [9], we obtain
Ha(τ, f, f1) = Aθ Aθ eiπ [( f 2− f 21 ) cot θ]e−i2π( f − f1)τ csc θ
×e−2π2k2( f − f1)2(csc θ)2/( f csc θ)2p
, (3.9)
and using (3.7) we can write
Ha(τ, f, f1) = e−[2π2k2( f − f1)
2(csc θ)2/( f csc θ)2p]Ka(t, f )Ka(t, f1). (3.10)
Also, the FRST can be defined as operations on the fractional Fourier domain
F RST au (τ, f ) =
∫
R
⎡⎣
∫
R
Fau ( f )Ka(t, f )d f
⎤⎦ g(τ − t, f )Ka(t, f )dt
=∫
R
Fau ( f1)Ha(τ, f, f1)d f1. (3.11)
By using (2.4) and (3.9) we can write (3.11 ) as follows:
F RST au (τ, f ) =
∫
R
Aθeiπ f 21 cot θ [eiπ t2 cot θ u(t)]( f1 csc θ)Aθ Aθ eiπ [( f 2− f 2
1 ) cot θ]
×e−i2π( f − f1)τ csc θ e−2π2k2( f − f1)2(csc θ)2/( f csc θ)2p
d f1
= Aθ |Aθ |2∫
R
eiπ f 21 cot θ [eiπ t2 cot θ u(t)]( f1 csc θ)eiπ [( f 2− f 2
1 ) cot θ]
×e−i2π( f − f1)τ csc θ e−2π2k2( f − f1)2(csc θ)2/( f csc θ)2p
d f1
= Aθ |Aθ |2eiπ f 2 cot θ∫
R
[eiπ t2 cot θ u(t)]( f1 csc θ)
×e−i2π( f − f1)τ csc θ e−2π2k2( f − f1)2(csc θ)2/( f csc θ)2p
d f1. (3.12)
256 S. K. Singh
The spaces of type W were first introduced by Gel’fand and Shilov [2] in the study of thetheory of differential equations. Pathak and Pandey [4] studied the wavelet transformon the spaces of type W . The S-transform has been studied on the spaces of type W bySingh [6]. In this paper, the continuity results for the fractional S-transform are obtainedon some suitably designed spaces of type W defined on R × (R\{0}), C × (R\{0}).
4 The spaces of type W
In this section we recall the definitions and properties of Gel’fand–Shilov spaces ofW -type.
Let μ(ξ) (0 ≤ ξ < ∞) and ω(η) (0 ≤ η < ∞) be continuous increasing functionssuch that μ(0) = 0, μ(ξ) → ∞ for ξ → ∞ and ω(0) = 0, ω(η) → ∞ for η → ∞.For x ≥ 0, y ≥ 0, we define
M(x) =x∫
0
μ(ξ)dξ, M(x) = M(−x), for x < 0 (4.1)
and
(y) =y∫
0
ω(η)dη, (y) = (−y), for y < 0. (4.2)
The functions M(x) and (y) are continuous, increasing and convex with M(0) = 0,M(x) → ∞ for x → ∞ and (0) = 0, (y) → ∞ for y → ∞. We have thefollowing fundamental convex inequalities:
M(x1) + M(x2) ≤ M(x1 + x2), (y1) + (y2) ≤ (y1 + y2). (4.3)
If the functions μ(ξ) and ω(η) are mutually inverse, that is, μ(ω(η)) = η, ω(μ(ξ)) =ξ , then the corresponding functions M(x) and (y) will be said to be dual in the senseof Young. In this case, we have the following Young inequality:
xy ≤ M(x) + (y), (4.4)
for x ≥ 0, y ≥ 0.
Definition 4.1 The space WM,α, α > 0, consists of all complex valued infinitelydifferentiable functions φ(x), (−∞ < x < ∞) which for any δ > 0 satisfy
∣∣∣φ(q)(x)
∣∣∣ ≤ Cqδe−M[(α−δ)x], q = 0, 1, 2, . . . . (4.5)
where positive constants Cqδ depend on function φ(x).
The fractional S-transform 257
Definition 4.2 The space W ,β, β > 0, consists of all entire analytic functions φ(z),(z = x + iy ∈ C) which for any ρ > 0 satisfy
∣∣∣zkφ(z)∣∣∣ ≤ Ckρe[(β+ρ)y], k = 0, 1, 2, . . . . (4.6)
where positive constants Ckρ depend on function φ(z).
Definition 4.3 The space W ,βM,α , α > 0, β > 0, consists of all entire analytic func-
tions φ(z), (z = x + iy ∈ C) which for any δ, ρ > 0 satisfy
|φ(z)| ≤ Cδρe−M[(α−δ)x]+[(β+ρ)y], (4.7)
where positive constants Cδρ depend on function φ(z).
Theorem 4.4 If the functions M(x) and (y) are mutually dual in the sense of Young,
then the Fourier operator F : WM,α → W , 1α ,F : W ,β → WM, 1
βis continuous
and WM,α = W , 1α , W ,β = WM, 1
β.
Theorem 4.5 Let 1(y) and M1(x) be the functions which are dual in the senseof Young to the functions M(x) and (y) respectively, then the Fourier operator
F : W ,βM,α → W
1,1α
M1,1β
is continuous and W ,βM,α = W
1,1α
M1,1β
.
In order to study the fractional S-transform on these spaces we need the followingsimilar test function spaces.
Definition 4.6 The space WM,α, α > 0 is defined to be the set of all complex valuedinfinitely differentiable functions φ f (τ ) = φ(τ, f ) ∈ C∞(R × (R \ {0})) which forany δ > 0 satisfy
∣∣∣∣ 1
(| f csc θ | + 1)t
(∂
∂τ
)t
φ(τ, f )
∣∣∣∣ ≤ Ctδθ e−M[(α−δ)τ ]; t = 0, 1, 2, . . . , (4.8)
where positive constants Ctδθ depend on function φ.
Definition 4.7 The space W ,β, β > 0, is defined to be the set of all functionsφ f (η) = φ(η, f ), (η, f ) ∈ C × (R\{0}), entire analytic with respect to η = τ + iμwhich for any ρ > 0 and 0 < p ≤ 1, satisfy
∣∣∣∣∣e−2π |μ f csc θ | ( f csc θ)2psηs(( f csc θ)2p + | f csc θ | + 1
)s φ(η, f )
∣∣∣∣∣≤ Csρθ e[(β+ρ)μ]; s = 0, 1, 2, . . . , (4.9)
where positive constants Csρθ depend on function φ.
258 S. K. Singh
Definition 4.8 The space W ,βM,α , α > 0, β > 0, is defined to be the set of all functions
φ f (η) = φ(η, f ), (η, f ) ∈ C × (R\{0}), entire analytic with respect to η = τ + iμwhich for any δ, ρ > 0 satisfy
∣∣∣e−2π |μ f csc θ | ηsφ(η, f )
∣∣∣≤Cδρθ e−M[(α−δ)τ ]+[(β+ρ)μ]; s =0, 1, 2, . . . , (4.10)
where positive constants Cδρθ depend on function φ.
5 The fractional S-transformation on spaces of type W
In this section we study the fractional S-transform on the spaces WM,α , W ,β andW ,β
M,α .
Theorem 5.1 For 0 < p ≤ 1, the fractional S-transform is a continuous linear mapof WM,α into WM,α where α ≥ 0.
Proof Let t ∈ N0 and I = (∂∂τ
)tF RST a
u (τ, f ). Then, after differentiation and inte-gration by parts, we obtain
I =(
∂
∂τ
)t
Aθ |Aθ |2eiπ f 2 cot θ∫
R
e−i2π( f − f1)τ csc θ
×e−2π2k2( f − f1)2(csc θ)2/( f csc θ)2p [eiπ t2 cot θu(t)]( f1 csc θ)d f1
= Aθ |Aθ |2eiπ f 2 cot θ∫
R
(−i2π( f − f1) csc θ)t e−i2π( f − f1)τ csc θ
×e−2π2k2( f − f1)2(csc θ)2/( f csc θ)2p [eiπ t2 cot θu(t)]( f1 csc θ)d f1
= Aθ |Aθ |2eiπ f 2 cot θ (−i2π)t∫
R
e−i2π( f − f1)τ csc θ (( f − f1) csc θ)t
×e−2π2k2( f − f1)2(csc θ)2/( f csc θ)2p [eiπ t2 cot θu(t)]( f1 csc θ)d f1
= Aθ |Aθ |2eiπ f 2 cot θ (−i2π)t∫
R
e−i2π( f −y sin θ)τ csc θ (( f − y sin θ) csc θ)t
×e−2π2k2( f −y sin θ)2(csc θ)2/( f csc θ)2p [eiπ t2 cot θu(t)](y) sin θdy
= Aθ |Aθ |2eiπ f 2 cot θ (−i2π)t sin θ
∫
R
e−i2π( f csc θ−y)τ ( f csc θ − y)t
×e−2π2k2( f csc θ−y)2/( f csc θ)2p [eiπ t2 cot θ u(t)](y)dy
= Aθ |Aθ |2eiπ f 2 cot θ (−i2π)t sin θ
∫
R
e−i2π( f csc θ−y)τt∑
r=0
(t
r
)( f csc θ)r (−y)t−r
×e−2π2k2( f csc θ−y)2/( f csc θ)2p [eiπ t2 cot θ u(t)](y)dy
The fractional S-transform 259
= Aθ |Aθ |2(−1)r (i2π)t sin θ eiπ f 2 cot θ e−i2π f csc θτt∑
r=0
(t
r
)( f csc θ)r
×∫
R
ei2πyτ yt−r e−2π2k2( f csc θ−y)2/( f csc θ)2p [eiπ t2 cot θ u(t)](y)dy
Using the technique of [2, p. 22], we can write
I = Aθ |Aθ |2(−1)r (i2π)t sin θ eiπ f 2 cot θ e−i2π f csc θτt∑
r=0
(t
r
)( f csc θ)r
×∫
R
ei2π(y+iλ)τ (y + iλ)t−r e−2π2k2( f csc θ−(y+iλ))2/( f csc θ)2p
× [eiπ t2 cot θ u(t)](y + iλ)dy
where λ = f csc θ/k, k > 0.
We use the inequality |z|q ≤ |z|q+2+|z|q1+y2 , z = y + iλ and condition u ∈ W , 1
α toobtain
|I | ≤ |Aθ |3(2π)tt∑
r=0
(t
r
)| f csc θ |r
×∫
R
e−2πτλ( |z|t−r+2+|z|t−r
1+y2
)| (eiπ t2 cot θu(t))(z)|dy
≤ |Aθ |3(2π)tt∑
r=0
(t
r
)| f csc θ |t−r
∫
R
e−2πτλ(C(t−r+2)ρ + C(t−r)ρ)
× e[( 1α+ρ)λ] 1
1+y2 dy
≤ Ctρθ e−2πτλ+[( 1α+ρ)λ](1 + | f csc θ |)t ,
where positive constant Ctρθ depends on function u.Let us now choose the sign of λ in such a manner that the equality τλ = |τ ||λ| be
satisfied. Thus we have
−2πτλ = −M[2πτ/( 1α
+ ρ)] − [( 1α
+ ρ)λ].
Therefore we obtain
|I | ≤ Ctρθ e−M[2πτ/( 1α+ρ)](1 + | f csc θ |)t .
260 S. K. Singh
Replacing 2π/( 1α
+ ρ) by α − δ where δ is arbitrary small together with ρ, we obtainthe estimate
∣∣∣∣ 1
(1 + | f csc θ |)t
(∂
∂τ
)t
F RST au (τ, f )
∣∣∣∣ ≤ Ctρθ e−M[(α−δ)τ ].
Hence the fractional S-transform is a continuous linear map of WM,α into WM,α . �Theorem 5.2 For 0 < p ≤ 1, the fractional S-transform is a continuous linear mapof W ,β into W ,β where β ≥ 0.
Proof Let s ∈ N0 and I = ηs F RST au (η, f ), η = τ + iμ. Then, after differentiation
and integration by parts, we obtain
I = ηs Aθ |Aθ |2eiπ f 2 cot θ∫
R
e−i2π( f − f1)η csc θ
×e−2π2k2( f − f1)2(csc θ)2/( f csc θ)2p [eiπ t2 cot θu(t)]( f1 csc θ)d f1
= Aθ |Aθ |2eiπ f 2 cot θ
(−i2π csc θ)s
∫
R
e−i2π( f − f1)η csc θ
×Dsf1
(e−2π2k2( f − f1)
2(csc θ)2/( f csc θ)2p [eiπ t2 cot θu(t)]( f1 csc θ)
)d f1
×s∑
m=0
(s
m
)Dm
f1
[e−2[πk csc θ( f − f1)/( f csc θ)p]2
]Ds−m
f1
[eiπ t2 cot θ u(t)]( f1 csc θ)d f1
= Aθ |Aθ |2eiπ f 2 cot θ
(−i2π csc θ)s
∫
R
e−i2π( f − f1)η csc θ
×s∑
m=0
(s
m
) (− 2πk csc θ
( f csc θ)p
)m
e−2[πk csc θ( f − f1)/( f csc θ)p]2
×[m/2]∑n=0
(−1)nm!(4πk csc θ)m−2n
n!(m − 2n)!(
f − f1
( f csc θ)p
)(m−2n)
×Ds−mf1
[eiπ t2 cot θ u(t)]( f1 csc θ)d f1
= Aθ |Aθ |2eiπ f 2 cot θ
(−i2π csc θ)s
s∑m=0
(s
m
) (− 2πk csc θ
( f csc θ)p
)m [m/2]∑n=0
(−1)nm!(4πk csc θ)m−2n
n!(m − 2n)!
× 1
( f csc θ)p(m−2n)
∫
R
e−i2π( f − f1)η csc θ e−2[πk csc θ( f − f1)/( f csc θ)p]2
×( f − f1)m−2n Ds−m
f1
[eiπ t2 cot θ u(t)]( f1 csc θ)d f1
The fractional S-transform 261
= Aθ |Aθ |2eiπ f 2 cot θ
(−i2π csc θ)s
s∑m=0
(s
m
) (− 2πk csc θ
( f csc θ)p
)m [m/2]∑n=0
(−1)nm!(4πk csc θ)m−2n
n!(m − 2n)!
× 1
( f csc θ)p(m−2n)
∫
R
e−i2π( f −y sin θ)η csc θ e−2[πk csc θ( f −y sin θ)/( f csc θ)p]2
×( f − y sin θ)m−2n(sin θ)−s+m+1 Ds−my
[eiπ t2 cot θ u(t)](y)dy
= Aθ |Aθ |2eiπ f 2 cot θ
(−i2π csc θ)s
s∑m=0
(s
m
) (− 2πk csc θ
( f csc θ)p
)m [m/2]∑n=0
(−1)nm!(4πk csc θ)m−2n
n!(m − 2n)!
× (sin θ)−s+2m−2n+1
( f csc θ)p(m−2n)
∫
R
e−i2π( f csc θ−y)ηe−2[πk( f csc θ−y)/( f csc θ)p]2
×( f csc θ − y)m−2n Ds−my
[eiπ t2 cot θu(t)](y)dy
= Aθ |Aθ |2eiπ f 2 cot θ sin θ
(−i2π)s
s∑m=0
(s
m
) (− 2πk
( f csc θ)p
)m [m/2]∑n=0
(−1)nm!(4πk)m−2n
n!(m − 2n)!
× 1
( f csc θ)p(m−2n)
∫
R
e−i2π( f csc θ−y)ηe−2[πk( f csc θ−y)/( f csc θ)p]2
×( f csc θ − y)m−2n Ds−my
[eiπ t2 cot θu(t)](y)dy
= Aθ |Aθ |2eiπ f 2 cot θ sin θ
(−i2π)s
s∑m=0
(s
m
) (− 2πk
( f csc θ)p
)m [m/2]∑n=0
(−1)nm!(4πk)m−2n
n!(m − 2n)!
× 1
( f csc θ)p(m−2n)
∫
R
e−i2π( f csc θ−y)ηe−2[πk( f csc θ−y)/( f csc θ)p]2
×m−2n∑r=0
(m − 2n
r
)( f csc θ)r (−y)m−2n−r Ds−m
y[eiπ t2 cot θ u(t)](y)dy
Use the property (4.5) and the following inequality [2, p. 13]
|x |e−M(ax) ≤ Cδ e−M[(a−δ)x], for any positive δ,
we obtain
|I | ≤ |Aθ |3(2π)s
s∑m=0
(s
m
)2πk
| f csc θ |pm
[m/2]∑n=0
m!(4πk)m−2n
n!(m − 2n)!1
| f csc θ |p(m−2n)
×∫
R
e2π( f csc θ−y)μm−2n∑r=0
(m − 2n
r
)| f csc θ |r
×|ym−2n−r Ds−my
[eiπ t2 cot θu(t)](y)|dy
262 S. K. Singh
≤ |Aθ |3(2π)s
s∑m=0
(s
m
)2πk
| f csc θ |pm
[m/2]∑n=0
m!(4πk)m−2n
n!(m − 2n)!1
| f csc θ |p(m−2n)
×e2πμ f csc θm−2n∑r=0
(m − 2n
r
)| f csc θ |r
×∫
R
e−2πyμ|y|m−2n−r C(s−m)δe−M[((1/β)−δ)y]dy
≤ C′sδθ e2πμ f csc θ
s∑m=0
(s
m
)1
| f csc θ |pm
[m/2]∑n=0
1
| f csc θ |p(m−2n)
×m−2n∑r=0
(m − 2n
r
)| f csc θ |r
∫
R
e2π |y||μ| e−M[((1/β)−δ1)y]dy.
By applying (4.3) and Young’s inequality (4.4), we obtain
|2πμy| − M[( 1β
− δ1)y] ≤ [2πμ/( 1β
− 2δ1)] + M[( 1β
− 2δ1)y] − M[( 1β
− δ1)y]≤ [2πμ/( 1
β− 2δ1)] − M [δ1 y] . (5.1)
Thus we get
|I | ≤ C′sδθ e2πμ f csc θ
s∑m=0
(s
m
)1
| f csc θ |pm
[m/2]∑n=0
1
| f csc θ |p(m−2n)
×m−2n∑r=0
(m − 2n
r
)| f csc θ |r
∫
R
e−M[δ1 y]e[2πμ/((1/β)−2δ1)]dy.
Since the integral∫R
e−M[δ1 y]dy has a finite value [2, p. 21], hence we get theestimate
|I | ≤ C′′sδθ e2πμ f csc θ
s∑m=0
(s
m
)1
| f csc θ |pm
[m/2]∑n=0
1
| f csc θ |p(m−2n)
×m−2n∑r=0
(m − 2n
r
)| f csc θ |r e[2πβμ/(1−2βδ1)]
= C′′sδθ e2πμ f csc θ e[2πβμ/(1−2βδ1)]
s∑m=0
(s
m
)1
| f csc θ |pm
The fractional S-transform 263
×[m/2]∑n=0
(1 + | f csc θ |)m−2n
| f csc θ |p(m−2n)
= C′′sδθ e2πμ f csc θ e[2πβμ/(1−2βδ1)]
s∑m=0
(s
m
)1
| f csc θ |pm
× (1 + | f csc θ |)m
| f csc θ |pm
[m/2]∑n=0
| f csc θ |2pn
(1 + | f csc θ |)2n
≤ Csδθ e2π |μ f csc θ |e[2πβμ/(1−2βδ1)]s∑
m=0
(s
m
)(1 + | f csc θ |)m
( f csc θ)2pm
= Csδθ e2π |μ f csc θ |e[2πβμ/(1−2βδ1)](
1 + 1 + | f csc θ |( f csc θ)2p
)s
.
Replacing 2πβμ1−2βδ1
by β + ρ where ρ is arbitrary small together with δ1, we obtain theestimate
∣∣∣∣e−2π |μ f csc θ | ( f csc θ)2ps(τ + iμ)s
(( f csc θ)2p + | f csc θ | + 1)sF RST a
u (η, f )(τ + iμ, f )
∣∣∣∣ ≤ Csδθ e[(β+ρ)μ],
where positive constant Csδθ depending on function u. �
Theorem 5.3 For 0 < p ≤ 1, the fractional S-transform is a continuous linear map
of W ,βM,α into W 1,β
M1,αwhere α, β ≥ 0.
Proof For η = τ + iμ, we can write by the definition of the fractional S-transform:
F RST au (η, f ) = Aθ |Aθ |2eiπ f 2 cot θ
∫
R
e−i2π( f − f1)η csc θ
×e−2π2k2( f − f1)2(csc θ)2/( f csc θ)2p [eiπ t2 cot θu(t)]( f1 csc θ)d f1
= Aθ |Aθ |2eiπ f 2 cot θ∫
R
e−i2π( f −y sin θ)η csc θ
×e−2π2k2( f −y sin θ)2(csc θ)2/( f csc θ)2p [eiπ t2 cot θu(t)](y) sin θdy
= Aθ |Aθ |2eiπ f 2 cot θ∫
R
e−i2π( f csc θ−y)η
×e−2π2k2( f csc θ−y)2/( f csc θ)2p [eiπ t2 cot θ u(t)](y) sin θdy.
264 S. K. Singh
Following the technique of [2, p. 22], we have
F RST au (η, f ) = Aθ |Aθ |2eiπ f 2 cot θ
∫
R
e−i2π( f csc θ−y−iλ)(τ+iμ)
×e−2π2k2( f csc θ−y−iλ)2/( f csc θ)2p [eiπ t2 cot θ u(t)](y + iλ) sin θdy,
where λ = f csc θ/k, k > 0.We use the property (4.7) and (5.1) to obtain
|F RST au (η, f )| ≤ |Aθ |3
∫
R
e−2πτλ−2πμ(y− f csc θ)| [eiπ t2 cot θ u(t)](y + iλ)|dy
≤ |Aθ |3∫
R
e−2πτλ−2πμ(y− f csc θ)Cδρe−M1[( 1β−δ)y]+1[( 1
α+ρ)λ]dy
= Cδρ |Aθ |3e−2πτλ+2πμ f csc θ e1[( 1α+ρ)λ]
∫
R
e−2πμye−M1[( 1β−δ)y]dy
≤ Cδρ |Aθ |3e−2πτλ+2πμ f csc θ e1
[( 1
α+ρ)λ
]e1[2πμ/( 1
β−2δ)]
∫
R
e−M1[δy]dy
≤ C ′δρ |Aθ |3e−2πτλ+2πμ f csc θ e
1
[( 1
α+ρ)λ
]e1
[2πμ/( 1
β−2δ)
].
Using the technique [2, p. 22], let us now choose the sign of λ in such a manner thatthe equality τλ = |τ ||λ| be satisfied, thus we have
−2πτλ = −M1 (2πτ/ ((1/α) + ρ)) − 1(( 1α
+ ρ)λ).
Therefore we get
|F RST au (η, f )| ≤ Cδρθ e2πμ f csc θ e
−M1
[2πτ/( 1
α+ρ)
]+1
[2πμ/( 1
β−2δ)
].
In the above, we set 2πτ
( 1α+ρ)
= α − ρ′ and 2πμ
( 1β−2δ)
= β − δ′ where the quantities ρ′, δ′
are arbitrary small together with ρ, δ. Hence, we obtain
∣∣∣e−2π |μ f csc θ |F RST au (η, f )
∣∣∣ ≤ Cδρθ e−M1[(α−ρ′)τ]+1
[(β−δ′)μ
].
Hence the fractional S-transform is a continuous linear map of W ,βM,α into W 1,β
M1,α. �
The fractional S-transform 265
6 Future directions
The results of this paper will be useful in the investigation of time-frequency behaviorof test functions and distributions. We may use these results to obtain new results inmodulation spaces, which may be defined by means of the fractional S-transform, byusing the techniques developed in [3].
Acknowledgments The author expresses his sincere thanks to Prof. R.S. Pathak for his help and encour-agement.
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