The fractional S-transform on spaces of type $W$

$: The fractional S-transform on spaces of type $W$$
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: [email protected]

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)


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]


d f1

= Aθ |Aθ |2eiπ f 2 cot θ∫

R

[eiπ t2 cot θ u(t)]( f1 csc θ)


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).


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


R

(−i2π( f − f1) csc θ)t e−i2π( f − f1)τ csc θ


= Aθ |Aθ |2eiπ f 2 cot θ (−i2π)t∫

R

e−i2π( f − f1)τ csc θ (( f − f1) csc θ)t


= 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


= 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


= Aθ |Aθ |2eiπ f 2 cot θ

(−i2π csc θ)s

∫

R


×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


(−i2π csc θ)s

∫

R


×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



(−i2π csc θ)s

s∑m=0

(s

m

) (− 2πk csc θ

( f csc θ)p

)m [m/2]∑n=0


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




(−i2π csc θ)s

s∑m=0

(s

m

) (− 2πk csc θ

( f csc θ)p

)m [m/2]∑n=0


n!(m − 2n)!

× 1


∫

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


(−i2π csc θ)s

s∑m=0

(s

m

) (− 2πk csc θ

( f csc θ)p

)m [m/2]∑n=0


n!(m − 2n)!

× (sin θ)−s+2m−2n+1


∫

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


∫

R


×( 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


∫

R


×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


×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


×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


×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


×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

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. �


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.

References

1. Almeida, L.B.: The fractional Fourier transform and time-frequency representations. IEEE Trans. SignalProcess. 42(11), 3084–3091 (1994)

2. Gel’fand, I.M., Shilov, G.E.: Generalized Functions, vol. 3. Academic Press, New York (1967)3. Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)4. Pathak, R.S., Pandey, G.: Wavelet Transform on spaces of type W. Rocky Mt. J. Math. 39(2), 619–631

(2009)5. Singh, S.K.: The S-Transform on spaces of type S. Integr. Transforms Special Funct. 23(7), 481–494

(2012)6. Singh, S.K.: The S-Transform on spaces of type W . Integr. Transforms Special Funct. 23(12), 891–899

(2012)7. Stockwell, R.G., Mansinha, L., Lowe, R.P.: Localization of the complex spectrum: The S transform.

IEEE Trans. Signal Process. 44(4), 998–1001 (1996)8. Ventosa, S., Simon, C., Schimmel, M., Dañobeitia, J., Mànuel, A.: The S-transform from a wavelet

point of view. IEEE Trans. Signal Process. 56(07), 2771–2780 (2008)9. Xu, D.P., Guo, K.: Fractional S transform—Part 1: Theory. Appl. Geophys. 9(1), 73–79 (2012)

The fractional S-transform on spaces of type $W$

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