The Fractional S-Transform on Spaces of Type 𝑆

Abstract

Certain properties of the S-transform are given and composition of S-transform is defined. We study the fractional S-transform on spaces of type 𝑆. It is shown that the fractional S-transform is a continuous linear map of the space of type 𝑆 defined on ℝ into the space of type 𝑆^∼ defined on ℝ×(ℝ⧵{0}).

1. Introduction

In this section we give the definition and some basic properties of the S-transform. The S-transform was first introduced by Stockwell et al. [1] in 1996 as invertible time-frequency spectral localization technique. The S-transform is an extension to the ideas of continuous wavelet transform and is based on a moving and scalable localizing Gaussian window and has characteristics superior to both of the Fourier transform and the wavelet transform [2–5].

The one-dimensional continuous S-transform of 𝑢(𝑡) is defined as [4] ()(𝑢(𝑡=𝑆𝑢𝜏,𝑓=𝑆))𝜏,𝑓_ℝ𝑢(𝑡)𝜔𝜏−𝑡,𝑓𝑒^{−𝑖2𝜋𝑓𝑡}𝑑𝑡, (1) where the window 𝜔 is assumed to satisfy the following: _ℝ𝜔𝑡,𝑓𝑑𝑡=1∀𝑓∈ℝ⧵{0}. (2) The most usual window 𝜔 is the Gaussian one 𝜔=𝑓𝑡,𝑓𝑘√𝑒2𝜋^{−(𝑓2𝑡2/2𝑘2)},𝑘>0, (3) where 𝑓 is the frequency, 𝑡 is the time variable, and 𝑘 is a scaling factor that controls the number of oscillations in the window.

Then, (1) can be rewritten as a convolution (𝑆𝑢)𝜏,𝑓=𝑢(⋅)𝑒^{−𝑖2𝜋𝑓⋅}∗𝜔⋅,𝑓(𝜏). (4) Applying the convolution property for the Fourier transform, we obtain (𝑆𝑢)𝜏,𝑓=ℱ⁻¹𝑢⋅+𝑓𝜔⋅,𝑓(𝜏), (5) where ℱ⁻¹ is the inverse Fourier transform. For the Gaussian window case (3), 𝜔𝛼,𝑓=𝑒^{−2(𝜋𝑘𝛼/𝑓)2}. (6) Thus we can write the S-transform in the following form: (𝑆𝑢)𝜏,𝑓=_ℝ𝑢𝛼+𝑓𝑒^{−2(𝜋𝑘𝛼/𝑓)2}𝑒^{𝑖2𝜋𝛼𝜏}𝑑𝛼. (7) Also, if 𝑢(𝑓) and (𝑆𝑢)(𝜏,𝑓) are the Fourier transform and S-transform of 𝑢, respectively, then 𝑢𝑓=_ℝ(𝑆𝑢)𝜏,𝑓𝑑𝜏, (8) so that 𝑢(𝑡)=ℱ⁻¹_ℝ(𝑆𝑢)(𝜏,⋅)𝑑𝜏(𝑡). (9)

2. Some Important Properties of S-Transform

Some basic properties of S-transform can be found in [6, 7] and certain properties of S-transform are obtained in this section.

Theorem 1. If 𝑢(𝑡)∈𝐿¹(ℝ), then lim_|𝜏|→∞|(𝑆𝑢)(𝜏,𝑓)|=0.

Proof. We have by definition (𝑆𝑢)𝜏,𝑓=_ℝ𝑢𝛼+𝑓𝑒^{−2(𝜋𝑘𝛼/𝑓)2}𝑒^{𝑖2𝜋𝛼𝜏}𝑑𝛼=−_ℝ𝑢𝛼+𝑓𝑒^{−2(𝜋𝑘𝛼/𝑓)2}𝑒^{𝑖2𝜋(𝛼+1/2𝜏)𝜏}𝑑𝛼=−_ℝ𝑢1𝑥+𝑓−𝑒2𝜏^{−2[𝜋𝑘(𝑥−1/2𝜏)/𝑓]2}𝑒^{𝑖2𝜋𝜏𝑥}𝑑𝑥. (10) Thus ()=1𝑆𝑢𝜏,𝑓2_ℝ𝑢𝛼+𝑓𝑒^{−2(𝜋𝑘𝛼/𝑓)2}𝑒^{𝑖2𝜋𝛼𝜏}−𝑑𝛼_ℝ𝑢1𝛼+𝑓−𝑒2𝜏^{−2[𝜋𝑘(𝛼−1/2𝜏)/𝑓]2}𝑒^{𝑖2𝜋𝜏𝛼}𝑑𝛼. (11)
Clearly, lim_|𝜏|→∞)≤1(𝑆𝑢𝜏,𝑓2lim_{|𝜏|→∞ℝ}𝑢−𝑢1𝛼+𝑓𝛼+𝑓−2𝜏𝑑𝛼=0. (12)

Theorem 2. If 𝑢(𝑡)∈𝐿¹(ℝ), then lim_|𝑓|→0|(𝑆𝑢)(𝜏,𝑓)|=0.

Theorem 3. If 𝑒^{−2(𝜋𝑘𝛼/𝑓)2}𝑢(𝛼+𝑓)∈𝐿²(ℝ), then ‖(𝑆𝑢)(𝜏,𝑓)‖_𝐿2≤‖𝑢‖_𝐿².

Theorem 4. If 𝑢(𝑡)∈𝐿¹(ℝ), then (𝑆𝑢)(𝜏,𝑓) is continuous with respect to the variables 𝜏 and 𝑓.

Proof. By the definition )−()=(𝑆𝑢𝜏+ℎ,𝑓𝑆𝑢𝜏,𝑓_ℝ𝑢𝛼+𝑓𝜔𝛼,𝑓𝑒^{𝑖2𝜋𝛼ℎ}−1𝑒^{𝑖2𝜋𝛼𝜏}≤𝑑𝛼_ℝ𝑢𝛼+𝑓𝑒^{𝑖2𝜋𝛼ℎ}−1𝑑𝛼,⟶0asℎ⟶0. (13) Hence the S-transform is continuous with respect to the first variables 𝜏. Similarly we can show that it is also continuous with respect to the second variables 𝑓.

2.1. Composition of S-Transforms

Let 𝑆₁ and 𝑆₂ be two S-transforms defined as 𝑆₁𝑢𝜏₁,𝑓₁=_ℝ𝑢𝛼+𝑓₁𝜔₁𝛼,𝑓₁𝑒^{𝑖2𝜋𝛼𝜏1}𝑆𝑑𝛼,₂𝑢𝜏₂,𝑓₂=_ℝ𝑢𝛼+𝑓₂𝜔₂𝛼,𝑓₂𝑒^{𝑖2𝜋𝛼𝜏2}𝑑𝛼, (14) provided the integrals exist. Then their composition 𝑆₁∘𝑆₂ is defined by 𝑆^∗𝑢𝜏₁,𝑓₁,𝑓₂=𝑆₁∘𝑆₂𝑢𝜏₁,𝑓₁,𝑓₂=_ℝℱ𝑆₂𝑢𝜏₂,𝑓₂𝛼+𝑓₁𝜔₁𝛼,𝑓₁𝑒^{𝑖2𝜋𝛼𝜏1}=𝑑𝛼_ℝ𝑢𝛼+𝑓₁+𝑓₂𝜔₂𝛼+𝑓₁,𝑓₂𝜔₁𝛼,𝑓₁𝑒^{𝑖2𝜋𝛼𝜏1}=𝑑𝛼_ℝ𝑢𝛼+𝑓₁+𝑓₂𝜔𝛼,𝑓₁,𝑓₂𝑒^{𝑖2𝜋𝛼𝜏1}𝑑𝛼, (15) where 𝜔(𝛼,𝑓₁,𝑓₂)=𝜔₂(𝛼+𝑓₁,𝑓₂)𝜔₁(𝛼,𝑓₁), provided the integral exists. The composition 𝑆₁∘𝑆₂ is an S-transform of three variables mapping from ℝ into ℝ×(ℝ⧵{0})×(ℝ⧵{0}) and analogous to the composition of the wavelet transform [8, pages 52-53].

Theorem 5. If 𝑢(𝑡)∈𝐿¹(ℝ), then (𝑆^∗𝑢)(𝜏,𝑓₁,𝑓₂) is continuous with respect to the variables 𝜏, 𝑓₁ and 𝑓₂.

3. Fractional Fourier Transform

The fractional Fourier transform (FRFT) has played an important role in signal processing. The 𝑎th order FRFT of a signal 𝑢(𝑡) is defined as [9] 𝐹^𝑎_𝑢𝑓=_ℝ𝑢(𝑡)𝐾_𝑎𝑡,𝑓𝑑𝑡, (16) where the transform kernel 𝐾_𝑎(𝑡,𝑓) is defined as 𝐾_𝑎=𝐴𝑡,𝑓_𝜃𝑒^{𝑖𝜋(𝑓2cot𝜃−2𝑓𝑡csc𝜃+𝑡2cot𝜃)},if𝛿,𝜃≠𝑛𝜋𝑡−𝑓if𝛿,𝜃=2𝑛𝜋𝑡+𝑓if𝜃+𝜋=2𝑛𝜋, (17) where 𝐴_𝜃=√1−𝑖cot𝜃,  𝜃=𝑎𝜋/2,  𝑎∈[0,4), 𝑖 is the complex unit, 𝑛 is an integer, and 𝑓 is the fractional Fourier frequency (FRFfr). The inverse FRFT of (16) is 𝑢(𝑡)=_ℝ𝐹^𝑎_𝑢𝑓𝐾_𝑎𝑡,𝑓𝑑𝑓. (18) We can write (16) as 𝐹^𝑎_𝑢𝑓=_ℝ𝑢(𝑡)𝐴_𝜃𝑒^{𝑖𝜋(𝑓2cot𝜃−2𝑓𝑡csc𝜃+𝑡2cot𝜃)}𝑑𝑡=𝐴_𝜃𝑒^{𝑖𝜋𝑓2cot𝜃}_ℝ𝑢(𝑡)𝑒^{−𝑖2𝜋𝑓𝑡csc𝜃}𝑒^{𝑖𝜋𝑡2cot𝜃}𝑑𝑡=𝐴_𝜃𝑒^{𝑖𝜋𝑓2cot𝜃}𝑒^{𝑖𝜋𝑡2cot𝜃}𝑢(𝑡)𝑓csc𝜃. (19) Replacing 𝑢(𝑡) by 𝑒^{−𝑖𝜋𝑡2cot𝜃}𝜙(𝑡), we have 𝐹^𝑎_𝑢𝑓=𝐴_𝜃𝑒^{𝑖𝜋𝑓2cot𝜃}[𝜙(𝑡)]𝑓csc𝜃. (20) Putting 𝑓=𝜉sin𝜃, we have 𝐹^𝑎_𝑢(𝜉sin𝜃)=𝐴_𝜃𝑒^{𝑖𝜋𝜉2sin2𝜃/2}𝜙(𝜉),𝜙(𝜉)=1𝐴_𝜃𝑒^{−𝑖𝜋𝜉2sin2𝜃/2}𝐹^𝑎_𝑢(𝜉sin𝜃), (21) where 𝑢(𝑡)=𝑒^{−𝑖𝜋𝑡2cot𝜃}𝜙(𝑡).

4. The Fractional S-Transform

The fractional S-transform (FRST) is a generalization of the S-transform. The 𝑎th order continuous fractional S-transform of 𝑢(𝑡) is defined as [10] FRST^𝑎_𝑢𝜏,𝑓=_ℝ𝑢(𝑡)𝑔𝜏−𝑡,𝑓𝐾_𝑎𝑡,𝑓𝑑𝑡, (22) where the window 𝑔 is 𝑔=𝑓𝑡,𝑓csc𝜃^𝑝𝑘√𝑒2𝜋^{−𝑡2(𝑓csc𝜃)2𝑝/2𝑘2},𝑘,𝑝>0 (23) and satisfies the condition _ℝ𝑔𝑡,𝑓𝑑𝑡=1∀𝑓∈ℝ⧵{0}. (24) Inverse fractional S-transform is defined by 𝑢(𝑡)=_ℝℝFRST^𝑎_𝑢𝜏,𝑓𝑑𝜏𝐾_𝑎𝑡,𝑓𝑑𝑓. (25) Note that the fractional S-transform depends on a parameter 𝜃 and can be interpreted as a rotation by an angle 𝜃 in the time-frequency plane. An FRST with 𝜃=𝜋/2 corresponds to the S-transform, and an FRST with 𝜃=0 corresponds to the zero operator. The parameters 𝑝 and 𝑘 can be used to adjust the window function space.

Let ℎ𝑡,𝜏,𝑓=𝑔𝜏−𝑡,𝑓𝐾_𝑎𝑡,𝑓,𝐻_𝑎𝜏,𝑓,𝑓₁=_ℝℎ𝑡,𝜏,𝑓𝐾_𝑎𝑡,𝑓₁=_ℝ𝑔𝜏−𝑡,𝑓𝐾_𝑎𝑡,𝑓𝐾_𝑎𝑡,𝑓₁𝑑𝑡. (26)(27) Since 𝐾_𝑎𝑡,𝑓𝐾_𝑎𝑡,𝑓₁=𝐴_𝜃𝐴_𝜃𝑒^{𝑖𝜋[(𝑓2−𝑓21)cot𝜃−2(𝑓−𝑓1)𝑡csc𝜃]}. (28) By using (23) and (28) in (27) we obtain 𝐻_𝑎𝜏,𝑓,𝑓₁=_ℝ𝑓csc𝜃^𝑝𝑘√𝑒2𝜋^{−(𝜏−𝑡)2(𝑓csc𝜃)2𝑝/2𝑘2}×𝐴_𝜃𝐴_𝜃𝑒^{𝑖𝜋[(𝑓2−𝑓21)cot𝜃−2(𝑓−𝑓1)𝑡csc𝜃]}=𝑓𝑑𝑡csc𝜃^𝑝𝑘√𝐴2𝜋^𝜃𝐴_𝜃𝑒^{𝑖𝜋[(𝑓2−𝑓21)cot𝜃]}×_ℝ𝑒^{−(𝜏−𝑡)2(𝑓csc𝜃)2𝑝/2𝑘2}𝑒^{−𝑖2𝜋(𝑓−𝑓1)𝑡csc𝜃}𝑑𝑡. (29) By using the technique of [10], we obtain 𝐻_𝑎𝜏,𝑓,𝑓₁=𝐴_𝜃𝐴_𝜃𝑒^{𝑖𝜋[(𝑓2−𝑓21)cot𝜃]}𝑒^{−𝑖2𝜋(𝑓−𝑓1)𝜏csc𝜃}×𝑒^{−2𝜋2𝑘2(𝑓−𝑓1)2(csc𝜃)2/(𝑓csc𝜃)2𝑝}, (30) and by using (28) we can write 𝐻_𝑎𝜏,𝑓,𝑓₁=𝑒^{−[2𝜋2𝑘2(𝑓−𝑓1)2(csc𝜃)2/(𝑓csc𝜃)2𝑝]}𝐾_𝑎𝑡,𝑓𝐾_𝑎𝑡,𝑓₁. (31) Also, the FRST can be also defined as operations on the fractional Fourier domain: FRST^𝑎_𝑢𝜏,𝑓=_ℝℝ𝐹^𝑎_𝑢𝑓𝐾_𝑎𝐾𝑡,𝑓𝑑𝑓×𝑔𝜏−𝑡,𝑓_𝑎=𝑡,𝑓𝑑𝑡_ℝ𝐹^𝑎_𝑢𝑓₁𝐻_𝑎𝜏,𝑓,𝑓₁𝑑𝑓₁. (32) By using (19) and (30) we can write (32) as FRST^𝑎_𝑢=𝜏,𝑓_ℝ𝐴_𝜃𝑒^{𝑖𝜋𝑓21cot𝜃}𝑒^{𝑖𝜋𝑡2cot𝜃}𝑢(𝑡×𝑓₁)csc𝜃𝐴_𝜃𝐴_𝜃𝑒^{𝑖𝜋[(𝑓2−𝑓21)cot𝜃]}×𝑒^{−𝑖2𝜋(𝑓−𝑓1)𝜏csc𝜃}×𝑒^{−2𝜋2𝑘2(𝑓−𝑓1)2(csc𝜃)2/(𝑓csc𝜃)2𝑝}𝑑𝑓₁=𝐴_𝜃𝐴_𝜃²_ℝ𝑒^{𝑖𝜋𝑓21cot𝜃}𝑒^{𝑖𝜋𝑡2cot𝜃}𝑢(𝑡×𝑓₁)csc𝜃𝑒^{𝑖𝜋[(𝑓2−𝑓21)cot𝜃]}×𝑒^{−𝑖2𝜋(𝑓−𝑓1)𝜏csc𝜃}×𝑒^{−2𝜋2𝑘2(𝑓−𝑓1)2(csc𝜃)2/(𝑓csc𝜃)2𝑝}𝑑𝑓₁=𝐴_𝜃𝐴_𝜃²𝑒^{𝑖𝜋𝑓2cot𝜃}×_ℝ𝑒^{𝑖𝜋𝑡2cot𝜃}𝑢(𝑡)𝑓₁csc𝜃×𝑒^{−𝑖2𝜋(𝑓−𝑓1)𝜏csc𝜃}×𝑒^{−2𝜋2𝑘2(𝑓−𝑓1)2(csc𝜃)2/(𝑓csc𝜃)2𝑝}𝑑𝑓₁. (33)

5. The Fractional S-Transform on Spaces of Type 𝑆

The spaces of type 𝑆 play an important role in the theory of linear partial differential equations as intermediate spaces between the spaces of the 𝐶^∞ and the analytic functions. The Fourier transform has been studied on the spaces of type 𝑆 by Friedman [11] and Gel’fand and Shilov [12]. Pathak and Singh [13] studied the wavelet transform on the spaces of type 𝑆. Also, Singh [6] studied the S-transform on the spaces of type 𝑆. In this section we study the fractional S-transform (22) on these spaces.

Let us recall the definitions of these spaces.

Definition 6. The space 𝑆_𝛼(𝛼≥0) consists of all infinitely differentiable functions 𝜙(𝑥)  (−∞<𝑥<∞), satisfying the inequalities 𝛾_𝑘,𝑞𝜙∶=sup_𝑥∈ℝ𝑥^𝑘𝜙^(𝑞)(𝑥)≤𝐶_𝑞𝐴^𝑘𝑘^𝑘𝛼𝑘,𝑞=0,1,2,…, (34) where the constants 𝐴 and 𝐶_𝑞 depend on the function 𝜙. For 𝑘=0, the expression 𝑘^𝑘𝛼 is considered to be equal to 1.

Definition 7. The space 𝑆^𝛽(𝛽≥0) consists of all infinitely differentiable functions 𝜙(𝑥)  (−∞<𝑥<∞), satisfying the inequalities 𝛾_𝑘,𝑞𝜙∶=sup_𝑥∈ℝ𝑥^𝑘𝜙^(𝑞)(𝑥)≤𝐶_𝑘𝐵^𝑞𝑞^𝑞𝛽𝑘,𝑞=0,1,2,…, (35) where the constants 𝐵 and 𝐶_𝑘 depend on the function 𝜙.

Definition 8. The space 𝑆^𝛽_𝛼(𝛼≥0,𝛽≥0) consists of all infinitely differentiable functions 𝜙(𝑥)  (−∞<𝑥<∞), satisfying the inequalities 𝛾_𝑘,𝑞𝜙∶=sup_𝑥∈ℝ𝑥^𝑘𝜙^(𝑞)(𝑥)≤𝐶𝐴^𝑘𝐵^𝑞𝑘^𝑘𝛼𝑞^𝑞𝛽𝑘,𝑞=0,1,2,…, (36) where the constants 𝐴,𝐵, and 𝐶 depend on the function 𝜙.

The spaces of type 𝑆 are closely interrelated by means of the Fourier transformation; namely, the formulas 𝑆_𝛼=𝑆^𝛼,𝑆^𝛽=𝑆_𝛽,𝑆^𝛽_𝛼=𝑆^𝛼_𝛽 (37) hold.

We will make use of the following inequalities in our investigation: 𝑞!!𝑞𝑘𝑞−𝑘=𝑘!≤𝑘!^𝑞_𝑘=0𝑞𝑘=𝑘!2^𝑞,𝑚+𝑞^{(𝑚+𝑞)𝛽}≤𝑚^𝑚𝛽𝑞^𝑞𝛽𝑒^𝑚𝛽𝑒^𝑞𝛽11,page265. (38)(39) In what follows we will also need the following similar test function space defined on ℝ×(ℝ⧵{0}), the space of type 𝑆^∼.

Definition 9. The space 𝑆^∼𝛽_𝛼(ℝ×(ℝ⧵{0})),  𝛼,𝛽≥0, is defined to be the space of all functions 𝜙∈𝐶^∞(ℝ×(ℝ⧵{0})) such that for all 𝑠,𝑡∈ℕ₀, 𝑓csc𝜃^2𝑝𝑠𝜏^𝑠1+𝑓csc𝜃^𝑡2𝑓csc𝜃^2𝑝+𝑓csc𝜃+1^𝑠𝜕𝜕𝜏^𝑡𝜙𝜏,𝑓≤𝐶_{𝜃,𝑠,𝑡}𝐴^𝑠𝐵^𝑡𝑠^𝑠𝛼𝑡^𝑡𝛽, (40) where the constants 𝐴,𝐵, and 𝐶_{𝜃,𝑠,𝑡} depend on 𝜙.

We show that the fractional S-transform is a continuous linear map of the space of type 𝑆 into the space of type 𝑆^∼ defined mentioned previously.

Theorem 10. For 0<𝑝≤1, the fractional S-transform is a continuous linear map of 𝑆_𝛼(ℝ) into 𝑆^{∼∼𝛽∼}_𝛼(ℝ×(ℝ⧵{0})) where 𝛼^∼=(𝛼+3) and 𝛽^∼=0.

Proof. Let 𝑠,𝑡∈ℕ and 𝐼=𝜏^𝑠(𝜕/𝜕𝜏)^𝑡FRST^𝑎_𝑢(𝜏,𝑓). Then, after differentiation and integration by parts, we obtain 𝐼=𝜏^𝑠𝜕𝜕𝜏^𝑡𝐴_𝜃𝐴_𝜃²𝑒^{𝑖𝜋𝑓2cot𝜃}×_ℝ𝑒^{−𝑖2𝜋(𝑓−𝑓1)𝜏csc𝜃}×𝑒^{−2𝜋2𝑘2(𝑓−𝑓1)2(csc𝜃)2/(𝑓csc𝜃)2𝑝}×𝑒^{𝑖𝜋𝑡2cot𝜃}𝑢(𝑡)𝑓₁csc𝜃𝑑𝑓₁=𝜏^𝑠𝐴_𝜃𝐴_𝜃²𝑒^{𝑖𝜋𝑓2cot𝜃}×_ℝ−𝑖2𝜋𝑓−𝑓₁csc𝜃^𝑡𝑒^{−𝑖2𝜋(𝑓−𝑓1)𝜏csc𝜃}×𝑒^{−2𝜋2𝑘2(𝑓−𝑓1)2(csc𝜃)2/(𝑓csc𝜃)2𝑝}×𝑒^{𝑖𝜋𝑡2cot𝜃}𝑢(𝑡)𝑓₁csc𝜃𝑑𝑓₁=𝐴_𝜃𝐴_𝜃²𝑒^{𝑖𝜋𝑓2cot𝜃}(−𝑖2𝜋csc𝜃)^𝑡×_ℝ1(−𝑖2𝜋csc𝜃)^𝑠𝑒^{−𝑖2𝜋(𝑓−𝑓1)𝜏csc𝜃}×𝐷^𝑠_𝑓1𝑓−𝑓₁^𝑡𝑒^{−2𝜋2𝑘2(𝑓−𝑓1)2(csc𝜃)2/(𝑓csc𝜃)2𝑝}×𝑒^{𝑖𝜋𝑡2cot𝜃}𝑢(𝑡)𝑓₁csc𝜃𝑑𝑓₁=𝐴_𝜃𝐴_𝜃²𝑒^{𝑖𝜋𝑓2cot𝜃}(−𝑖2𝜋csc𝜃)^𝑡−𝑠×_ℝ𝑒^{−𝑖2𝜋(𝑓−𝑓1)𝜏csc𝜃𝑠}_𝑙=0𝑠𝑙𝐷_𝑓^𝑠−𝑙₁𝑓−𝑓₁^𝑡×𝐷^𝑙_𝑓1𝑒^{−2𝜋2𝑘2(𝑓−𝑓1)2(csc𝜃)2/(𝑓csc𝜃)2𝑝}×𝑒^{𝑖𝜋𝑡2cot𝜃}𝑢(𝑡)𝑓₁csc𝜃𝑑𝑓₁=𝐴_𝜃𝐴_𝜃²𝑒^{𝑖𝜋𝑓2cot𝜃}(−𝑖2𝜋csc𝜃)^𝑡−𝑠×^𝑠_𝑙=0𝑠𝑙𝑡!(−1)^𝑠−𝑙()!×𝑡−𝑠+𝑙_ℝ𝑒^{−𝑖2𝜋(𝑓−𝑓1)𝜏csc𝜃}𝑓−𝑓₁^{𝑡−𝑠+𝑙}×^𝑙_𝑚=0𝑙𝑚𝐷^𝑚_𝑓1𝑒^{−2[𝜋𝑘csc𝜃(𝑓−𝑓1)/(𝑓csc𝜃)𝑝]2}×𝐷_𝑓^𝑙−𝑚₁𝑒^{𝑖𝜋𝑡2cot𝜃}𝑢(𝑡)𝑓₁csc𝜃𝑑𝑓₁=𝐴_𝜃𝐴_𝜃²𝑒^{𝑖𝜋𝑓2cot𝜃}(−𝑖2𝜋csc𝜃)^𝑡−𝑠×^𝑠_𝑙=0𝑠𝑙𝑡!(−1)^𝑠−𝑙(𝑡−𝑠+𝑙)!^𝑙_𝑚=0𝑙𝑚×_ℝ𝑒^{−𝑖2𝜋(𝑓−𝑓1)𝜏csc𝜃}𝑓−𝑓₁^{𝑡−𝑠+𝑙}×−2𝜋𝑘csc𝜃𝑓csc𝜃^𝑝𝑚𝑒^{−2[𝜋𝑘csc𝜃(𝑓−𝑓1)/(𝑓csc𝜃)𝑝]2}×^[𝑚/2]_𝑛=0(−1)^𝑛𝑚!(4𝜋𝑘csc𝜃)^𝑚−2𝑛()!×𝑛!𝑚−2𝑛𝑓−𝑓₁𝑓csc𝜃^{𝑝(𝑚−2𝑛)}×𝐷_𝑓^𝑙−𝑚₁𝑒^{𝑖𝜋𝑡2cot𝜃}𝑢(𝑡)𝑓₁csc𝜃𝑑𝑓₁=𝐴_𝜃𝐴_𝜃²𝑒^{𝑖𝜋𝑓2cot𝜃}(−𝑖2𝜋csc𝜃)^𝑡−𝑠×^𝑠_𝑙=0𝑠𝑙𝑡!(−1)^𝑠−𝑙(𝑡−𝑠+𝑙)!^𝑙_𝑚=0𝑙𝑚×−2𝜋𝑘csc𝜃𝑓csc𝜃^𝑝𝑚×^[𝑚/2]_𝑛=0(−1)^𝑛𝑚!(4𝜋𝑘csc𝜃)^𝑚−2𝑛()!1𝑛!𝑚−2𝑛𝑓csc𝜃^{𝑝(𝑚−2𝑛)}×_ℝ𝑒^{−𝑖2𝜋(𝑓−𝑓1)𝜏csc𝜃}𝑓−𝑓₁^{𝑡−𝑠+𝑙+𝑚−2𝑛}×𝑒^{−2[𝜋𝑘csc𝜃(𝑓−𝑓1)/(𝑓csc𝜃)𝑝]2}×𝐷_𝑓^𝑙−𝑚₁𝑒^{𝑖𝜋𝑡2cot𝜃}𝑢(𝑡)𝑓₁csc𝜃𝑑𝑓₁=𝐴_𝜃𝐴_𝜃²𝑒^{𝑖𝜋𝑓2cot𝜃}(−𝑖2𝜋csc𝜃)^𝑡−𝑠×^𝑠_𝑙=0𝑠𝑙𝑡!(−1)^𝑠−𝑙(𝑡−𝑠+𝑙)!^𝑙_𝑚=0𝑙𝑚×−2𝜋𝑘csc𝜃𝑓csc𝜃^𝑝𝑚×^[𝑚/2]_𝑛=0(−1)^𝑛𝑚!(4𝜋𝑘csc𝜃)^𝑚−2𝑛()!1𝑛!𝑚−2𝑛𝑓csc𝜃^{𝑝(𝑚−2𝑛)}×_ℝ𝑒^{−𝑖2𝜋(𝑓−𝑦sin𝜃)𝜏csc𝜃}𝑓−𝑦sin𝜃^{𝑡−𝑠+𝑙+𝑚−2𝑛}×𝑒^{−2[𝜋𝑘csc𝜃(𝑓−𝑦sin𝜃)/(𝑓csc𝜃)𝑝]2}×(sin𝜃)^{−𝑙+𝑚+1}𝐷_𝑦^𝑙−𝑚𝑒^{𝑖𝜋𝑡2cot𝜃}𝑢(𝑡)𝑦𝑑𝑦=𝐴_𝜃𝐴_𝜃²𝑒^{𝑖𝜋𝑓2cot𝜃}(−𝑖2𝜋csc𝜃)^𝑡−𝑠×^𝑠_𝑙=0𝑠𝑙𝑡!(−1)^𝑠−𝑙(𝑡−𝑠+𝑙)!^𝑙_𝑚=0𝑙𝑚×−2𝜋𝑘csc𝜃𝑓csc𝜃^𝑝𝑚×^[𝑚/2]_𝑛=0(−1)^𝑛𝑚!(4𝜋𝑘csc𝜃)^𝑚−2𝑛()!1𝑛!𝑚−2𝑛𝑓csc𝜃^{𝑝(𝑚−2𝑛)}×(sin𝜃)^{𝑡−𝑠+2𝑚−2𝑛+1}×_ℝ𝑒^{−𝑖2𝜋(𝑓csc𝜃−𝑦)𝜏}𝑓csc𝜃−𝑦^{𝑡−𝑠+𝑙+𝑚−2𝑛}×𝑒^{−2[𝜋𝑘(𝑓csc𝜃−𝑦)/(𝑓csc𝜃)𝑝]2}×𝐷_𝑦^𝑙−𝑚𝑒^{𝑖𝜋𝑡2cot𝜃}𝑢(𝑡)𝑦𝑑𝑦=𝐴_𝜃𝐴_𝜃²𝑒^{𝑖𝜋𝑓2cot𝜃}(−𝑖2𝜋csc𝜃)^𝑡−𝑠×^𝑠_𝑙=0𝑠𝑙𝑡!(𝑡−𝑠+𝑙)!^𝑙_𝑚=0𝑙𝑚×−2𝜋𝑘csc𝜃𝑓csc𝜃^𝑝𝑚×^[𝑚/2]_𝑛=0(−1)^𝑛𝑚!(4𝜋𝑘csc𝜃)^𝑚−2𝑛()!1𝑛!𝑚−2𝑛𝑓csc𝜃^{𝑝(𝑚−2𝑛)}×(sin𝜃)^{𝑡−𝑠+2𝑚−2𝑛+1}×_ℝ𝑒^{−𝑖2𝜋(𝑓csc𝜃−𝑦)𝜏}×^{𝑡−𝑠+𝑙+𝑚−2𝑛}_𝑟=0𝑟×𝑓𝑡−𝑠+𝑙+𝑚−2𝑛csc𝜃^{𝑡−𝑠+𝑙+𝑚−2𝑛−𝑟}−𝑦^𝑟×𝑒^{−2[𝜋𝑘(𝑓csc𝜃−𝑦)/(𝑓csc𝜃)𝑝]2}×𝐷_𝑦^𝑙−𝑚𝑒^{𝑖𝜋𝑡2cot𝜃}𝑢(𝑡)𝑦𝑑𝑦. (41) By using the convergence factor 1+𝑦², we obtain |𝐼|≤𝐴_𝜃³(2𝜋)^{𝑠𝑡−𝑠}_𝑙=0𝑠𝑙𝑡!(𝑡−𝑠+𝑙)!^𝑙_𝑚=0𝑙𝑚×2𝜋𝑘𝑓csc𝜃^𝑝𝑚×^[𝑚/2]_𝑛=0𝑚!(4𝜋𝑘)^𝑚−2𝑛()!1𝑛!𝑚−2𝑛𝑓csc𝜃^{𝑝(𝑚−2𝑛)}×^{𝑡−𝑠+𝑙+𝑚−2𝑛}_𝑟=0𝑟×𝑓𝑡−𝑠+𝑙+𝑚−2𝑛csc𝜃^{𝑡−𝑠+𝑙+𝑚−2𝑛−𝑟}×_ℝ1+𝑦²𝑦^𝑟𝐷_𝑦^𝑙−𝑚𝑒^{𝑖𝜋𝑡2cot𝜃}𝑢(𝑡𝑦×1)1+𝑦²𝑑𝑦. (42) Using the property (38) we obtain |𝐼|≤𝐶^′𝑠_{𝜃,𝑠,𝑡𝑙=0}𝑠𝑙(𝑠−𝑙)^𝑠−𝑙2^𝑡𝑙_𝑚=0𝑙𝑚×1𝑓csc𝜃^𝑝𝑚×^[𝑚/2]_𝑛=0(4𝜋𝑘)^𝑚−2𝑛(2𝑛)^2𝑛2^𝑚1𝑛!𝑓csc𝜃^{𝑝(𝑚−2𝑛)}×^{𝑡−𝑠+𝑙+𝑚−2𝑛}_𝑟=0𝑟×𝑓𝑡−𝑠+𝑙+𝑚−2𝑛csc𝜃^{𝑡−𝑠+𝑙+𝑚−2𝑛−𝑟}×_ℝ1+𝑦²𝑦^𝑟𝐷_𝑦^𝑙−𝑚×𝑒^{𝑖𝜋𝑡2cot𝜃}𝑢(𝑡𝑦1)1+𝑦²𝑑𝑦. (43) Use the properties (35) and (39) to obtain |𝐼|≤𝐶^′𝑠_{𝜃,𝑠,𝑡𝑙=0}𝑠𝑙𝑠^𝑠2^𝑡𝑙_𝑚=0𝑙𝑚×1𝑓csc𝜃^𝑝𝑚×^[𝑚/2]_𝑛=0(4𝜋𝑘)^𝑚−2𝑛(2𝑛)^2𝑛2^𝑚1𝑛!𝑓csc𝜃^{𝑝(𝑚−2𝑛)}×^{𝑡−𝑠+𝑙+𝑚−2𝑛}_𝑟=0𝑟×𝑓𝑡−𝑠+𝑙+𝑚−2𝑛csc𝜃^{𝑡−𝑠+𝑙+𝑚−2𝑛−𝑟}×𝐶_𝑟𝐵^𝑙−𝑚(𝑙−𝑚)^{(𝑙−𝑚)𝛼}≤𝐶^′′_{𝜃,𝑠,𝑡}𝑠^𝑠2^𝑡𝑠_𝑙=0𝑠𝑙^𝑙𝑚=0𝑙𝑚×1𝑓csc𝜃^𝑝𝑚×^[𝑚/2]_𝑛=0(4𝜋𝑘)^𝑚−2𝑛(2𝑛)^2𝑛2^𝑚1𝑛!𝑓csc𝜃^{𝑝(𝑚−2𝑛)}×^{𝑡−𝑠+𝑙+𝑚−2𝑛}_𝑟=0𝑟×𝑓𝑡−𝑠+𝑙+𝑚−2𝑛csc𝜃^{𝑡−𝑠+𝑙+𝑚−2𝑛−𝑟}𝐵^𝑙𝑙^𝑙𝛼≤𝐶^′′_{𝜃,𝑠,𝑡}𝐴^𝑠₁𝐵^𝑡₁𝑠^{𝑠(𝛼+3)𝑠}_𝑙=0𝑠𝑙^𝑙𝑚=0𝑙𝑚×1𝑓csc𝜃^𝑝𝑚×^[𝑚/2]_𝑛=01𝑓csc𝜃^{𝑝(𝑚−2𝑛)}1+𝑓csc𝜃^{𝑡−𝑠+𝑙+𝑚−2𝑛}=𝐶^′′_{𝜃,𝑠,𝑡}𝐴^𝑠₁𝐵^𝑡₁𝑠^{𝑠(𝛼+3)𝑠}_𝑙=0𝑠𝑙^𝑙𝑚=0𝑙𝑚×1𝑓csc𝜃^𝑝𝑚1+𝑓csc𝜃^{𝑡−𝑠+𝑙+𝑚}𝑓csc𝜃^𝑝𝑚×^[𝑚/2]_𝑛=0𝑓csc𝜃^2𝑝1+𝑓csc𝜃^2𝑛≤𝐶^′′_{𝜃,𝑠,𝑡}𝐴^𝑠₁𝐵^𝑡₁𝑠^{𝑠(𝛼+3)𝑠}_𝑙=0𝑠𝑙×^𝑙_𝑚=0𝑙𝑚1+𝑓csc𝜃^{𝑡−𝑠+𝑙+𝑚}𝑓csc𝜃^2𝑝𝑚𝑚2≤𝐶_{𝜃,𝑠,𝑡}𝐴^𝑠₁𝐵^𝑡₁𝑠^(𝛼+3)𝑠1+𝑓csc𝜃^𝑡×^𝑠_𝑙=0𝑠𝑙^𝑙𝑚=0𝑙𝑚1+𝑓csc𝜃^𝑚𝑓csc𝜃^2𝑝𝑚=𝐶_{𝜃,𝑠,𝑡}𝐴^𝑠₁𝐵^𝑡₁𝑠^(𝛼+3)𝑠1+𝑓csc𝜃^𝑡×^𝑠_𝑙=0𝑠𝑙𝑓1+1+csc𝜃𝑓csc𝜃^2𝑝𝑙=𝐶_{𝜃,𝑠,𝑡}𝐴^𝑠₁𝐵^𝑡₁𝑠^(𝛼+3)𝑠1+𝑓csc𝜃^𝑡×𝑓2+1+csc𝜃𝑓csc𝜃^2𝑝𝑠=𝐶_{𝜃,𝑠,𝑡}𝐴^𝑠₁𝐵^𝑡₁𝑠^(𝛼+3)𝑠1+𝑓csc𝜃^𝑡×2𝑓csc𝜃^2𝑝+𝑓csc𝜃+1𝑓csc𝜃^2𝑝𝑠. (44) Hence, we obtain the estimate 𝑓csc𝜃^2𝑝𝑠𝜏^𝑠1+𝑓csc𝜃^𝑡2𝑓csc𝜃^2𝑝+𝑓csc𝜃+1^𝑠×𝜕𝜕𝜏^𝑡FRST^𝑎_𝑢𝜏,𝑓≤𝐶_{𝜃,𝑠,𝑡}𝐴^𝑠₁𝐵^𝑡₁𝑠^(𝛼+3)𝑠, (45) where 𝐴₁ and 𝐵₁ are positive constants.

Theorem 11. For 0<𝑝≤1, the fractional S-transform is a continuous linear map of 𝑆^𝛼(ℝ) into 𝑆^{∼∼𝛽∼}_𝛼(ℝ×(ℝ⧵{0})) where 𝛼^∼=(𝛼+3) and 𝛽^∼=𝛼.

Theorem 12. For 0<𝑝≤1, the fractional S-transform is a continuous linear map of 𝑆^𝛽_𝛼(ℝ) into 𝑆^{∼∼𝛽∼}_𝛼(ℝ×(ℝ⧵{0})) where 𝛼^∼=(𝛼+3) and 𝛽^∼=𝛽.

Proof. By inequality (43), we have |𝐼|≤𝐶^′𝑠_{𝜃,𝑠,𝑡𝑙=0}𝑠𝑙(𝑠−𝑙)^𝑠−𝑙2^𝑡𝑙_𝑚=0𝑙𝑚×1𝑓csc𝜃^𝑝𝑚×^[𝑚/2]_𝑛=0(4𝜋𝑘)^𝑚−2𝑛(2𝑛)^2𝑛2^𝑚1𝑛!𝑓csc𝜃^{𝑝(𝑚−2𝑛)}×^{𝑡−𝑠+𝑙+𝑚−2𝑛}_𝑟=0𝑟×𝑓𝑡−𝑠+𝑙+𝑚−2𝑛csc𝜃^{𝑡−𝑠+𝑙+𝑚−2𝑛−𝑟}×_ℝ1+𝑦²𝑦^𝑟𝐷_𝑦^𝑙−𝑚×𝑒^{𝑖𝜋𝑡2cot𝜃}𝑢(𝑡𝑦1)1+𝑦²𝑑𝑦. (46) Next, using the properties (36) and (39) we have |𝐼|≤𝐶^′𝑠_{𝜃,𝑠,𝑡𝑙=0}𝑠𝑙𝑠^𝑠2^𝑡𝑙_𝑚=0𝑙𝑚×1𝑓csc𝜃^𝑝𝑚×^[𝑚/2]_𝑛=0(4𝜋𝑘)^𝑚−2𝑛(2𝑛)^2𝑛2^𝑚1𝑛!𝑓csc𝜃^{𝑝(𝑚−2𝑛)}×^{𝑡−𝑠+𝑙+𝑚−2𝑛}_𝑟=0𝑟×𝑓𝑡−𝑠+𝑙+𝑚−2𝑛csc𝜃^{𝑡−𝑠+𝑙+𝑚−2𝑛−𝑟}×𝐶𝐴^𝑟₁𝑟^𝑟𝛽+𝐴₁^(𝑟+2)(𝑟+2)^(𝑟+2)𝛽×𝐵₁^{(𝑙−𝑚)}(𝑙−𝑚)^{(𝑙−𝑚)𝛼}≤𝐶^′𝑠_{𝜃,𝑠,𝑡𝑙=0}𝑠𝑙𝑠^𝑠2^𝑡𝑙_𝑚=0𝑙𝑚×1𝑓csc𝜃^𝑝𝑚×^[𝑚/2]_𝑛=0(4𝜋𝑘)^𝑚−2𝑛(2𝑛)^2𝑛2^𝑚1𝑛!𝑓csc𝜃^{𝑝(𝑚−2𝑛)}×^{𝑡−𝑠+𝑙+𝑚−2𝑛}_𝑟=0𝑟×𝑓𝑡−𝑠+𝑙+𝑚−2𝑛csc𝜃^{𝑡−𝑠+𝑙+𝑚−2𝑛−𝑟}×𝐶𝐴^𝑟₂(𝑟+2)^(𝑟+2)𝛽𝐵^𝑙₁𝑙^𝑙𝛼≤𝐶^′𝑠_{𝜃,𝑠,𝑡𝑙=0}𝑠𝑙𝑠^𝑠2^𝑡𝑙_𝑚=0𝑙𝑚×1𝑓csc𝜃^𝑝𝑚×^[𝑚/2]_𝑛=0(4𝜋𝑘)^𝑚−2𝑛(2𝑛)^2𝑛2^𝑚1𝑛!𝑓csc𝜃^{𝑝(𝑚−2𝑛)}×^{𝑡−𝑠+𝑙+𝑚−2𝑛}_𝑟=0𝑟×𝑓𝑡−𝑠+𝑙+𝑚−2𝑛csc𝜃^{𝑡−𝑠+𝑙+𝑚−2𝑛−𝑟}×𝐶𝐴^𝑟₂𝑟^𝑟𝛽2^2𝛽𝑒^𝑟𝛽𝑒^2𝛽𝐵^𝑙₁𝑙^𝑙𝛼≤𝐶_{𝜃,𝑠,𝑡}𝐴^𝑠𝐵^𝑡𝑠^𝑠(𝛼+3)𝑡^𝑡𝛽×^𝑠_𝑙=0𝑠𝑙𝑠^𝑠2^𝑡𝑙_𝑚=0𝑙𝑚×1𝑓csc𝜃^𝑝𝑚×^[𝑚/2]_𝑛=0(4𝜋𝑘)^𝑚−2𝑛(2𝑛)^2𝑛2^𝑚1𝑛!𝑓csc𝜃^{𝑝(𝑚−2𝑛)}×^{𝑡−𝑠+𝑙+𝑚−2𝑛}_𝑟=0𝑟×𝑓𝑡−𝑠+𝑙+𝑚−2𝑛csc𝜃^{𝑡−𝑠+𝑙+𝑚−2𝑛−𝑟}. (47) Next, by using the technique of Theorem 10, we obtain |𝐼|≤𝐶_{𝜃,𝑠,𝑡}𝐴^𝑠𝐵^𝑡𝑠^𝑠(𝛼+3)𝑡^𝑡𝛽1+𝑓csc𝜃^𝑡×2𝑓csc𝜃^2𝑝+𝑓csc𝜃+1𝑓csc𝜃^2𝑝𝑠. (48) Hence, we obtain the estimate 𝑓csc𝜃^2𝑝𝑠𝜏^𝑠1+𝑓csc𝜃^𝑡2𝑓csc𝜃^2𝑝+𝑓csc𝜃+1^𝑠×𝜕𝜕𝜏^𝑡FRST^𝑎_𝑢𝜏,𝑓≤𝐶_{𝜃,𝑠,𝑡}𝐴^𝑠𝐵^𝑡𝑠^𝑠(𝛼+3)𝑡^𝑡𝛽, (49) where 𝐴 and 𝐵 are positive constants.

6. Future Directions

The results of this paper will be useful in the investigation of time-frequency behaviour of test functions and distributions. We may use these results to obtain new results in modulation spaces, which may be defined by means of the fractional S-transform, by using the techniques developed in [14, Chapter 11–13].

Acknowledgment

The author would like to express their sincere thanks to Professor R. S. Pathak for his help and encouragement.