Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 623294, 4 pages http://dx.doi.org/10.1155/2014/623294

Research Article The S-Transform of Distributions Sunil Kumar Singh Department of Mathematics, Rajiv Gandhi University, Doimukh, Arunachal Pradesh 791112, India Correspondence should be addressed to Sunil Kumar Singh; sks [email protected] Received 30 August 2013; Accepted 10 October 2013; Published 2 January 2014 Academic Editors: B. Carpentieri and A. Ibeas Copyright © 2014 Sunil Kumar Singh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Parseval’s formula and inversion formula for the S-transform are given. A relation between the S-transform and pseudodifferential operators is obtained. The S-transform is studied on the spaces S(R𝑛 ) and S󸀠 (R𝑛 ).

1. Introduction The S-transform was first used by Stockwell et al. [1] in 1996. If 𝜔(𝑡, 𝜉) is a window function, then the continuous S-transform of 𝜙(𝑡) with respect to 𝜔 is defined as [2] (S𝜔 𝜙) (𝜏, 𝜉) = ∫ 𝜙 (𝑡) 𝜔 (𝜏 − 𝑡, 𝜉) 𝑒−𝑖2𝜋⟨𝑡,𝜉⟩ 𝑑𝑡, R𝑛

(1)

for 𝑥, 𝜉 ∈ R𝑛 . In signal analysis, at least in dimension 𝑛 = 1, R2𝑛 is called the time-frequency plane, and in physics R2𝑛 is called the phase space. Equation (1) can be rewritten as a convolution as (S𝜔 𝜙) (𝜏, 𝜉) = (𝜙 (⋅) 𝑒−𝑖2𝜋⟨⋅,𝜉⟩ ∗ 𝜔 (⋅, 𝜉)) (𝜏) .

(2)

Now, we recall the definitions of Fourier transform on R𝑛 . Definition 1. If 𝜙(𝑡) is defined on R𝑛 , then the Fourier transform of 𝜙 is given by −𝑖2𝜋⟨𝑡,𝜉⟩

F [𝜙 (𝑡)] (𝜉) = ∫ 𝜙 (𝑡) 𝑒 R𝑛

where ⟨𝑡, 𝜉⟩ =

∑𝑛𝑗=1 𝑡𝑗 𝜉𝑗

𝑑𝑡,

(3) 𝑛

is the usual inner product on R .

Definition 2. If 𝜙(𝑡, 𝑥) is defined on R2𝑛 , then the partial Fourier transform of 𝜙(𝑡, 𝑥) with respect to the first coordinate is given by F1 [𝜙 (𝑡, 𝑥)] (𝜉, 𝑥) = ∫ 𝜙 (𝑡, 𝑥) 𝑒−𝑖2𝜋⟨𝑡,𝜉⟩ 𝑑𝑡, R𝑛

(4)

and the partial Fourier transform of 𝜙(𝑡, 𝑥) with respect to the second coordinate is given by F2 [𝜙 (𝑡, 𝑥)] (𝑡, 𝜂) = ∫ 𝜙 (𝑡, 𝑥) 𝑒−𝑖2𝜋⟨𝑥,𝜂⟩ 𝑑𝑥. R𝑛

(5)

Applying the convolution property for the Fourier transform in (2), we obtain (S𝜔 𝜙) (𝜏, 𝜉) = F−1 1 [F1 (𝜙) (𝛼 + 𝜉) F1 (𝜔) (𝛼, 𝜉)] (𝜏, 𝜉) , (6) where F−1 1 is the inverse Fourier transform. Now, we define the translation, modulation, and involution operators, respectively, by 𝑇𝜏 𝜙 (𝑡) = 𝜙 (𝑡 − 𝜏) 𝑀𝜉 𝜙 (𝑡) = 𝑒𝑖2𝜋⟨𝜉,𝑡⟩ 𝜙 (𝑡) I𝜙 (𝑡) = 𝜙 (−𝑡)

(translation) (modulation)

(7)

(involution) ,

𝑛

where 𝑡, 𝜏, 𝜉 ∈ R . Definition 3 (the Dirac delta). The Dirac delta function is defined by ∬ 𝜙 (𝑡, 𝑥) 𝛿 (𝑡, 𝑥) 𝑑𝑡 𝑑𝑥 = 𝜙 (0, 0) . R𝑛

(8)

Definition 4 (tempered distribution). A function 𝜙 ∈ 𝐶∞ (R𝑛 ) is said to be rapidly decreasing if 󵄨 󵄨 𝛾𝛼,𝛽 (𝜙) = sup 󵄨󵄨󵄨󵄨𝑥𝛼 𝐷𝛽 𝜙 (𝑥)󵄨󵄨󵄨󵄨 < ∞, (9) 𝑛 𝑥∈R

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for all pairs of multi-indices 𝛼, 𝛽 ∈ N𝑛0 . The space of all rapidly decreasing functions on R𝑛 is denoted by S(R𝑛 ) or simply S. Elements in the dual space S󸀠 of S are called tempered distribution.

2. Some Important Properties of S-Transform Some properties of S-transform can be found in [3–8] and certain properties of S-transform are obtained in this section. By definition, we have

This immediately implies the Plancherel formula 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩S𝜔 𝜙󵄩󵄩󵄩𝐿2 (R𝑛 ×R𝑛 ) = 󵄩󵄩󵄩𝜙󵄩󵄩󵄩𝐿2 (R𝑛 ) . Proof. Consider ∬

R𝑛

(S𝜔1 𝜙1 ) (𝜏, 𝜉) (S𝜔2 𝜙2 )(𝜏, 𝜉) 𝑑𝜏 𝑑𝜉

=∬

R𝑛

(S𝜔 𝜙) (𝜏, 𝜉) = ∫ 𝜔 (𝜏 − 𝑡, 𝜉) 𝜙 (𝑡) 𝑒−𝑖2𝜋⟨𝑡,𝜉⟩ 𝑑𝑡

= ∭ 𝜙1 (𝑡) 𝜙2 (𝑥) 𝑒𝑖2𝜋⟨(𝑥−𝑡),𝜉⟩ R𝑛

(10)

= ∫ 𝜔 (−𝑥, 𝜉) 𝑀−𝜉 𝜙 (𝜏 + 𝑥) 𝑑𝑥

R𝑛

= ∫ 𝜔 (−𝑥, 𝜉) 𝑇−𝜏 𝑀−𝜉 𝜙 (𝑥) 𝑑𝑥.

= ∭ 𝜙1 (𝑡) 𝜙2 (𝑥) 𝑒𝑖2𝜋⟨(𝑥−𝑡),𝜉⟩ 𝛿 (𝑥 − 𝑡, 0) 𝑑𝜉 𝑑𝑡 𝑑𝑥

R𝑛

R𝑛

Thus, the S-transform S𝜔 appears as a superposition of timefrequency shifts as follows:

R𝑛

= ∫ 𝜙1 (𝑡) 𝜙2 (𝑡)𝑑𝑡. R𝑛

(11)

Example 5. If 𝜔(𝑡, 𝜉) = 𝑚(𝜉), that is, independent of 𝑡, then (S𝜔 𝜙) (𝜏, 𝜉) = ∫ 𝑚 (𝜉) 𝜙 (𝑡) 𝑒−𝑖2𝜋⟨𝑡,𝜉⟩ 𝑑𝑡

(12)

= 𝑚 (𝜉) (F𝜙) (𝜉) .

Theorem 8 (inversion formula). If 𝜙 ∈ 𝐿2 (R𝑛 ) and window function 𝜔 satisfy the condition (14) of the previous theorem, then 𝜙 (𝑡) = ∬

R𝑛

(S𝜔 𝜙) (𝜏, 𝜉) 𝜔 (𝜏 − 𝑡, 𝜉)𝑒𝑖2𝜋⟨𝑡,𝜉⟩ 𝑑𝜏 𝑑𝜉.

So S𝜔 is a multiplication operator. In particular, if 𝜔(𝑡, 𝜉) = 1, then (S𝜔 𝜙)(𝜏, 𝜉) = (F𝜙)(𝜉).

Proof. By the previous theorem we can write

Example 6. If 𝜔(𝑡, 𝜉) = 𝑚(𝑡), then

⟨𝜙1 , 𝜙2 ⟩ = ∬

(S𝜔 𝜙) (𝜏, 𝜉) = ∫ 𝑚 (𝜏 − 𝑡) 𝜙 (𝑡) 𝑒−𝑖2𝜋⟨𝑡,𝜉⟩ 𝑑𝑡

=∬

R𝑛

= ∫ 𝑇−𝜏 𝑚 (−𝑡) 𝜙 (𝑡) 𝑒−𝑖2𝜋⟨𝑡,𝜉⟩ 𝑑𝑡 R𝑛

(S𝜔 𝜙1 ) (𝜏, 𝜉)

R𝑛

× (∫ 𝜙2 (𝑥) 𝜔 (𝜏 − 𝑥, 𝜉) 𝑒−𝑖2𝜋⟨𝑥,𝜉⟩ 𝑑𝑥) 𝑑𝜏 𝑑𝜉 R𝑛

= ∫ (∬

R𝑛

R𝑛

= F (𝜙𝑇−𝜏 I𝑚) (𝜉) .

R𝑛

Let 𝜙1 , 𝜙2 ∈ 𝐿2 (R𝑛 ) and let (S𝜔1 𝜙1 ) and (S𝜔2 𝜙2 ) be the Stransforms of 𝜙1 and 𝜙2 , respectively. Then ∬

R𝑛

(S𝜔1 𝜙1 ) (𝜏, 𝜉) (S𝜔2 𝜙2 ) (𝜏, 𝜉)𝑑𝜏 𝑑𝜉 (15)

= ∫ 𝜙1 (𝑡) 𝜙2 (𝑡) 𝑑𝑡. R𝑛

R𝑛

(S𝜔 𝜙1 ) (𝜏, 𝜉) × 𝜔 (𝜏 − 𝑥, 𝜉)𝑒𝑖2𝜋⟨𝑥,𝜉⟩ 𝑑𝜏 𝑑𝜉) 𝜙2 (𝑥) 𝑑𝑥.

Theorem 7 (Parseval’s formula). Let 𝜔1 and 𝜔2 be the window functions such that (14)

(18)

(S𝜔 𝜙1 ) (𝜏, 𝜉) (S𝜔 𝜙2 ) (𝜏, 𝜉) 𝑑𝜏 𝑑𝜉

R𝑛

(13)

= ∫ 𝑇−𝜏 I𝑚 (𝑡) 𝜙 (𝑡) 𝑒−𝑖2𝜋⟨𝑡,𝜉⟩ 𝑑𝑡

∫ 𝜔1 (𝜏 − 𝑡, 𝜉) 𝜔2 (𝜏 − 𝑥, 𝜂) 𝑑𝜏 = 𝛿 (𝑥 − 𝑡, 𝜉 − 𝜂) .

(17)

× (∫ 𝜔1 (𝜏 − 𝑡, 𝜉) 𝜔2 (𝜏 − 𝑥, 𝜉)𝑑𝜏) 𝑑𝜉 𝑑𝑡 𝑑𝑥

R𝑛

R𝑛

R𝑛

R𝑛

= ∫ 𝜔 (−𝑥, 𝜉) 𝜙 (𝜏 + 𝑥) 𝑒−𝑖2𝜋⟨𝜏+𝑥,𝜉⟩ 𝑑𝑥

S𝜔 := ∫ 𝜔 (−𝑥, 𝜉) 𝑇−𝜏 𝑀−𝜉 𝑑𝑥.

(∫ 𝜙1 (𝑡) 𝜔1 (𝜏 − 𝑡, 𝜉) 𝑒−𝑖2𝜋⟨𝑡,𝜉⟩ 𝑑𝑡) × (∫ 𝜙2 (𝑥)𝜔2 (𝜏 − 𝑥, 𝜉)𝑒−𝑖2𝜋⟨𝑥,𝜉⟩ 𝑑𝑥) 𝑑𝜏 𝑑𝜉

R𝑛

R𝑛

(16)

(19) Hence 𝜙1 (𝑡) = ∬

R𝑛

(S𝜔 𝜙1 ) (𝜏, 𝜉) 𝜔 (𝜏 − 𝑡, 𝜉)𝑒𝑖2𝜋⟨𝑡,𝜉⟩ 𝑑𝜏 𝑑𝜉.

(20)

Definition 9. Let 𝜔 be a window function and S𝜔 is the Stransform. Then the transform S∗𝜔 defined by ⟨S𝜔 𝜙, 𝜓⟩ = ⟨𝜙, S∗𝜔 𝜓⟩

(21)

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is called the adjoint of S𝜔 . If 𝜙 ∈ 𝐿2 (R𝑛 ) and 𝜓 ∈ 𝐿2 (R𝑛 × R𝑛 ), then (21) implies that (S∗𝜔 𝜓) (𝑡) = ∬ 𝜓 (𝜏, 𝜉) 𝜔 (𝜏 − 𝑡, 𝜉)𝑒𝑖2𝜋⟨𝑡,𝜉⟩ 𝑑𝜏 𝑑𝜉, R𝑛

(22)

where 𝑡, 𝜏, 𝜉 ∈ R𝑛 . Theorem 10 (Parseval’s formula for S∗𝜔 ). Let 𝜔1 and 𝜔2 be the window functions that satisfy the condition (14). If 𝜓1 , 𝜓2 ∈ 𝐿2 (R𝑛 × R𝑛 ), then

(23) = ∬ 𝜓1 (𝜏, 𝜉) 𝜓2 (𝜏, 𝜉) 𝑑𝜏 𝑑𝜉, R𝑛

and the Plancherel formula is 󵄩 󵄩 󵄩󵄩 ∗ 󵄩󵄩 󵄩󵄩S𝜔 𝜓󵄩󵄩𝐿2 (R𝑛 ) = 󵄩󵄩󵄩𝜓󵄩󵄩󵄩𝐿2 (R𝑛 ×R𝑛 ) .

(24)

Proof. Consider ∫ (S∗𝜔1 𝜓1 ) (𝑡) (S∗𝜔2 𝜓2 ) (𝑡) 𝑑𝑡 R𝑛

= ∫ (∬ 𝜓1 (𝜏, 𝜉) 𝜔1 (𝜏 − 𝑡, 𝜉)𝑒𝑖2𝜋⟨𝑡,𝜉⟩ 𝑑𝜏 𝑑𝜉) R𝑛

R𝑛

× (∬ 𝜓2 (𝛼, 𝜂) 𝜔2 (𝛼 − R𝑛

S∗𝜔 = S−1 𝜔 . This proves the theorem.

Definition 12 (pseudodifferential operator). Let 𝜎 be a (measurable) function or a tempered distribution on R𝑛 . Then the operator 𝐾𝜎 𝜙 (𝑡) = ∫ 𝜎 (𝑡, 𝜉) 𝜙̂ (𝜉) 𝑒𝑖2𝜋⟨𝑡,𝜉⟩ 𝑑𝜉 R𝑛

(29)

The pseudodifferential operator plays an important role in the theory of partial differential equations. The pseudodifferential operator has been studied on function and distribution spaces by many authors. Details of the concept can be found in [9, 10]. 2.1. Relation between the S-Transform and Pseudodifferential Operator. Here we give a direct relation between S-transform and pseudodifferential operator which will may be very useful in the study of S-transform of distribution spaces. The continuous S-transform of a function 𝜙 with respect to a window function 𝜔 is given by (S𝜔 𝜙) (𝜏, 𝜉) = ∫ 𝜙 (𝑡) 𝜔 (𝜏 − 𝑡, 𝜉) 𝑒−𝑖2𝜋⟨𝑡,𝜉⟩ 𝑑𝑡 R𝑛

𝑡, 𝜂)𝑒𝑖2𝜋⟨𝑡,𝜂⟩ 𝑑𝛼 𝑑𝜂) 𝑑𝑡

= ∫ 𝜙 (𝜏 − 𝑥) 𝜔 (𝑥, 𝜉) 𝑒−𝑖2𝜋⟨(𝜏−𝑥),𝜉⟩ 𝑑𝑥 R𝑛

= ∫∭ 𝜓1 (𝜏, 𝜉) 𝜓2 (𝛼, 𝜂)𝑒𝑖2𝜋⟨𝑡,𝜉−𝜂⟩

= 𝑒−𝑖2𝜋⟨𝜏,𝜉⟩ ∫ 𝑇−𝜏 𝜙 (−𝑥) 𝜔 (𝑥, 𝜉) 𝑒𝑖2𝜋⟨𝑥,𝜉⟩ 𝑑𝑥

R𝑛

× (∫ 𝜔1 (𝜏 − 𝑡, 𝜉)𝜔2 (𝛼 − 𝑡, 𝜂) 𝑑𝑡) 𝑑𝜏 𝑑𝜉 𝑑𝛼 𝑑𝜂 R𝑛

R𝑛

= 𝑒−𝑖2𝜋⟨𝜏,𝜉⟩ ∫ F [F (𝑇−𝜏 𝜙)] (𝑥) R𝑛

= ∫∭ 𝜓1 (𝜏, 𝜉) 𝜓2 (𝛼, 𝜂)𝑒𝑖2𝜋⟨𝑡,𝜉−𝜂⟩

× 𝜔 (𝑥, 𝜉) 𝑒𝑖2𝜋⟨𝑥,𝜉⟩ 𝑑𝑥

R𝑛

× 𝛿 (𝜏 − 𝛼, 𝜉 − 𝜂) 𝑑𝜏 𝑑𝜉 𝑑𝛼 𝑑𝜂

= 𝑒−𝑖2𝜋⟨𝜏,𝜉⟩ 𝐾𝜎 [F (𝑇−𝜏 𝜙)] (𝜉) , (30)

= ∬ 𝜓1 (𝜏, 𝜉) 𝜓2 (𝜏, 𝜉) 𝑑𝜏 𝑑𝜉. R𝑛

(28)

is called the pseudodifferential operator.

∫ (S∗𝜔1 𝜓1 ) (𝑡) (S∗𝜔2 𝜓2 ) (𝑡) 𝑑𝑡 R𝑛

Thus

(25)

where 𝜎(𝜉, 𝑥) = 𝜔(𝑥, 𝜉).

This proves the theorem.

3. The S-Transform of Distributions

Theorem 11. If the window function 𝜔 satisfies the condition (14), then

In this section we will investigate the S-transform of tempered distribution by means of the Fourier transform.

S𝜔 S∗𝜔 = 𝐼 = S∗𝜔 S𝜔 ,

Theorem 13. If 𝜔 ∈ S(R2𝑛 ), then S𝜔 maps S(R𝑛 ) into S(R2𝑛 ).

(26)

where 𝐼 is the identity operator.

Proof. By (6) we have (S𝜔 𝜙) (𝜏, 𝜉) = F−1 1 [F1 (𝜙) (𝛼 + 𝜉) F1 (𝜔) (𝛼, 𝜉)] (𝜏, 𝜉) . (31)

Proof. By definition S∗𝜔 [S𝜔1 𝜓] (𝑡) = ∬

R𝑛

(S𝜔1 𝜓) (𝜏, 𝜉) 𝜔 (𝜏 − 𝑡, 𝜉)e𝑖2𝜋⟨𝑡,𝜉⟩ 𝑑𝜏 𝑑𝜉

= S−1 𝜔 [S𝜔1 𝜓] (𝑡) . (27)

Thus, (S𝜔 𝜙) ∈ S(R2𝑛 ), since the Fourier transform is continuous isomorphism from S(R𝑛 ) to S(R𝑛 ), and its inverse is also a continuous isomorphism from S(R𝑛 ) to S(R𝑛 ) (see [11], page 66-67).

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Theorem 14. If 𝜔 ∈ S(R2𝑛 ), then S𝜔 maps S󸀠 (R𝑛 ) into S󸀠 (R2𝑛 ). Proof. For any 𝑓 ∈ S󸀠 (R𝑛 ) and 𝜓 ∈ S(R2𝑛 ), we have ⟨(S𝜔 𝑓) (𝜏, 𝜉) , 𝜓 (𝜏, 𝜉)⟩ =∬

R𝑛

(∫ 𝑓 (𝑡) 𝜔 (𝜏 − 𝑡, 𝜉) 𝑒−𝑖2𝜋⟨𝑡,𝜉⟩ 𝑑𝑡) 𝜓(𝜏, 𝜉) 𝑑𝜏 𝑑𝜉 R𝑛

(in fact 𝜓 = 𝜓, since 𝜓 ∈ S) = ∫ 𝑓 (𝑡) (∬ 𝜓(𝜏, 𝜉)𝜔 (𝜏 − 𝑡, 𝜉) 𝑒−𝑖2𝜋⟨𝑡,𝜉⟩ 𝑑𝜏 𝑑𝜉) 𝑑𝑡 R𝑛

R𝑛

= ⟨𝑓, 𝜙⟩ , (32) where 𝜙 (𝑡) = ∬ 𝜓 (𝜏, 𝜉) 𝜔 (𝜏 − 𝑡, 𝜉)𝑒𝑖2𝜋⟨𝑡,𝜉⟩ 𝑑𝜏 𝑑𝜉 R𝑛

= ∫ (𝜓 ∗ (I𝜔)) (𝑡, 𝜉) 𝑒𝑖2𝜋⟨𝑡,𝜉⟩ 𝑑𝜉

(33)

R𝑛

𝑛 = F−1 2 (𝜓 ∗ (I𝜔)) (𝑡) ∈ S (R ) .

Thus, (S𝜔 𝑓)(𝜏, 𝜉) ∈ S󸀠 (R2𝑛 ). Theorem 15. If 𝜔 ∈ S󸀠 (R2𝑛 ), then S𝜔 maps S(R𝑛 ) into S󸀠 (R2𝑛 ). Proof. If 𝜙 ∈ S(R𝑛 ) and 𝜓 ∈ S(R2𝑛 ), then 𝑈𝜙,𝜓 (𝜏, 𝜉) := F (𝜙) (𝛼 + 𝜉)𝜓 (𝛼, 𝜉) ∈ S (R2𝑛 ) .

(34)

Thus for any 𝜔 ∈ S󸀠 (R2𝑛 ), we have ⟨(F1 𝜔) (𝛼, 𝜉) , 𝑈𝜙,𝜓 (𝛼, 𝜉)⟩ =∬ =∬

R𝑛

R𝑛

(F1 𝜔) (𝛼, 𝜉) (F𝜙) (𝛼 + 𝜉) 𝜓 (𝛼, 𝜉) 𝑑𝛼 𝑑𝜉 (35) (F1 (S𝜔 𝜙)) (𝛼, 𝜉) 𝜓 (𝛼, 𝜉) 𝑑𝛼 𝑑𝜉

= ⟨(F1 (S𝜔 𝜙)) (𝛼, 𝜉) , 𝜓 (𝛼, 𝜉)⟩ . Thus F1 (S𝜔 𝜙) ∈ S󸀠 (R2𝑛 ) and hence (S𝜔 𝜙) ∈ S󸀠 (R2𝑛 ).

Conflict of Interests The authors declare that there is no conflict of interests.

Acknowledgment The author expresses his sincere thanks to Professor R. S. Pathak for his help and encouragement.

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