The S-Transform of Distributions

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).


Introduction
The S-transform was first used by Stockwell et al. [1] in 1996. If ( , ) is a window function, then the continuous S-transform of ( ) with respect to is defined as [2] (S ) ( , ) = ∫ R ( ) ( − , ) − 2 ⟨ , ⟩ , for , ∈ R . (1) In signal analysis, at least in dimension = 1, R 2 is called the time-frequency plane, and in physics R 2 is called the phase space.
Now, we recall the definitions of Fourier transform on R .
Definition 1. If ( ) is defined on R , then the Fourier transform of is given by where ⟨ , ⟩ = ∑ =1 is the usual inner product on R .
Definition 2. If ( , ) is defined on R 2 , then the partial Fourier transform of ( , ) with respect to the first coordinate is given by and the partial Fourier transform of ( , ) with respect to the second coordinate is given by Applying the convolution property for the Fourier transform in (2), we obtain where F −1 1 is the inverse Fourier transform. Now, we define the translation, modulation, and involution operators, respectively, by where , , ∈ R .
Definition 4 (tempered distribution). A function ∈ ∞ (R ) is said to be rapidly decreasing if The Scientific World Journal for all pairs of multi-indices , ∈ N 0 . The space of all rapidly decreasing functions on R is denoted by S(R ) or simply S. Elements in the dual space S of S are called tempered distribution.

Some Important Properties of S-Transform
Some properties of S-transform can be found in [3][4][5][6][7][8] and certain properties of S-transform are obtained in this section. By definition, we have Thus, the S-transform S appears as a superposition of timefrequency shifts as follows: So S is a multiplication operator. In particular, if ( , ) = 1, then (S )( , ) = (F )( ).
Theorem 7 (Parseval's formula). Let 1 and 2 be the window functions such that Let 1 , 2 ∈ 2 (R ) and let (S 1 1 ) and (S 2 2 ) be the Stransforms of 1 and 2 , respectively. Then This immediately implies the Plancherel formula Theorem 8 (inversion formula). If ∈ 2 (R ) and window function satisfy the condition (14) of the previous theorem, then Proof. By the previous theorem we can write The Scientific World Journal 3 is called the adjoint of S . If ∈ 2 (R ) and ∈ 2 (R × R ), then (21) implies that where , , ∈ R .
Theorem 10 (Parseval's formula for S * ). Let 1 and 2 be the window functions that satisfy the condition (14).
and the Plancherel formula is This proves the theorem.
This proves the theorem.
Definition 12 (pseudodifferential operator). Let be a (measurable) function or a tempered distribution on R . Then the operator is called the pseudodifferential operator.
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].

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 with respect to a window function is given by where ( , ) = ( , ).

The S-Transform of Distributions
In this section we will investigate the S-transform of tempered distribution by means of the Fourier transform.
Proof. By (6) we have Thus, (S ) ∈ S(R 2 ), since the Fourier transform is continuous isomorphism from S(R ) to S(R ), and its inverse is also a continuous isomorphism from S(R ) to S(R ) (see [11], page 66-67).

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The Scientific World Journal Theorem 14. If ∈ S(R 2 ), then S maps S (R ) into S (R 2 ).

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