Research Article The S-Transform of Distributions

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1 e Scientific World Journal, Article ID , 4 pages Research Article The S-Transform of Distributions Sunil Kumar Singh Department of Mathematics, Rajiv Gandhi University, Doimukh, Arunachal Pradesh , India Correspondence should be addressed to Sunil Kumar Singh; sks math@yahoo.com 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(R n ) and S (R n ). 1. Introduction The S-transform was first used by Stockwell et al. [1]in1996.If ω(t, ξ) is a window function, then the continuous S-transform of φ(t) with respect to ω is defined as [2] (S ω φ) (τ, ξ) = R n φ (t) ω (τ t,ξ) e i2π t,ξ dt, for x, ξ R n. In signal analysis, at least in dimension n=1, R 2n is called the time-frequency plane, and in physics R 2n is called the phase space. Equation (1)canberewrittenasaconvolutionas (1) (S ω φ) (τ, ξ) =(φ( ) e i2π,ξ ω(, ξ)) (τ). (2) and the partial Fourier transform of φ(t, x) with respect to the second coordinate is given by F 2 [φ (t, x)](t, η) = R n φ (t, x) e i2π x,η dx. (5) Applying the convolution property for the Fourier transform in (2), we obtain (S ω φ) (τ, ξ) = F 1 1 [F 1 (φ) (α+ξ) F 1 (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 T τ φ (t) =φ(t τ) (translation) Now, we recall the definitions of Fourier transform on R n. M ξ φ (t) =e i2π ξ,t φ (t) (modulation) (7) Definition 1. If φ(t) is defined on R n, then the Fourier transform of φ is given by F [φ (t)] (ξ) = R n φ (t) e i2π t,ξ dt, (3) where t, ξ = n j=1 t jξ j is the usual inner product on R n. Definition 2. If φ(t, x) is defined on R 2n, then the partial Fourier transform of φ(t, x) with respect to the first coordinate is given by F 1 [φ (t, x)] (ξ, x) = R n φ (t, x) e i2π t,ξ dt, (4) where t, τ, ξ R n. Iφ (t) = φ ( t) (involution), Definition 3 (the Dirac delta). The Dirac delta function is defined by R n φ (t, x) δ (t, x) dt dx = φ (0, 0). (8) Definition 4 (tempered distribution). A function φ C (R n ) is said to be rapidly decreasing if γ α,β (φ) = sup x R n xα D β φ (x) <, (9)

2 2 The Scientific World Journal for all pairs of multi-indices α, β N n 0.Thespaceofallrapidly decreasing functions on R n is denoted by S(R n ) 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 (S ω φ) (τ, ξ) = R n ω (τ t,ξ) φ (t) e i2π t,ξ dt This immediately implies the Plancherel formula S ωφ L 2 (R n R n ) = φ L 2 (R n ). (16) Proof. Consider (S ω1 φ 1 ) (τ, ξ) (S ω2 φ 2 )(τ, ξ) dτ dξ R n = φ 1 (t) ω 1 (τ t,ξ) e R ( R i2π t,ξ dt) n n φ 2 (x)ω 2 (τ x, ξ)e ( R i2π x,ξ dx) dτ dξ n = R n ω ( x, ξ) φ (τ+x) e i2π τ+x,ξ dx = R n ω ( x, ξ) M ξ φ (τ+x) dx (10) = R n φ 1 (t) φ 2 (x) e i2π (x t),ξ ( R n ω 1 (τ t,ξ) ω 2 (τ x, ξ)dτ) dξ dt dx (17) = R n ω ( x, ξ) T τ M ξ φ (x) dx. = R n φ 1 (t) φ 2 (x) e i2π (x t),ξ δ (x t,0) dξ dt dx Thus, the S-transform S ω appearsasasuperpositionoftimefrequency shifts as follows: = R n φ 1 (t) φ 2 (t)dt. S ω := R n ω ( x, ξ) T τ M ξ dx. (11) Example 5. If ω(t, ξ) = m(ξ), that is, independent of t,then (S ω φ) (τ, ξ) = R n m (ξ) φ (t) e i2π t,ξ dt =m(ξ) (Fφ) (ξ). (12) So S ω is a multiplication operator. In particular, if ω(t, ξ) = 1, then (S ω φ)(τ, ξ) = (Fφ)(ξ). Example 6. If ω(t, ξ) = m(t),then (S ω φ) (τ, ξ) = R n m (τ t) φ (t) e i2π t,ξ dt Theorem 8 (inversion formula). If φ L 2 (R n ) and window function ω satisfy the condition (14) of the previous theorem, then φ (t) = R n (S ω φ) (τ, ξ) ω (τ t,ξ)e i2π t,ξ dτ dξ. (18) Proof. By the previous theorem we can write φ 1,φ 2 = R n (S ω φ 1 ) (τ, ξ) (S ω φ 2 ) (τ, ξ) dτ dξ = R n (S ω φ 1 ) (τ, ξ) = R n T τ m ( t) φ (t) e i2π t,ξ dt = R n T τ Im (t) φ (t) e i2π t,ξ dt (13) ( R n φ 2 (x) ω (τ x,ξ) e i2π x,ξ dx) dτ dξ = R n ( R n (S ω φ 1 ) (τ, ξ) = F (φt τ Im) (ξ). Theorem 7 (Parseval s formula). Let ω 1 and ω 2 be the window functions such that R n ω 1 (τ t,ξ) ω 2 (τ x, η) dτ = δ (x t,ξ η). (14) Let φ 1,φ 2 L 2 (R n ) and let (S ω1 φ 1 ) and (S ω2 φ 2 ) be the S- transforms of φ 1 and φ 2,respectively.Then ω (τ x,ξ)e i2π x,ξ dτ dξ) φ 2 (x) dx. (19) Hence φ 1 (t) = (S ω φ 1 ) (τ, ξ) ω (τ t,ξ)e R i2π t,ξ dτ dξ. (20) n R n (S ω1 φ 1 ) (τ, ξ) (S ω2 φ 2 ) (τ, ξ)dτ dξ = R n φ 1 (t) φ 2 (t) dt. (15) Definition 9. Let ω be a window function and S ω is the S- transform. Then the transform S ω defined by S ω φ, ψ = φ, S ωψ (21)

3 The Scientific World Journal 3 is called the adjoint of S ω.ifφ L 2 (R n ) and ψ L 2 (R n R n ), then (21)impliesthat (S ω ψ) (t) = R n ψ (τ, ξ) ω (τ t,ξ)e i2π t,ξ dτ dξ, (22) Thus This proves the theorem. S ω = S 1 ω. (28) where t, τ, ξ R n. Theorem 10 (Parseval s formula for S ω ). Let ω 1 and ω 2 be the window functions that satisfy the condition (14). Ifψ 1,ψ 2 L 2 (R n R n ),then R n (S ω 1 ψ 1 ) (t) (S ω 2 ψ 2 ) (t) dt and the Plancherel formula is = R n ψ 1 (τ, ξ) ψ 2 (τ, ξ) dτ dξ, (23) S ω ψ L 2 (R n ) = ψ L 2 (R n R n ). (24) Proof. Consider R n (S ω 1 ψ 1 ) (t) (S ω 2 ψ 2 ) (t) dt = R n ( R n ψ 1 (τ, ξ) ω 1 (τ t,ξ)e i2π t,ξ dτ dξ) ( R n ψ 2 (α, η) ω 2 (α t, η)e i2π t,η dα dη) dt = R n ψ 1 (τ, ξ) ψ 2 (α, η)e i2π t,ξ η ( R n ω 1 (τ t,ξ)ω 2 (α t, η) dt) dτ dξ dα dη = R n ψ 1 (τ, ξ) ψ 2 (α, η)e i2π t,ξ η δ(τ α,ξ η)dτdξdαdη = R n ψ 1 (τ, ξ) ψ 2 (τ, ξ) dτ dξ. This proves the theorem. (25) Theorem 11. If the window function ω satisfies the condition (14),then where I is the identity operator. Proof. By definition S ω S ω =I=S ω S ω, (26) S ω [S ω 1 ψ] (t) = R n (S ω1 ψ) (τ, ξ) ω (τ t,ξ)e i2π t,ξ dτ dξ = S 1 ω [S ω 1 ψ] (t). (27) Definition 12 (pseudodifferential operator). Let σ be a (measurable) function or a tempered distribution on R n.thenthe operator K σ φ (t) = R n σ (t, ξ) φ (ξ) e i2π t,ξ dξ (29) 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 φ with respect to a window function ω is given by (S ω φ) (τ, ξ) = R n φ (t) ω (τ t,ξ) e i2π t,ξ dt where σ(ξ, x) = ω(x, ξ). = R n φ (τ x) ω (x, ξ) e i2π (τ x),ξ dx =e i2π τ,ξ R n T τ φ ( x) ω (x, ξ) e i2π x,ξ dx =e i2π τ,ξ R n F [F (T τ φ)] (x) ω(x, ξ) e i2π x,ξ dx =e i2π τ,ξ K σ [F (T τ φ)] (ξ), 3. The S-Transform of Distributions (30) In this section we will investigate the S-transform of tempered distribution by means of the Fourier transform. Theorem 13. If ω S(R 2n ),thens ω maps S(R n ) into S(R 2n ). Proof. By (6)wehave (S ω φ) (τ, ξ) = F 1 1 [F 1 (φ) (α+ξ) F 1 (ω)(α, ξ)] (τ, ξ). (31) Thus, (S ω φ) S(R 2n ), since the Fourier transform is continuous isomorphism from S(R n ) to S(R n ), and its inverse is also a continuous isomorphism from S(R n ) to S(R n ) (see [11], page 66-67).

4 4 The Scientific World Journal Theorem 14. If ω S(R 2n ),thens ω maps S (R n ) into S (R 2n ). Proof. For any f S (R n ) and ψ S(R 2n ),wehave (S ω f) (τ, ξ),ψ(τ, ξ) where = R n ( R n f (t) ω (τ t,ξ) e i2π t,ξ dt) ψ(τ, ξ) dτ dξ (in fact ψ=ψ,since ψ S) = R n f (t) ( R n ψ(τ, ξ)ω (τ t,ξ) e i2π t,ξ dτ dξ) dt = f,φ, φ (t) = R n ψ (τ, ξ) ω (τ t,ξ)e i2π t,ξ dτ dξ = R n (ψ (Iω)) (t, ξ) e i2π t,ξ dξ = F 1 2 (ψ (Iω)) (t) S (Rn ). Thus, (S ω f)(τ, ξ) S (R 2n ). (32) (33) Theorem 15. If ω S (R 2n ),thens ω maps S(R n ) into S (R 2n ). [2] S. Ventosa, C. Simon, M. Schimmel, J. J. Danobeitia, and A. Manuel, The S-transform from a wavelet point of view, IEEE Transactions on Signal Processing, vol.56,no.7,pp , [3] S. K. Singh, The S-Transform on spaces of type S, Integral Transforms and Special Functions, vol.23,no.7,pp , [4] S. K. Singh, The S-Transform on spaces of type W, Integral Transforms and Special Functions, vol. 23, no. 12, pp , [5] S.K.Singh, ThefractionalS-Transformoftemperedultradistibutions, Investigations in Mathematical Sciences,vol.2,no.2, pp , [6] S. K. Singh, The fractional S-Transform on spaces of type S, JournalofMathematics,vol.2013,ArticleID105848,9pages, [7] S. K. Singh, The fractional S-Transform on spaces of type W, Pseudo-Differential Operators and Applications,vol.4, no. 2, pp , [8] S. K. Singh, A new integral transform: theory part, Investigations in Mathematical Sciences,vol.3,no.1,pp ,2013. [9] K. Grochenig, Foundations of Time-Frequency Analysis, Birkhäauser, Boston, Mass, USA, [10] S. Zaidman, Distributions and Pseudo-Differential Operators, Logman, Essex, UK, [11] R. S. Pathak, ACourseinDistributionTheoryandApplications, Narosa Publishing House, New Delhi, India, Proof. If φ S(R n ) and ψ S(R 2n ),then U φ,ψ (τ, ξ) := F (φ) (α+ξ)ψ (α, ξ) S (R 2n ). (34) Thus for any ω S (R 2n ),wehave (F 1 ω) (α, ξ),u φ,ψ (α, ξ) = R n (F 1 ω) (α, ξ) (Fφ) (α+ξ) ψ (α, ξ) dα dξ = R n (F 1 (S ω φ)) (α, ξ) ψ (α, ξ) dα dξ (35) = (F 1 (S ω φ)) (α, ξ),ψ(α, ξ). Thus F 1 (S ω φ) S (R 2n ) and hence (S ω φ) S (R 2n ). 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. References [1] R. G. Stockwell, L. Mansinha, and R. P. Lowe, Localization of the complex spectrum: the S transform, IEEE Transactions on Signal Processing, vol. 44, no. 4, pp , 1996.

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