Fractional S-Transform for Boehmians

Transcription

1 J. Ana. Num. Theor. 3, No., (015) 103 Journal of Analysis & Number Theory An International Journal Fractional S-Transform for Boehmians Abhishek Singh DST-Centre for Interdisciplinary Mathematical Sciences, Faculty of Science, Banaras Hindu University, Varanasi , India eceived: 7 Feb. 015, evised: 30 Mar. 015, Accepted: 31 Mar. 015 Published online: 1 Jul. 015 Abstract: In this paper we extend the results of ultradistributions for the fractional S-transform given by Singh [18] to the Boehmian spaces namely, tempered Boehmian and ultraboehmian spaces. Keywords: Fractional Fourier transform, fractional S-transform, S-transform, wavelet transform, ultradistributions, tempered distribution, tempered Boehmian, ultraboehmian. Subject Classifications: 46E15, 46E40, 46F1 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. [19] 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 [4, 0, 1]. The one-dimensional continuous S-transform of u(t) is defined as [0] (Su)(τ, f)s(u(t))(τ, f) u(t)ω(τ t, f)e iπ ft dt, (1) where the window ω is assumed to satisfy the following: ω(t, f)dt 1 for all f \{0}. () The most usual window ω is the Gaussian one ω(t, f) f k f t π e k, 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, Equation (1) can be rewritten as a convolution (Su)(τ, f) ( u( )e iπ f ω(, f) ) (τ). (4) Applying the convolution property for the Fourier transform, we obtain (Su)(τ, f)f 1 {û( + f) ˆω(, f)}(τ), (5) where F 1 is the inverse Fourier transform. For the Gaussian window case (3), F{ω(t, f)}(α, f) ˆω(α, f)e (πkα/ f). (6) Thus we can write the S-transform in the following form: (Su)(τ, f) û(α+ f)e (πkα/ f) e iπατ dα. (7) Also, if û( f) and (Su)(τ, f) are the Fourier transform and S-transform of u respectively, then û( f) (Su)(τ, f)dτ, (8) so that ( ) u(t)f 1 (t). (9) (Su)(τ, )dτ Some basic properties of S-transform can be found in [16, 17]. Fractional Fourier transform The fractional Fourier transform(fft) has played an important role in signal processing. The a th order FFT Corresponding author mathdras@gmail.com

2 104 A. Singh: Fractional S-Transform for Boehmians of a signal u(t) is defined as [1]: Fu( a f) u(t)k a (t, f)dt, (10) where the transform kernel K a (t, f) is defined as A θ e iπ( f cotθ ft cscθ+t cotθ), if θ nπ K a (t, f) δ(t f), if θ nπ δ(t+ f), if θ+ π nπ, (11) where A θ 1 icotθ, θ aπ/, a [0,4), i is the complex unit, n is an integer, and f is the fractional Fourier frequency(fffr). The inverse FFT of equation (10) is : u(t) Fu a ( f)k a(t, f)d f. (1) We can write (10) as Fu( a f) u(t)a θ e iπ( f cotθ ft cscθ+t cotθ) dt A θ e iπ f cotθ u(t)e iπ ft cscθ e iπt cotθ dt A θ e iπ f cotθ [e iπt cotθ u(t)]( f cscθ). eplacing u(t) by e iπt cotθ φ(t), we have (13) F a u( f) A θ e iπ f cotθ [φ(t)]( f cscθ). (14) Put f ξ sinθ, we have F a u (ξ sin θ)a θ e iπξ sinθ/ ˆφ(ξ). (15) ˆφ(ξ) 1 e iπξ sinθ/ Fu a (ξ sinθ), (16) A θ where u(t)e iπt cotθ φ(t). 3 The fractional S-transform The fractional S-transform(FST) is a generalization of the S-transform. The a th order continuous fractional S-transform of u(t) is defined as [5]: FST a u (τ, f) where the window g is: u(t)g(τ t, f)k a (t, f)dt, (17) g(t, f) f cscθ p k π e t ( f cscθ)p /k ; k, p>0, (18) and satisfy the condition: g(t, f)dt 1 for all f \{0}. (19) Inverse fractional S-transform is defined by [ ] u(t) FSTu(τ, a f)dτ K a (t, f)d f. (0) 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 FST with θ π corresponds to the S-transform, and an FST with θ 0 corresponds to the zero operator. The parameters p and k can be used to adjust the window function space. Let and Since H a (τ, f, f 1 ) h(t,τ, f)g(τ t, f)k a (t, f), (1) h(t,τ, f)k a (t, f 1 ) g(τ t, f)k a (t, f)k a (t, f 1 )dt. () K a (t, f)k a (t, f 1 )A θ A θ e iπ[( f f 1 )cotθ ( f f 1)t cscθ]. (3) By using (18) and (3) in () we obtain H a (τ, f, f 1 ) f cscθ p k π e (τ t) ( f cscθ)p /k Aθ A θ e iπ[( f f 1 )cotθ ( f f 1)t cscθ] dt f cscθ p k π A θ A θ e iπ[( f f 1 )cotθ] e (τ t) ( f cscθ) p /k e iπ( f f 1)t cscθ dt. (4) By using the technique of [5], we obtain H a (τ, f, f 1 )A θ A θ e iπ[( f f 1 )cotθ] e iπ( f f 1)τ cscθ e π k ( f f 1 ) (cscθ) /( f cscθ) p, and by using (3) we can write (5) H a (τ, f, f 1 )e [π k ( f f 1 ) (csc θ) /( f cscθ) p ] Ka (t, f)k a (t, f 1 ). (6) Also, the FST can be also defined as operations on the fractional Fourier domain FSTu(τ, a f) [ ] Fu( a f)k a (t, f)d f g(τ t, f)k a (t, f)dt F a u( f 1 )H a (τ, f, f 1 )d f 1. (7)

3 J. Ana. Num. Theor. 3, No., (015) / By using (13) and (5) we can write (7 ) as FSTu(τ, a f) A θ e iπ f 1 cotθ [e iπt cotθ u(t)]( f 1 cscθ)a θ A θ e iπ[( f f1 )cotθ] e iπ( f f 1)τ cscθ e π k ( f f 1 ) (cscθ) /( f csc θ) p d f 1 A θ A θ [e iπt cotθ u(t)]( f 1 cscθ)e iπ[( f f1 )cotθ] e iπ f 1 cotθ e iπ( f f 1)τ cscθ e π k ( f f 1 ) (cscθ) /( f csc θ) p d f 1 A θ A θ e iπ f cotθ [e iπt cotθ u(t)]( f 1 cscθ) e iπ( f f 1)τ cscθ e π k ( f f 1 ) (cscθ) /( f csc θ) p d f 1. (8) The S-transform has been studied on the spaces of type S and spaces of tempered ultradistributions by S. K. Singh [16]. Pathak and Singh [11] have studied the wavelet transform of Tempered Ultradistributions. The fractional S-transform on ultradistribution space is studied by Singh [18], which is, in this paper, extended to tempered and ultra Boehmians. 4 Tempered Boehmians for the fractional S-transform The concept of Boehmian is motivated by the so called egular operators, introduced by Boehme [3]. egular operators form a subalgebra of the field of Mikusinski operators and hence they include only such functions whose support is bounded from the left. Boehmians contain all regular operators, all distributions and some objects which are neither operators nor distributions, they have an algebraic character of Mikusinski operators, and at the same time do not have any restrictions on the support, and they contain all Schwartz distributions, oumieu ultradistributions, regular operators and tempered distribution. Mikusinski [9] introduced the space of tempered Boehmians, and its Fourier transform is defined as a classical distribution. In another paper [10], he enlarged the space of tempered Boehmians, by introducing larger class of delta sequence, which is identified with the space of ultradistribution. Tempered Boehmians : The pair of sequence ( f n,ϕ n ) is called a quotient of sequence, denoted by f n /ϕ n, whose numerator belongs to some set G and the denominator is a delta sequence such that f n ϕ m f m ϕ n, n,m N. (9) Two quotients of sequence f n /ϕ n and g n /ψ n are said to be equivalent if f n ψ n g n ϕ n, n N. (30) The equivalence classes are called the Boehmians. The space of Boehmians is denoted by B, an element of which is written as x f n /ϕ n. Application of construction of Boehmians to function spaces with the convolution product yields various spaces of generalized functions. The spaces, so obtained, contain the standard spaces of generalized functions defined as dual spaces. For example, if G C( N ) and a delta sequence defined as sequence of functions ϕ n D such that (i) ϕ n dx1, a N (ii) ϕ n dx C, for some constant C and n N, (iii) supp ϕ n (x) 0, as, then the space of Boehmian that is obtained, contains properly the space of Schwartz distributions. Similarly, this space of Boehmians also contains properly the space of tempered distributions S, when G is the space of slowly increasing functions with delta sequence. The fractional S-transform of tempered Boehmian form a proper subspace of Schwartz distribution D. Since D is dense in S there cannot more than one element to S (dual of S) having the same restriction to D. Therefore, this type of correspondence between S and D is one to one and so we express it by saying that S D. Boehmian space have two types of convergence, namely, the δ- and - convergences, which are stated as: (i) A sequence of Boehmians (x n ) in the Boehmian space B is said to be δ- convergent to a Boehmian x in B, which is denoted by x n δ x if there exists a delta sequence (δ n ) such that (x n δ n ),(x δ n ) G, n N and (x n δ k ) (x δ k ) as in G, k N. (ii) A sequence of Boehmians (x n ) in B is said to be - convergent to a Boehmian x in B, denoted by x n δ x if there exists a delta sequence (δ n ) such that (x n x) δ n G, n N and(x n x) δ n 0 as in G. For details of the properties and convergence of Boehmians one can refer to [7]. We have employed following notations and definitions. A complex valued infinitely differentiable function f, defined on N, is called rapidly decreasing, if ( sup sup 1+x 1 + x + + ) m D x N α f(x) <, x N α m for every non-negative integer m. Here α α α N, and D α α x α x α. 1 1 xα N N The space of all rapidly decreasing functions on N is denoted by S. The delta sequence, i.e., sequence of real valued functions ϕ 1,ϕ,... S, is such that (i) ϕ n dx1, a N (ii) ϕ n dx C, for some constant C and n N, (iii) lim x ε x k ϕ n dx0, for every k N,ε > 0.

4 106 A. Singh: Fractional S-Transform for Boehmians If ϕ S and ϕ 1, then the sequence of functions ϕ n is a delta sequence. A complex-valued function f on N is called slowly increasing if there exists a polynomial p on N such that f(x)/p(x) is bounded. The space of all slowly increasing continuous functions on N is denoted by I. If f n I, {ϕ n } is a delta sequence under usual notion, then the space of equivalence classes of quotients of sequence will be denoted by B I, elements of which will be called tempered Boehmians. For F [ f n /ϕ n ] B I, define D α F [( f n D α ϕ n )/(ϕ n ϕ n )]. If F is a Boehmian corresponding to differentiable function, then D α F B I. If F [ f n /ϕ n ] B I and f n S, for all n N, then F is called a rapidly decreasing Boehmian. The space of all rapidly decreasing Boehmian is denoted by B S. If F [ f n /ϕ n ] B I and G[g n /ψ n ] B S,then the convolution is F G[( f n g n )/(ϕ n ψ n )] B I. The convolution quotient is denoted by f/ϕ and f ϕ denotes a usual quotient. Let f I. Then the fractional S-transformation of f, denoted as f, is defined for distribution spaces of slowly increasing function f in the following form: f,ϕ f, ϕ, ϕ S(). Moreover, as we know that the Fourier transform of the convolution of two functions is the product of their Fourier transform. Whereas, in the present case, when we consider this condition for fractional S-transform, it does not seem to be as nice or as practical. Actually, the convolution operation defined by ( f g)(x) f(t)g(x t)dt (31) is not the right sort of convolution for the fractional S-transform. Therefore, we define fractional S convolution in terms of basic function [1, p.119]: ( f g)(z) D(x, y, z) f(x)g(y)dy, (3) usually, translation τ is defined by (τ y f)(x) f (x,y) D(x,y,z) f(z)dz. (33) We assume D(x,y,z) 1 for fractional S-transform and define ( f g) f g. (34) Theorem 1. If [ f n /ϕ n ] B I, then the sequence { f n }, n 1,,..., converges in D. Moreover, if [ f n /ϕ n ] [g n /ψ n ],then { f n } and { g n } converges to the same limit for the fractional S-transform of tempered Boehmians. The proof is similar to that of Theorem 1; see [9]. Definition 1. By the fractional S-transform F of a tempered Boehmian F [ f n /ϕ n ] β I we mean the limit of the sequence { f n } is in D. The fractional S-transform is thus a mapping from β I to D, which is a linear mapping. Theorem. Let F [ f n /ϕ n ] B I and G[g n /ψ n ] B S. Then (i)( G) is an infinitely differentiable function (ii)[f G] F G and (iii) ( F) ( ϕ n )( f n ), n N Proof. (i) Let G[g n /γ n ] B S and let U be the bounded open subset of. Then there exists n N such that{ γ n }> 0 on U. We have, thus [ G] { g n } { g n }{ γ p } { γ p } { g p }{ γ n } { γ p } { g p} { γ p } { g p} { γ p } lim { γ n} on U. Since { g p },{ γ p } S and { γ p }>0 on U, thus { G} is an infinitely differentiable function on U. (ii) Let F [ f n /ϕ n ] B I and G [g n /γ n ] B S. If ϕ D, then there exists p N such that { γ p }>0 on the support of ϕ. We have (F G) {ϕ} ( f n g n ) (ϕ) {( f n g n )}(ϕ) ( f n )( g n,ϕ) { } gn γ p ( f n ) ϕ, n, p N γ p { } gp γ n ( f n ) ϕ γ p { } gp ( f n ) ϕ( γ n ) γ p ( f n ){( G)ϕ( γ n )} ; ( G) lim {( f n )( γ n )}(ϕ) ( G) lim ( f n γ n ) (ϕ) ( F G)(ϕ). from (i) The last equality follows from the fact that [ f n /ϕ n ] [( f n ϕ n )/(ϕ n γ n )]. (iii) Let ϕ D. Then

5 J. Ana. Num. Theor. 3, No., (015) / F ϕ p,ϕ F, ϕ p ϕ, p N f n, ϕ p ϕ f n ϕ p,ϕ f p ϕ n,ϕ f p, ϕ n ϕ i.e. f p,ϕ Hence F ϕ p f p. Thus, the fractional S-transform of an arbitrary distribution can be defined as a tempered Boehmian. The theorem is, therefore, completely proved. Theorem 3. A distribution f is the fractional S-transform of tempered Boehmian if and only if there exists a delta sequence {δ n } such that { f( δ n )} is a tempered distribution for every n N. Proof. Let F [ f n /ϕ n ] B I and f { F}. Then f{ ϕ n } { F}{ ϕ n }. Thus, f( ϕ n ) is a tempered distribution. Now let f D, and (δ n ) be a delta sequence such that f( δ n ) is tempered distribution for every n N. We define [ { f( F δ ] n )} δ n (35) (δ n δ n ) where { f( δ n )} is the inverse fractional S-transform of { f( δ n )}. Since { f( δ n )} is a tempered distribution, therefore,{ f( δ n )} is also a tempered distribution. 5 UltraBoehmians for the fractional S-transform Consider a complex valued infinitely differentiable function f, defined on n, which satisfies z k f(z) C k e a y, z, Im(z)y, (36) where k is a non-negative integer. Such a space of all entire function over the complex z-plane is denoted by Z and the delta sequence is such that and (i) ϕ n 1 (ii) z k ϕ n (z) C k e a y, k N. Considering G as Z, which satisfies (9) and (30) and the properties of the delta sequence, is called the ultraboehmian. It is denoted by B Z, where f n Z. For [ f n /ϕ n ] B Z, and f n Z, n N, F is called the space of entire function of Boehmians, and is denoted by B Z. If [ f n /ϕ n ] B Z and G[g n /ψ n ] B Z, then the convolution, given by [ ] ( fn g n ) F G B (ϕ n ψ n ) Z, (37) is well defined. Theorem 4. If [ f n /ϕ n ] B Z then the fractional S-transform converges in D. Moreover, if [ f n /ϕ n ] [g n /ψ n ] B Z, then the fractional S-transform converges to the same limit of ultraboehmians. Proof. Let ϕ D (testing function space) and k N be such that ϕ k > 0 on the support of ϕ. Since f n ϕ m f m ϕ n, m,n N, we have f n ϕ m f m ϕ n. Thus, f n,ϕ f n, ϕ ϕ k ϕ k f n ϕ k, ϕ ϕ k i.e. { } ϕ ϕn ϕ k f k ϕ n, ϕ ϕ k f k, ϕ ϕ n. ϕ k Since the sequence converges to ϕ ϕ k in D, the sequence { f n,ϕ} converges in D. This proves that the sequence { f n } converges in D (dual of space D). Now, we consider[ f n /ϕ n ][g n /γ n ] B I, and define and f n+1 γ n+1,if n is odd h n. g n ϕ n,if n is even ϕ n+1 γ n+1,if n is odd δ n ϕ n γ n,if n is even. Then [h n /δ n ][ f n /ϕ n ] [g n /γ n ], and the sequence { h n } converges in D. Moreover, h n 1,ϕ ( fn γ n ),ϕ i.e lim f n γ n,ϕ f n, γ n ϕ f n,ϕ. Thus, this proves that sequences { f n } and { h n } have the same limit. Similarly, it can be shown that the sequences { h n } and{ g n } will have the same limit.

6 108 A. Singh: Fractional S-Transform for Boehmians emark.(1) For the classical Fourier transform, Z S S Z [1, p. 01], and owing to this fact, the elements of ultraboehmians will be contained in the tempered Boehmians. Therefore, the fractional S-transform of ultraboehmians and tempered Boehmians, discussed in this paper, shows that B Z B S B I B Z []. () Integrable Boehmians of fractional S-transform : Let G be the space of complex-valued Lebesgue integrable functions on real line and δ be the delta sequence. Then the equivalence class of quotients is called the integrable Boehmians, space of which is denoted by B L1. One may refer to Mikusinski [8], where Fourier transform for integrable Boehmian is investigated. By using the relation between fraction Fourier transform, fractional S-transform given, one can define and find fractional S-transform for integrable Boehmians. Acknowledgement This work is partially supported by the University Grants Commission, Govt. of India under the Dr. D.S. Kothari Post-Doctoral Fellowship, Sanction No. F.4-/006(BS)/13-663/01(BS). The author is grateful to the anonymous referee for their valuable comments and suggestions. [1]. S. Pathak, The Wavelet Transform, Atlantis Press/World Scientific, Amsterdam, Paris, 009 [13] Abhishek Singh, D. Loonker, and P.K. Banerji, Fourier- Bessel transform for tempered Boehmians, Int. Journal of Math. Analysis 4 (45), (010). [14] Abhishek Singh and P.K. Banerji, Fractional integrals for fractional Fourier transform of integrable Boehmains. (Submitted) [15] Abhishek Singh and S. K. Singh, Integral transform for ultradistributions and their extensions to Boehmians, Integration: Theory and applications, (To appear). [16] S. K. Singh, The S-Transform on spaces of type S, Integ. Trans. Spl. Funct. 3 (7), (01). [17] S. K. Singh, The S-Transform on spaces of type W, Integ. Trans. Spl. Funct. 3 (1), (01). [18] S. K. Singh, The fractional S-Transform for tempered ultradistributions, Invest. Math. Sci. (), (01). [19]. G. Stockwell, L. Mansinha and. P. Lowe, Localization of the complex spectrum: The S transform, IEEE Trans. Signal Process. 44 (4), (1996). [0] S. Ventosa, C. Simon, M. Schimmel, J. Dañobeitia,and A. Mànuel, The S-transform from a wavelet point of view, IEEE Trans. Signal Process., 56 (07), (008). [1] A. H. Zemanian, Generalized Integral Transformations, Interscience Publishers, New York, eferences [1] L. B. Almeida, The fractional Fourier transform and timefrequency representations, IEEE Trans. Signal Process 4 (11), (1994). [] S.K.Q. Al-Omari, D. Loonker, P.K. Banerji and S.L. Kalla, Fourier sine (cosine) transform for ultradistributions and their extensions to tempered and ultraboehmian spaces, Integ. Trans. Spl. Funct. 19 (6), (008). [3] T. K. Boehme, Support of Mikusinski operators, Trans. Amer. Math. Soc. 176, (1973). [4] C. K. Chui, An Introduction to Wavelets, Academic Press, New York, 199. [5] D. P. Xu and K. Guo, Fractional S transform -Part 1:Theory, Applied Geophysics 9 (1), (01). [6] I. M. Gel fand and G. E. Shilov, Generalized Functions, Volume, Academic Press, New York, [7] P. Mikusinski, Convergence of Boehmains, Japan. J. Math. 9 (1), (1983). [8] P. Mikusiński, Fourier transform for integrable Boehmians, ocky Mountain J. Math. 17, (1987). [9] P. Mikusinski, The Fourier transform of tempered Bohemians, Fourier Analysis, Lecture Notes in Pure and Applied Math., Marcel Dekker, New York, (1994). [10] P. Mikusinski, Tempered Boehmians and Ultradistributions, Proc. Amer. Math. Soc. 13 (3), (1995). [11]. S. Pathak and S. K. Singh, The Wavelet Transform on spaces of type S, Proceedings of the oyal Society of Edinburgh, 136A, (006).