Product of two generalized pseudo-differential operators involving fractional Fourier transform

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1 J. Pseudo-Diffe. Ope. Appl : DOI /s Poduct of two genealized pseudo-diffeential opeatos involving factional Fouie tansfom Akhilesh Pasad Manish Kuma eceived: 21 Febuay 2011 / evised: 14 Apil 2011 / Accepted: 20 Apil 2011 / Published online: 6 May 2011 Spinge Basel AG 2011 Abstact A genealized pseudo-diffeential opeato involving factional Fouie tansfom associated with symbol ax, y is defined. The poduct of two genealized pseudo-diffeential opeatos is shown to be a genealized pseudo-diffeential opeato. Keywods Genealized pseudo-diffeential opeato Fouie tansfom Factional Fouie tansfom Mathematics Subject Classification 2000 Pimay 47G30; Seconday 46F12 1 Intoduction The Fouie tansfom of a function φ L 1, is defined by φy = Fφy = 1 e ixy φx, y, 1.1 The authos ae thankful to Pof.. S. Pathak fo his suggestions which helped in impoving the pape. This wok has been suppoted by CSI New Delhi, Govt. of India, unde Gant No. F. No. 09/ /2010-EM-I. A. Pasad M. Kuma B Depatment of Applied Mathematics, Indian School of Mines, Dhanbad , India manish.math.bhu@gmail.com A. Pasad ap_bhu@yahoo.com

2 356 A. Pasad, M. Kuma and if φ L 1, then the invese Fouie tansfom is given by φx = F 1 φx = 1 e ixy φydy, x. 1.2 Factional Fouie tansfom is a otation opeation on the time-fequency distibution and it can tansfom a function eithe in the time domain o fequency domain into the domain between time and fequency. The factional Fouie tansfom is a genealization of the odinay Fouie tansfom with a paamete α, has many applications in seveal aeas, including signal pocessing, optics and quantum physics [1]. The one dimensional factional Fouie tansfom [5,6] with paamete α of φx denoted by F α φξ = φˆ α ξ is given by whee the kenel F α φξ = φˆ α ξ = K α x,ξφx, 1.3 { C K α x,ξ= α e ix2 +ξ 2 cot α 2 ixξ csc α if α = nπ 1 e ixξ if α = π 2 whee n is an intege and C α = i sin α 1 2 e iα 2 = 1 i cot α and its invesion fomula: φx = 1 K α x,ξf α φξdξ, 1.4 whee K α x,ξ = C α e ix2 +ξ 2 cot α 2 +ixξ csc α and C α = i sin α 1 2 sinα e iα 2 = 1 + i cot α. Definition 1.1 A tempeed distibution φ belongs to the Sobolev space H s, and s, if its factional Fouie tansfom F α φ coesponding to a locally integable

3 Pseudo-diffeential opeato 357 function F α φξ ove such that φ H s = 1 + ξ 2 s/2 F α φξ 2 dξ This space is complete with espect to the nom φ H s. 1/2 < Popeties of factional Fouie tansfom Fist we ecall the definition of the Schwatz space S. Definition 2.1 The space S, the so-called space of smooth functions of apid descent, is defined as follows: φ is membe of S iff it is a complex valued C -function on and fo evey choice of β and γ of non-negative integes, it satisfies Ɣ β,γ φ = sup x β D γ φx <. 2.1 x Poposition 2.1 Let K α x,ξbe the kenel of factional Fouie tansfom and x = d ix cot α, then x K αx,ξ= iξ csc α K α x,ξ, N 0. Poof so that d K d αx, y = C α e ix2 +ξ 2 cot α 2 ixξ csc α = K α x,ξix cot α ξ csc α; d ix cot α K α x,ξ= iξ csc αk α x,ξ. Continuing in this way, we get d ix cot α K α x,ξ= iξ csc α K α x,ξ. Thus, we have the desied esult. Poposition 2.2 Fo all φ S, we have x K αx,ξφx = K α x, ξ x φx, N 0,

4 358 A. Pasad, M. Kuma whee x = d + ix cot α. Poof At fist we pove x K α x,ξφx = K α x, ξ xφx. Using integation by pats, we have d ix cot α K α x,ξφx d = K α x,ξ + ix cot α φx. Theefoe, x K α x,ξφx = K α x, ξ xφx. In geneal, we have x K αx,ξφx = K α x, ξ x φx. Poposition 2.3 Let φ S, then F α x φxξ = iξ csc α F α φxξ, N 0. Poof Using Poposition 2.2 and Poposition 2.1, wehave F α x φxξ = K α x, ξ x φx = x K αx,ξφx = iξ csc α K α x,ξφx = iξ csc α F α φxξ.

5 Pseudo-diffeential opeato Poduct of two genealized pseudo-diffeential opeatos We have aleady defined the Sobolev space H s by 1.5. In the following we shall make use of vaiant of this, denoted by H s, whee s, defined as follows [2]: A tempeed distibution φ S 2 is said to belong to H s,iff α φξ, η is locally integable on 2 and 1 + ξ η 2 s/2 F α φξ, η L 2 2. A nom in this space is defined by φ H s = 1 + ξ 2 s/2 1 + η 2 s/2 F α φξ, η 2 dξdη 1/2 <, φ S. 3.1 Definition 3.1 The function ax, y : C C belongs to class S m if and only if q, ν,β N 0, thee exist D β,ν,m > 0 such that 1 + x q D β x D ν y ax, y Dβ,ν,m 1 + y m ν. 3.2 Lemma 3.1 Fo any symbol a S m, m and l > 1 N, thee exists a positive constant C m such that Poof we know that so that F α aξ, η C m 1 + η m 1 + ξ 2 csc 2 α l F α aξ, η = C α 1 + iξ csc α l F α aξ, η = ξ,η, l > 1 N. Now, 1 x l ax,η = = = l l l e ix2 +ξ 2 cot α 2 ixξ csc α ax,η, K α x,ξ1 x l ax,η, 3.4 l 1 x ax,η l 1 l 1 Px, kdx k ax,η k=0 k=0 l 1 =0 k d l1 x l 1 Dx k ax,η,

6 360 A. Pasad, M. Kuma whee Px, k is a polynomial of maximum degee. Using3.2 in the above, we have 1 x l ax,η l l Hence by 3.4 and 3.5, we get k=0 l 1 =0 k d l1 D k,m 1 + η m 1 + x l. 3.5 l 1 + iξ csc α l l k F α aξ, η C α d l1 k=0 l 1 =0 D k,m 1 + η m 1 + x l. Since x integal is convegent fo lage value of l, thee exists a constant C m > 0 depending on l, α, k,, m such that F α aξ, η C m 1 + η m 1 + ξ 2 csc 2 α l 2. A linea patial diffeential opeato Ax, x on is given by Ax, x = m a x x, 3.6 whee the coefficient a x ae functions defined on and x = d + ix cot α. If we eplace x in 3.6 by monomial iξ csc α in, then we obtain the so called symbol Ax,ξ= m a x iξ csc α. 3.7 In ode to get anothe epesentation of the opeato Ax, x, let us take any function φ S, then by 1.3, 1.4 and Poposition 2.3, wehave Ax, xφx = = m m = 1 a xfα 1 F α x φx a xfα 1 iξ csc α F α φxξ K α x,ξax,ξf α φξdξ,

7 Pseudo-diffeential opeato 361 whee K α x,ξ as 1.4. If we eplace the symbol Ax,ξ by moe geneal symbol ax,ξwhich is no longe polynomial in ξ, we get the genealized pseudo-diffeential opeato A a,α defined below. Fo pseudo-diffeential opeato involving Fouie tansfom we may efe to [3,4]. Definition 3.2 Let ax, y be a complex valued function belonging to the space C, and let its deivatives satisfies cetain gowth conditions such as 3.2. Then the genealized pseudo-diffeential opeato A a,α associated with the symbol ax, y is defined by A a,α φx = 1 K α x, yax, yf α φydy. 3.8 whee F α φy is defined in 1.3. Theoem 3.1 Fo any symbol a S m, the associated opeato A a,α φx, y admits the epesentation A a,α φx, y = C 2 α e ix2 +ξ 2 +y 2 +η 2 cot α 2 e ixξ+yη csc α F α aξ, ηf α φηdξ dη, whee φ S. 3.9 Poof Using 1.3 and 1.4 in3.8, we have the desied esult. Coollay 3.1 Fo any symbol a S m, the associated opeato F α [A a,α φ]ξ, η admits the epesentation F α [A a,α φ]ξ, η = F α aξ, ηf α φη, 3.10 whee φ S, the factional Fouie tansfom is taken with espect to all the vaiables x and y. Definition 3.3 Let σx,ξ S m 1 and τy,ξ S m 2, then the poduct of two genealized pseudo-diffeential opeatos B τ,α and A σ,α associated with symbol τy,ξ and σx,ξespectively is defined by B τ,α A σ,α φx, y = 1 K α x,ξτy,ξf α [A σ,α φ]ξ, ydξ, 3.11 povided the integal is convegent. Theoem 3.2 Let σx,ξ S m 1 and τy,ξ S m 2, then the poduct of two genealized pseudo-diffeential opeatos B τ,α and A σ,α is again a genealized pseudo-diffeential opeato whose symbol is in S m 1+m 2.

8 362 A. Pasad, M. Kuma Poof Let φ S. Then in view of Definition 3.3 and 3.10, we have B τ,α A σ,α φx, y = 1 K α x,ξτy,ξf α σy,ξ F α φξdξ This show that τy,ξf α σy,ξ is symbol of the poduct of B τ,α A σ,α.now,we show that this symbol is in S m 1+m 2. Fo α 1,β 1 N 0,wehave Now, Dy α 1 D ξ β 1 τy,ξf α σy,ξ α 1 α 2 =0 β1 α1 α 2 β 2 =0 β1 β 2 Dy α 1 α 2 D ξ β 1 β 2 τy,ξ Dy α 2 D ξ β 2 F α σy,ξ D α 2 y Dβ 2 ξ F ασy,ξ = C α Theefoe, D α 2 y D β 2 ξ σy,ξ α 2 = C α α 3 =0 e ix2 +y 2 cot α 2 ixycsc α D α 2 α 3 α y 3 e ix2 +y 2 cot α 2 D α 3 y e ixycsc α D β 2 ξ σy,ξ α 2 = C α e ix2 +y 2 cot α 2 P α α 3 y, i cot α 3 2 α 3 =0 ix csc α α 3 e ixycsc α D β 2 ξ σy,ξ α 2 α3 = C α a α s cot αy s i csc α α 3 3 α 3 =0 s=0 e ix2 +y 2 cot α 2 ixycsc α x α 3 D β 2 ξ σy,ξ. D α 2 y Dβ 2 ξ F α 2 α3 ασy,ξ C α a α s cot α 3 α 3 =0 s=0 y s csc α α 3 x α 3 D β 2 ξ σy,ξ

9 Pseudo-diffeential opeato 363 C α α 2 α 3 =0 D β3,m C α α3 a α s cot α 1 + y s csc α α 3 3 s=0 1 + x α 3 D β 2 ξ σy,ξ α 2 α 3 =0 α3 a α s cot α 1 + y s 3 s=0 csc α α ξ m 1 β x α 3 q. The x integal being convegent fo q >α 3 + 1, we have D α 2 y Dβ 2 ξ F ασy,ξ L1 + y s 1 + ξ m 1 β 2, 3.14 whee L is a positive constant depending on α, α 2,α 3, s,β 2, m and q. By 3.13, 3.14 and inequality 3.2, the ight hand side can be bounded by L α 1 α 2 =0 β1 α1 α 2 β 2 =0 β1 L ξ m 1+m 2 β 1, β 2 C α1 α 2,β 1 β 2,m 2,s1 + ξ m 1+m 2 β 1 whee L 1 is a positive constant depending on α 1,α 2,β 1,β 2, m 2, s and L. We see that the symbol of the poduct of B τ,α A σ,α is in S m 1+m 2. This complete the poof of the Theoem. Theoem 3.3 Let σx,ξ S m 1 and τy,ξ S m 2. Then fo cetain C 1 > 0, m 1, m 2 +, B τ,α A σ,α φx, y 2 H s C 1 φ 2 H s+m 1 +m 2, φ S. Poof By Definition 3.3 and fom 3.12 it follows that B τ,α A σ,α φx, y has the factional Fouie tansfom equal to τy,ξf α σy,ξ. Theefoe, B τ,α A σ,α φx, y 2 H s = 1 + ξ 2 s/2 1 + y 2 s/2 F α B τ,α A σ,α ξ, y = 1 + ξ 2 s/2 1 + y 2 s/2 τy,ξ F α σy,ξf α φξ 2 dy dξ. 2 L 2

10 364 A. Pasad, M. Kuma Since fom 3.2 and fom 3.3 We know that Theefoe, τy,ξ C m2 1 + ξ m y 2 l, F α σy,ξ C m1 1 + ξ m y 2 csc 2 α l/ ξ m 2 m/2 1 + ξ 2 m/2 if m 0, 1 + ξ m 1 + ξ 2 m/2 if m < ξ m max1, 2 m/2 1 + ξ 2 m/2 = C m 1 + ξ 2 m/2, whee C m = max1, 2 m/2. Thus Bτ,α A σ,α φx, y 2 H s C m1 C m2 1 + y 2 s 2l 1 + y 2 csc 2 α l dy C 2 m 1 + ξ 2 s+m 1 +m 2 2 F α φξ 2 dξ, fo convegent of the fist integal of the ight hand side is 1 + y 2 s 2l 1 + y 2 csc 2 α l dy = y y 2 s 2l 1 + y 2 csc 2 α l dy 1 + y 2 s 2l 1 + y 2 csc 2 α l dy = I 1 + I 2. Since I 1 is bounded, we show convegent fo I 2, I 2 = 1 + y 2 s 2l 1 + y 2 csc 2 α l dy.

11 Pseudo-diffeential opeato 365 If s 2l > 0, then 2 s 2l y 2s 4l sin 2 α + y 2 l csc α 2l dy 2 s 2l y 2s 4l y 2l csc α 2l dy = 2 s 2l csc α 2l y 2s 6l dy <. The y-integal is convegent by choosing l > 1+2s 6.Now,ifs 2l < 0, let s 2l = p, sin 2l α 1 + y 2 p sin 2 α + y 2 l dy 1 y 2p dy y2l y 2p+2l dy <. The above integal is convegent by choosing 2p + 2l > 1. Theefoe, Bτ,α A σ,α φx, y 2 H s C ξ 2 s+m 1 +m 2 2 F α φξ 2 dξ C 1 φ 2 H s+m 1 +m 2, whee C 1 is a positive constant. efeences 1. Almeida, L.: The factional Fouie tansfom and time-fequency epesentations. IEEE Tans. Signal Pocess. 4211, Pathak,.S., Pasad, A.: A genealized pseudo-diffeential opeato on Gel fand-shilov space and Sobolev space. Indian J. Pue Appl. Math. 374, Wong, M.W.: An intoduction to pseudo-diffeential opeatos, 2nd edn. Wold Scientific, Singapoe Zaidman, S.: Distibutions and pseudo-diffeential opeatos. Longman, Essex, England Zayed, A.I.: A convolution and poduct theoem fo the factional Fouie tansfom. IEEE Signal Pocess. Lett. 54, Zayed, A.I.: Factional Fouie tansfom of genealized functions. Integal Tansfoms Spec. Funct. 73 4,