Pseudo-Differential Operators Involving Fractional Fourier Cosine (Sine) Transform

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1 ilomat 31:6 17, DOI 1.98/IL176791P Publishe b ault of Sienes an Mathematis, Universit of Niš, Serbia Available at: Pseuo-Differential Operators Involving rational ourier Cosine Sine Transform Akhilesh Prasa a, Manoj Kumar Singh b a Department of Applie Mathematis, Inian Institute of Tehnolog Inian Shool of Mines, Dhanba, Jharkhan-864, Inia b Department of Mathematis, St. Xavier s College, Ranhi Universit, Ranhi-8341, Inia Abstrat. A brief introution to the frational ourier osine transform as well as frational ourier sine transform an their basi properties are given. rational ourier osine frational ourier sine transform of tempere istributions is stuie. Pseuo-ifferential operators involving these transformations are investigate an isusse the ontinuit on ertain spaes S e an S o. 1. Introution The frational ourier osine sine transform is a generalization of orinar ourier osine sine transform with an angle, has man appliations in the several areas inluing Phsis, Signal Proessing, Mathematial Analsis an other fiels [3, 8]. Motivate b the above works, we have efine one imensional frational ourier osine frational ourier sine transform for < < π as follows: Definition 1.1. The frational ourier osine transform of a funtion f L 1 R + ; R +, is efine as fˆ f x K x, f xx, 1 where K x, C e ix + ot / osx s nπ, π os x, π, δx nπ, n Z, where C 1 i ot. π 3 1 Mathematis Subjet Classifiation. Primar 461, 43A3; Seonar 35S5 Kewors. Pseuo-ifferential operators, ourier osine transform, ourier sine transform, Shwartz spae. Reeive: 1 April 15; Aepte: 1 June 15 Communiate b Dragan S. Djorjević The first author of this work is supporte b NBHMDAE projet, Govt. of Inia, uner grant No. /4814/11/-R&D II/351. aresses: apr bhu@ahoo.om Akhilesh Prasa, manojksingh88@gmail.om Manoj Kumar Singh

2 A. Prasa, M. K. Singh / ilomat 31:6 17, The orresponing inverse frational ourier osine transform is given b 1 ˆ f x f x K x, fˆ, 4 an where K x, C e ix + ot / osx s K x,, 1 + i ot C C. π Similarl, we efine frational ourier sine transform as: Definition 1.. If f L 1 R +, then its frational ourier sine transform is efine as where fˆ s f x s K s x, K s x, f xx, 6 e i π/ C e ix + ot / sinx s nπ, π sin x, π, δx n Z. The orresponing inverse frational ourier sine transform is given b 1 ˆ s f s x f x K s x, fˆ s, 8 where K s x, Ks x,, an C is given in 3. Definition 1.3. [1, 4] The spae S e S o is the subset of all even o funtions in S Shwartz spae. Thus φ S e S o funtion an it satisfies γ β, φ sup x R + x β D xφx <, β, N. 9 If f is of polnomial growth an is loall integrable funtion on R +, then it generates a istribution in S as follows: f, φ f xφxx ; φ SR +. 1 Lemma 1.4. [5] A funtion φ C R + satisfies 9 if an onl if τ m,β φ sup 1 + x m/ D β xφx <, m, β N. 11 x R + The frational ourier osine an frational ourier sine transforms are powerful tools in Mathematial Analsis, Phsis, Signal Proessing et. Man funamental results of these transforms are alrea known, but appliations in Pure Mathematis are still missing. In this orresponene, the ontinuit of above sai transforms an the transforms of tempere istributions have been investigate. Moreover, we obtaine ifferential operators x an x an also efine the pseuo-ifferential operators. Then, we isusse the ontinuit of pseuo-ifferential operators on spaes S e an S o. 5 7

3 A. Prasa, M. K. Singh / ilomat 31:6 17, Properties of rational ourier Cosine Sine Transform Theorem.1. Let K x, be the kernel of frational ourier osine transform then an x ix ot x x x ot i ot, 1 x K x, s K x,. 13 Proof: Sine K x, C e ix + ot / osx s, then K x x, C e ix + ot / [ ix ot osx s sinx s s ]. 14 Rearranging 14, we obtain sinx s 1 [ ] e ix + ot / s 1 ix ot K x, C x K x,. 15 Differentiating 14 with respet to x an using 15, we have Hene [ Therefore, x K x, x ot K x, + i ot K x, s K x, +ix ot x K x, + x ot K x,. ix ot x x x ot i ot x K x, s K x,. The above theorem an be generalize as follows: ] K x, s K x,. Remark.. Let r N an K x, be the kernel of frational ourier osine transform, then r xk x, s r K x,. 16 Theorem.3. or all φ S e R +, we have x K x,, φx K x,, x φx, 17 where x is onjugate omplex of x. Proof: Using integration b parts, we have x K x,, φx x K x, φxx x K x, φxx ot K x, K x, x φx x i ot K x, x φxx K x,, x φx. The above theorem an be generalize as follows: x K x, ix ot φxx K x + ix ot x x ot + i ot x, φxx φxx

4 A. Prasa, M. K. Singh / ilomat 31:6 17, Remark.4. If φ S e, then r x K x,, φx K x,, r x φx, 18 where r N an x is given b 1 an x is onjugate omplex of x. Remark.5. Similar results of above Theorems an Remarks of this Setion an be foun for kernel K s x,. Theorem.6. Let φ S e R +, then x r φx s r φ. 19 Proof: B 1, we have r x φx K x, r x φxx r φx K x,, x x r K x,, φx s r K x,, φx s r φ. Theorem.7. If φ S e R +, then r φ [ x s r ] φx. Proof: Using Remark., we have r φ r K x, φxx r K x, φxx x s r K x, φxx [ x s r ] φx. Definition.8. The test funtion spae S is efine as follows: an infinitel ifferential omplex value funtion φ on R + belongs to S if an onl if for ever hoie of β an of non-negative integers, it satisfies Γ x β, φ sup β xφx <, x R + 1 where x is given in 1. Theorem.9. The mapping : S e R + S R + is linear an ontinuous. Proof: Linearit of is obvious. Let β, be an two non-negative integers an {φ n } n N S e R +. Using, we have sup β φ n sup β x s φn. R + R + Sine φ n S e R +, x s φn S e R + x s φn Se R +.

5 A. Prasa, M. K. Singh / ilomat 31:6 17, Hene sup β φ n if φ n in S as n, R + whih implies the ontinuit of. Similar results an be foun of above Theorems.6-.9 for frational ourier sine transform as follows: Theorem.1. Let φ S o R +, then i s r x φx s r s φx, r N, ii r s φ [ s x s r ] φx, r N, iii The mapping s : S o R + S R + is linear an ontinuous. Theorem.11. Let φ be a measurable funtion efine on R +. or an fixe a >, we efine the funtion Ta, φ x φx + a e iax ot, then i ii s Ta, φ + T a, φ e i a ot osa s φ, Ta, φ + s T a, φ e i a ot osa s s φ. Proof: Proof of this theorem is straight forwar an thus avoie. 3. rational ourier Cosine Sine Transform of Tempere Distribution Theorem 3.1. The frational ourier osine transform is a ontinuous linear map of S e R + onto itself. Proof: Let φ S e R + L 1 R +, then φ K x, φxx C e ix + ot / osx s φxx C e i ot / e ix ot / osx s φxx C e i ot / [ e ix ot / φx ] s, C e i ot / Φ, where Φ [ e ix ot / φx ] s. [ Sine φ S e R +, Φ e ix ot / φx ] s S e R +. Therefore, D φ [ C D e i ot / Φ ] C C D e i ot / D Φ e i ot / P, i ot / D Φ

6 A. Prasa, M. K. Singh / ilomat 31:6 17, where P, i ot / is a polnomial of maximum egree, an using the tehnique [7, 9]. Thus D φ C e i ot / a s ot s D Φ, s therefore hene, β D φ C γ,β [ φ ] C <, a s ot β+s D Φ, s a s ot sup β+s D Φ s R + 3 beause Φ S e R +. Thus φ S e R +. Also from 1 an 4, we observe that for all φ S e R +, 1 φ φ 1 φ. It follows that is an 1-1 funtion of S e R + onto itself. Clearl, is also a linear map of S e R + onto itself. Also for ever sequene {φ n } n N whih onverges to zero as n in S e R + then b 3, { φ n } in S e R + as n, whih implies the ontinuit of frational ourier osine transform. Theorem 3.. The frational ourier sine transform is a ontinuous linear map of S o R + onto itself. Theorem 3.3. Parseval ientit of frational ourier osine transform Let φ, ψ S e R +, then the following equalities hol an φ ψ φx ψxx, 4 φ φx x. 5 Proof: Sine φ, ψ S e R +, hene using, we have φ ψ [ C e ix + ot / osx s φxx] ψ φx [C as an appliation of inverse frational ourier-osine transform, we obtain φ ψ If φ ψ in the last expression, we obtain φ φx x. φx ψxx. e ix + ot / osx s ψ ] x,

7 A. Prasa, M. K. Singh / ilomat 31:6 17, Theorem 3.4. Parseval ientit of frational ourier sine transform Let φ, ψ S o R +, then the following equalities hol s φ s ψ φx ψxx, 6 an s φ φx x. 7 Definition 3.5. The generalize frational ourier osine transform f of f S er + is efine b f, φ f, φ, 8 where φ S e R +. B Theorem 3.1, φ S e R + φ S e R +, so R.H.S. of 8 is well efine. Definition 3.6. The inverse frational ourier osine transform 1 f of f S er + is efine as 1 f, φ f, 1 φ ; φ S e R +. 9 Similarl, we efine the generalize frational ourier sine transform an an its inverse for S or + as follows: Definition 3.7. The generalize frational ourier sine transform s f of f S or + is efine b s f, φ f, s φ, 3 where φ S o R +. Definition 3.8. The inverse frational ourier osine transform 1 s f of f S or + is efine as 1 s f, φ f, 1 s φ ; φ S o R Theorem 3.9. The generalize frational ourier osine transform itself. is a ontinuous linear map of S er + onto Proof: Let f S er + an if the sequene {φ n } n N onverges to zero in S e R +, then b ontinuit of frational ourier osine transform { φ n } as n. Hene f, φ n f, φ n as n. Therefore, is ontinuous on S er +. Also for f, S er +, we have f +, φ f +, φ f, φ +, φ f, φ +, φ.

8 Hene, is linear on S er +. A. Prasa, M. K. Singh / ilomat 31:6 17, Also, f, φ f, f, 1 φ 1 φ, so that Similarl, 1 Therefore, f f. 1 f f. an 1 are 1-1 map of S er + onto itself. Example 3.1. If x R +, a >, then i ii s [δx a] K a,, [δx a] K s a,. Proof: i Let φ S e R +, then [δx a], φ δx a, φ x φ a C e ia + ot / osa s φ C e ia + ot / osa s, φ K a,, φ, hene, [δx a] K a,. ii Similarl if φ S o R + an proeeing as in i, we obtain ii. Example If x R +, then i [δx] C e i ot /. ii s [δx]. Proof: Put a in Example 3.1, we get the esire results. 4. Pseuo-Differential Operators P.D.O. s A linear partial ifferential operator Ax, x on R + is given b m Ax, x a r x r x, r where the oeffiient a r x are funtions efine on R + an x is onjugate omplex of x, given in 1. If we replae x b monomial s in R +, then we obtain the so alle smbol Ax, m a r x s r. 33 r 3

9 A. Prasa, M. K. Singh / ilomat 31:6 17, In orer to get another representation of the operator Ax, x, let us take an funtion φ S e, then we have Ax, x φx m 1 a r x r x φx 34 r m r a r x 1 r s φ x 35 K x, Ax, φ, 36 where K x, is as in. If we replae the smbol Ax, b more general smbol ax,, whih is no longer a polnomial in neessaril, we get the pseuo-ifferential operator A efine below. or p..o., a, involving ourier transform, Hankel transform, frational ourier transform, ourier-jaobi transform an a singular ifferential operator, we ma refer respetivel [11, 13], [6], [7, 9], [1, 1] an []. Definition 4.1. Let m R, then we efine the smbol lass S m to be the set of all funtions ax, C R + R + suh that for µ, N, there exists a positive onstant C µ, epening upon µ an onl, suh that D µ x D ax, m Cµ, Definition 4.. Let a be a smbol satisfing 37, then the pseuo-ifferential operator p..o. involving frational ourier osine transform, A is efine b a, A a, φ x K x, ax, φ ; φ S e R Similarl, we an efine p..o. involving frational ourier sine transform as follows: Definition 4.3. Let a be a smbol satisfing 37, then the p..o. involving frational ourier sine transform, A s a, is efine b A s a, φ x K s x, ax, s φ ; φ S o R Theorem 4.4. Let a be a smbol belonging to the smbol lass S m, m < β + + 1, then A a, maps S er + into itself. Proof: Let φ S e R + an β, N. In orer to prove that A a, maps S er + into itself we nee to prove that sup x β D x A a, φx <. x R + Using 38, we have x β D x A a, φ x x β D x C x β K x, ax, φ D x e ix + ot / os x s D x ax, φ C x β 1 ξ ξ Ax s s ξ D x [ ξ+1 ] P ξ x, i ot / e ix + ot / ax, φ,

10 A. Prasa, M. K. Singh / ilomat 31:6 17, where P ξ x, i ot / is a polnomial an Ax s osx s or sinx s epening upon ξ is even or o. Thus x β D x A [ a, φ x C x β ξ+1 ] ξ 1 ξ a r ot x r ξ r e ix + ot / Ax s s ξ D x 1 ξ+β+r+ C D β+r Now, using integration b parts, we have Hene, x β D x A a, φ x C ξ ξ a r ot ξ r Ax s ξ s ξ r β D x ξ Ax s ξ ax, φ e ix + ot / ax, φ. ξ a r ot s ξ r β 1 ξ+β+r+ ξ r β+r β + r {D t e ix + ot / D x ax, t t D β+r t ξ φ } ξ C a r ot s ξ r β 1 ξ+β+r+ ξ β+r t β + r t t j t j r j e ix + ot / D t j x β D x Aa, φ x C ξ γ j a γ ot Ax s γ γ D x t j t j a γ ot j γ j γ γ D β+r t ξ C t j t j D β+r t j γ ξ j γ ax, D β+r t ξ a r ot s ξ r β ξ r ξ D t j D ξ φ. x ax, β+r β + r t t φ ξ β+r β + r a r ot s ξ r β t r t m t j+γ a γ ot C,t j 1 + ξ φ, sine ξ φ S e R +, so the last integral is onvergent. Hene sup x β D x A a, φx <. x R +

11 A. Prasa, M. K. Singh / ilomat 31:6 17, Theorem 4.5. Let a be a smbol belonging to the smbol lass S m, m < β + + 1, then A s a, maps S or + into itself. Proof: Proof of this theorem is similar to that of Theorem 4.4 an thus avoie. Referenes [1] P. K. Banerji, S. K. Al-Omari, L. Debnath, Tempere istributional sine osine transform, Integral Transforms Spe. unt. 17, 11 6, [] A. Dahraoui, K. Triméhe, Pseuo-ifferential operators assoiate with a singular ifferential operator in ], + [, Inian J. Pure Appl. Math. 3, , [3] S. Iqbal, S. M. Raza, L. R. Kamal,. Sarwar, rational ourier integral theorem an frational ourier sine an ourier osine transform, Si. Int. Lahore 4, 3 1, [4] E. O. Milton, ourier transforms of o an even tempere istributions, Paifi. J. Math 5, 1974, [5] R.S. Pathak, A Course in Distribution Theor an Appliations, Narosa Publiation House, New Delhi, 9. [6] R. S. Pathak, P. K. Pane, A lass of pseuo-ifferential operators assoiate with Bessel operator, J. Math. Anal. Appl. 196, 1995, [7] R. S. Pathak, A. Prasa, M. Kumar, rational ourier transform of tempere istributions an generalize pseuo-ifferential operator, J. Pseuo-Differ. Oper. Appl. 3, 1, [8] S. C. Pei, J. J. Ding, rational osine, sine an Hertle transforms, IEEE Trans. Signal Proess 5, 7, [9] A. Prasa, M. Kumar, Prout of two generalize pseuo-ifferential operators involving frational ourier transform, J. Pseuo- Differ. Oper. Appl., 3 11, [1] A. Prasa, Manoj K. Singh, Pseuo-ifferential operators assoiate with the Jaobi ifferential operator an ourier-osine wavelet transform, 8, 1 15, Artile ID: pp.. [11] L Roino, Linear Partial Differential Operators in Gevre Spaes, Worl Sientifi, Singapore, [1] N. B. Salem, A. Dahraoui, Pseuo-ifferential operators assoiate with the Jaobi ifferential operator, J. Math. Anal. Appl., , [13] M. W. Wong, An Introution to Pseuo-Differential Operators, 3 r Eition Worl Sientifi, Singapore, 14.