On Generalized Fractional Hankel Transform

Transcription

1 Int. ournal o Math. nalss Vol. 6 no On Generaled Fratonal ankel Transorm R. D. Tawade Pro.Ram Meghe Insttute o Tehnolog & Researh Badnera Inda rajendratawade@redmal.om. S. Gudadhe Dept.o Mathemats Got. Vdarbha Insttute o Sene & umantes mraat Inda alka_gudadhe@ahoo.om V. N. Mahalle Bar.R.D.I.K.N.K.D. ollege Badnera Ralwa Inda datawade@ahoo.om bstrat Namas [4] had dened ratonalaton o onentonal ankel transorm ug the method o egen alues and studed ts propertes. Ths paper studes the ratonal generalaton o generaled ankel transorm whh s the generalaton o generaled ankel transorm gen b Zemanan [5]. We smpl reerred t as ratonal ankel transorm. Frst we ntrodue ratonal ankel transorm n the generaled sense. Some propertes o the Kernel are dsussed and nerson ormula or ratonal ankel transorm s proed. Generaled operatonal relatons are dered that an be used to sole ertan lasses o ordnar and partal derental equatons. Lastl the alues o ratonal ankel transorm are obtaned or some speal untons. Mathemats Subjet lassaton: 46F 44

2 884 R. D. Tawade. S. Gudadhe and V. N. Mahalle Kewords: Fratonal transorm Fratonal ankel transorm Inerson ormula Generaled operatonal relatons I. Introduton Fourer analss s one o the most requentl used tools n sgnal proesg and s used n man other sent dsplnes. In the mathemats lterature a generalaton o the Fourer transorm known as the ratonal Fourer transorm was proposed some ears ago [] [3]. lthough potentall useul or sgnal proesg applatons the ratonal Fourer transorm has been ndependentl renented b a number o researhers. L.B.lmeda [] had brel ntrodued the ratonal Fourer transorm. e dsussed the man propertes and presented the new results nludng the ratonal Fourer transorm. lso represented smple relatonshp o the ratonal Fourer transorm wth seeral tme-requen representatons that supports the nterpretaton o t as a rotaton operator. Fona.Kerr [] had dened the ratonal ankel transorm wth parameter o denoted b perorm a lnear operaton gen b the ntegral transorm. [ ]. K d where K ep δ or & π π and ep ˆ ν ˆ sgn L R R and ν > The aboe ratonal ankel transorm s the generalaton o the ankel transorm d or the parameter π the ratonal ankel transorm redues to the aboe ankel transorm.

3 On generaled ratonal ankel transorm 885 Ths paper s organed as ollows. Seton II presents the ratonal ankel transorm wth parameter n the sense o generaled unton and ts nterpretaton as rotaton operator. In seton III we ge some useul propertes o Kernel. Inerson ormula or ths transorm s dered n seton IV. Seton V lsts some operaton transorm ormulae or the ratonal ankel transorm.some propertes o ratonal ankel transorm are proed n seton VI. In seton VII we ge ratonal ankel transorm o some smple untons lastl seton VIII onludes.notaon and termnolog s as used n Zemanan[5]. II. Frantonal ankel transorm n the generaled sense Frst we dene.: The testng unton spae E: n nntel derentable omple alued n n unton ψ on R belongs to E R or E or eah ompat set K Sa where n S R a a > k N a { } n k γ K k ψ Sup D ψ < K learl E s omplete and so a Frehet spae. Moreoer we sa that s a ratonal transormable t s a member o E the dual spae o E..: The generaled ratonal ankel transorm: It s easl seen that or eah R n and < < π the unton K belongs to E as a unton o. ene the ratonal ankel transorm o E an be dened b [ ] K where K ep δ or & π 3 ˆ sgn π ep ˆ and

4 886 R. D. Tawade. S. Gudadhe and V. N. Mahalle then the rght hand sde o has a meanng as a applaton o E to K E. III. Propertes o Kernel: The kernel K gen n 3 satses the ollowng propertes. K K. * K K where * denotes the onjugaton 3. K K 4. For π the kernel ondes wth the kernel o the ankel transorm gen n [5] 5. K o 6. o K 7. K K d K Proo: Frst s propertes are smple to proe hene we proe the last propert. 7 K K d K L..S. K K d ep d ep 4 let rst ealuate ep d

5 On generaled ratonal ankel transorm 887 where d d d os ep ep π Equaton 4 ges L..S. ep ep π ater some straght orward steps we obtan

6 888 R. D. Tawade. S. Gudadhe and V. N. Mahalle L..S. ˆ ˆ ep π ep ep K R..S. IV. Inerson Formula : In ths seton we dere Inerson Formula or generaled ratonal ankel transormaton. It s possble to reoer the unton b means o the nerson ormula. d K where ep. K and K s omple onjugate o K Proo: The one dmensonal ratonal ankel transorm s gen b [ ] d K 5 where the kernel ep ˆ ep π K

7 On generaled ratonal ankel transorm 889 ep ep where thereore rom 5 d g ep [ ][ ] g where G sa where ep g 6 [ ][ ] ep G g s the ankel transorm o g wth argument. Inokng ankel nerson we an wrte d G g where ep G puttng the alue o g rom 6 and on smplng we get nerson ormulae d K where ep K

8 89 R. D. Tawade. S. Gudadhe and V. N. Mahalle π and ˆ ep V. Generaled operatonal relaton o ratonal ankel transorm s s well known an operatonal alulus an be based on the usual ankel transorm. We dere operatonal relatons nolng rst derates and operatonal relatons hang seond derates. 5. Operatonal relatons nolng rst derates : We dere operatonal transorm nolng d obtan b nsertng d d nstead o n the ntegral representaton we then ntegrate b parts d d d d d d ep ep ep ep d ep d

9 On generaled ratonal ankel transorm 89 ep ep ater some straght orward steps we obtan d d d ep d s the operatonal relaton nolng rst derate n 7 replae b d d ges d d d d replang b 3 d d n 3 ges d d 7 8 d d 9 5. Operatonal relatons nolng seond derates : We now alulate seond derate the equaton 7 we obtan d d b nsertng d d n plae o n

10 89 R. D. Tawade. S. Gudadhe and V. N. Mahalle d d d d d d d d. 4 VI: Propertes o ratonal ankel transorm: We proe the ollowng propertes o ratonal ankel transorm [ ] ep 4 where Proo : [ ] ep d ep dt t t t t where t dt t t t t. ep

11 On generaled ratonal ankel transorm 893 where [ ] ep 4 [ ] ep 4 ep ep the proo s tral hene omtted. VII: Transorm o some ommon untons: The ankel transorm o some ommon untons are proed. Result: ˆ ep τ τ τ π τ δ the proo s tral & hene omtted. Result : a a a 8 ep ep ep Re a> Re >-.

12 894 R. D. Tawade. S. Gudadhe and V. N. Mahalle Proo: ep ep ep ep a a d where ep ep d B where a B a a 8 ep ep Result 3: ep ep d a a Proo: ep ep d a a

13 On generaled ratonal ankel transorm 895 os d a d a ep π a a Result 4: 4 4 ep ep μ μ μ μ μ Proo s tral hene omtted. VIII: onluson We hae ntrodued an etenson o ankel transorm that s desgnated ratonal ankel transorm. Ths lnear transorm depends on a parameter and an be nterpreted as a rotaton b an angle. When π the ratonal ankel transorm ondes wth the onentonal ankel transorm. Inerson ormula or ths transorm s also establshed. We dere operatonal relaton or rst and seond order derate or ratonal ankel transorm. Some propertes o the ratonal ankel transorm are gen whh ondes wth orrespondng

14 896 R. D. Tawade. S. Gudadhe and V. N. Mahalle propertes or. ankel transorm n speal ase. Fratonal ankel transorm o some smple untons are also obtaned. Reerenes. L.B.lmeda : n ntroduton to the angular Fourer transorm n pro. 993 IEEE Int. on. oust. speeh sgnal proesg Mnneapols MN prl Fona.Kerr. : Fratonal power theor or ankel transorm Int.. o Mathematal nalss and pplaton V.Namas : The ratonal order Fourer transorm and ts applaton to quantum mehans.inst. Math. ppl. Vol V.Namas : Fratonalaton o ankel transorm.inst. Math. ppl. Vol Zemanan.. : Generaled ntegral transormaton Inter Sene Publshers New York Reeed: Otober