GENERALIZED OPERATIONAL RELATIONS AND PROPERTIES OF FRACTIONAL HANKEL TRANSFORM

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S. Res. Chem. Commu.: (3 8-88 ISSN 77-669 GENERLIZED OPERTIONL RELTIONS ND PROPERTIES OF FRCTIONL NKEL TRNSFORM R. D. TYWDE *. S. GUDDE d V. N. MLLE b Pro. Rm Meghe Isttute o Teholog & Reserh Bder MRVTI (M.S. INDI Deprtmet o MthemtsGot. Vdrbh Isttute o See & umtes MRVTI (M.S. INDI b Br R. D. I. K. N. K. D. College Bder MRVTI (M.S. INDI (Reeed : 8..; Resed : 5.3.; epted :.3. BSTRCT Nms hd deed rtolzto o oetol kel trsorm ug the method o ege lues d studed d ope the door or the reserh rtol tegrl trsorm. Ths pper studes the rtolzto o geerlzed kel trsorm s ge b Zem 5. We reerred t s rtol kel trsorm. Frst we trodue rtol kel trsorm the geerlzed sese. Geerlzed opertol reltos re dered tht be used to sole ert lsses o ordr d prtl deretl equtos. Lstl the lues o rtol kel trsorm re obted or some spel utos. INTRODUCTION Fourer lss s oe o the most requetl used tools sgl proesg d s used m other set dsples. I the mthemts lterture geerlzto o the Fourer trsorm kow s the rtol Fourer trsorm ws proposed some ers go 3. lthough potetll useul or sgl proesg ppltos the rtol Fourer trsorm hs bee depedetl reeted b umber o reserhers. L. B. lmed hd brel trodued the rtol Fourer trsorm. e dsussed the m propertes d preseted the ew results ludg the rtol Fourer trsorm. lso represeted smple reltoshp o the rtol Fourer trsorm wth seerl tme-reque represettos tht supports the terpretto o t s rotto opertor. Fo. Kerr hd deed the rtol kel trsorm wth prmeter o ( deoted b ( perorm ler operto ge b the tegrl trsorm. [ ] (. K ( d lble ole t www.sdgurupubltos.om * uthor or orrespodee; E-ml: rdtwde@redml.om lk_guddhe@hoo.om

S. Res. Chem. Commu.: (3 83 where K ( δ ( ep ( or & π d π ep ˆ ( ν ˆ ( L ( R R d ν > The boe rtol kel trsorm s the geerlzto o the kel trsorm ge s 5 ( d For the prmeter π the rtol kel trsorm redues to the boe kel trsorm. Ths pper s orgzed s ollows. Seto II presets the rtol kel trsorm wth prmeter the sese o geerlzed uto d ts terpretto s rotto opertor. I seto III we ge geerlzed opertol relto. Seto IV lsts some propertes o rtol kel trsorm. I seto V we ge rtol kel trsorm o some utos lstl seto VI oludes. Noto d termolog s s used Zem 5. Frtol hkel trsorm the geerlzed sese Frst we dee The testg uto spe E: tel deretble omple lued uto ψ o to E ( R or E or eh ompt set K where S { R > } k N γ k ( ψ Sup D < K K k ψ S Clerl E s omplete d so Frehet spe. R belogs Moreoer we s tht s rtol trsormble t s member o E (the dul spe o E. The geerlzed rtol kel trsorm It s esl see tht or eh R d < < π the uto K ( belogs to E s uto o. ee the rtol kel trsorm o E be deed b [ ] K ( (

8 R. D. Twde et l.: Geerlzed Opertol Reltos d. where K ( s s ge b ( the the rght hd sde o ( hs meg s pplto o E to K ( E. Geerlzed opertol relto o rtol hkel trsorm s s well kow opertol lulus be bsed o the usul kel trsorm. We dere opertol reltos olg rst dertes d opertol reltos hg seod dertes. Opertol reltos olg rst dertes Theorem: 3..: I deotes rtol kel trsorm o (t the - d d ( Proo: We dere opertol relto trsorm olg o ( the tegrl represetto ( we the tegrte b prts d d obt b sertg sted d d d d ep ep d d d ep ep d ep ep ep ( d d ep d

S. Res. Chem. Commu.: (3 85 ter some strght orwrd steps we obt d d ( ( s the opertol relto olg rst derte ( reple b d d d ges d d d 3 ( ( (5 replg b d d (3 ges d d d d ( (6 Opertol reltos olg seod dertes : obt We ow lulte seod derte d d b sertg d d ple o the equto ( we d d d d d d d d (. Propertes o rtol hkel trsorm ( We proe the ollowg propertes o rtol kel trsorm ep [ ] ( where

R. D. Twde et l.: Geerlzed Opertol Reltos d. 86 Proo : [ ] ep ( d ep ( dt t t t where t dt C tz tz z z t t ep ( Where C z [ ] ep z [ ] ep ( ep ep the proo s trl hee omtted. Trsorm o some ommo utos The kel trsorm o some ommo utos re proed. Result ˆ ep π δ

S. Res. Chem. Commu.: (3 87 the proo s trl & hee omtted. Result : 8 ep ep ep Re > Re > -. Proo: ep ep ep ep d where ep ep d B where B 8 ep ep Result 3: ep ep d

R. D. Twde et l.: Geerlzed Opertol Reltos d. 88 Proo: ep ep d os d d ep π CONCLUSION We he trodued eteso o kel trsorm tht s desgted rtol kel trsorm. Ths ler trsorm depeds o prmeter d be terpreted s rotto b gle slelog phse modulto ple. Whe π the rtol kel trsorm odes wth the oetol geerlzed kel trsorm. We dere opertol relto or rst d seod order derte or rtol kel trsorm. Some propertes o the rtol kel trsorm re ge whh odes wth orrespodg propertes or kel trsorm spel se. Frtol kel trsorm o some smple utos re lso obted. REFERENCES. L. B. lmed troduto to the gulr Fourer Trsorm Pro. IEEE It. Co. oust. Speeh Sgl Proesg (Mepols MN prl (993.. Fo. Kerr Frtol Power Theor or kel Trsorm. Mthemtl lss d pplto 58-3 (99 3. V. Nms The Frtol Order Fourer Trsorm d ts pplto to Qutum Mehs. Ist. Mth. ppl. 5-65 (98.. V. Nms The Frtol Order Fourer Trsorm d ts pplto to Qutum Mehs. Ist. Mth. ppl. 5 87-97 (98. 5... Zem Geerlzed Itegrl Trsormto Iter See Publshers New York (968.

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