ANGULAR COMPLEX MELLIN TRANSFORM

Transcription

1 Sc. Revs. Che. Co.: ISSN ANGULAR COMPLEX MELLIN TRANSFORM V. N. MAHALLE * A. S. GUDADHE a a R. D. TAYWADE b Ba. R. D. I.. N.. D. College Baea Ralway BADNERA M.S. INDIA a Deptt. o Matheatcs Govt. Vabha Isttte o Scece & Hates AMRAVATI M.S. INDIA b Po. Ra Meghe Isttte o Techology & Reseach BADNERA M.S. INDIA Receve : 8.0.0; Revse : ; Accepte : ABSTRACT The agla actoal cople Mell taso whch s a geealzato o the cople ell taso has ay applcatos seveal aeas clg sgal pocessg a optcs. I ths pape we have gve veso theoe o the geealze Factoal Cople Mell taso. Fo that we st pove two leas. Lastly we have scsse soe applcatos o ths taso. ey wos: Cople ell taso Agla ell taso. INTRODUCTION The actoal Foe taso s tately elate to seveal spesable cocepts appeag vese aeas. It plays a potat ole the sty o optcal syste ow as Foe optcs. The be accot o ts applcato s scsse 4. The actoal Foe taso R α s a eteso o the oay Foe taso a epes o the paaete α the posto-eqecy plae. The oe esoal actoal Foe taso wth paaete α o eote by R α peos a lea opeato gve by the tegal taso α [ R ] ξ F ξ ξ α α Whee α ξ s the eel as gve. Bhosale 3 etee actoal Foe taso to the stbtos o copact sppot. The actoal cople Mell taso toce 5 s the geealzato o the cople Mell taso s as ollows. Avalable ole at * Atho o coespoece; E-al: vyataywae@yahoo.co ala_gahe@yahoo.co ajeataywae@eal.co

2 300 V. N. Mahalle et al.: Agla Cople Mell. I the agla actoal Mell taso whch s a geealzato o the cople Mell taso ha etee to the stbto o copact sppot sg eel etho. The testg cto space E ee 5 s as ollows - The testg cto space E A tely eetable cople vale cto o R belogs to ER o E o each copact set S a S whee { } a R a a > 0 N γ Sp D < E Clealy E s coplete a so a Fechet space. Moeove we say that s a actoal Mell tasoable t s a ebe o E The al space o E. The actoal Mell taso o E It s easly see that o each s R π a 0 the cto s belogs to E as a cto o. Hece the actoal Mell taso o E ca be ee by [ M ] s M s s. π s cot π l cot π l s.csc whee s cot e. e. the the ght ha se o. has a eag as the applcato o E to s E. The pape s ogaze as ollows. Secto II gves the veso theoe wth two leas. Soe applcatos ae gve Secto III a Secto IV cocles the pape Notatos a teology se as Zeaa 6. Secto II: Iveso theoe Let E R 0 < π : R a a > 0 be the geealze actoal cople Mell tasoato o ee by [ ] S a a let M < a spp S a whee { } M M as pe The o each E we have M π whee s / ep ep l [ cos l. ] π π s s

3 Sc. Revs. Che. Co.: Poo : To pove the veso theoe we have establshe the ollowg leas to be se the seqel. Lea : Let [ ] M M o π < < 0 a spp S a whee { } 0 : > a a R S a o E θ θ The o ay e be - < <. whee C a s estcte to a copact sbset o R. Poo : The case 0 θ s tval hece cose 0 θ. It ca be easly see that s a C - cto o a t belogs to E. Hece the ght ha se o. s eagl. To pove the eqalty we costct the Rea-s o ths tegal a wte l Δ. l Δ We show that the last sato coveges E to the tegal o the ght ha se o.. Cayg the opeato D wth the tegal a sato sg whch s easly jste we get { } { }. Δ Δ γ D Sp As. l D Δ It ths ollows that o evey the sato s a ebe o E a t coveges E to the tegal o the ght ha se o.. Hece the poo.

4 30 V. N. Mahalle et al.: Agla Cople Mell. Lea : Fo θ E set as lea above o C s estcte to a copact sbset o R the M π π θ.3 coveges E to θ as. Poo : We shall show that θ E as. That s to show M [ M θ ] Sp{ D [ M θ ]} 0 γ as. We ote that o 0 θ π 0 That s to say that θ M l θ θ Sce the tega s a C - cto o a E we ca epeately eetate e the tegal sg a the tegals ae oly coveget we have π 0 D θ θ o all Hece the cla. Poo o veso theoe Now let θ π E. We shall show that M θ tes to M o C a the act that θ as..4 Fo the aalytcty o has a copact sppot R t ollows that the let se epesso.4 s eely a epeate tegal wth espect to a a the tegal.4 s a cotos cto o as the close boe oa o the tegato. Theeoe we wte.4 as - θ M θ π 0 π 0

5 Sc. Revs. Che. Co.: π θ 0 the oe o tegato ay be chage. The chage the oe o tegato s jste by appeal to lea. Sce θ s o copact sppot a the tega s a cotos cto o Thsyels θ π 0 whee s as lea. M π Ths s eqal to.5 π Aga by lea eqato.5 coveges to θ as. Ths copletes the poo o the theoe. Applcato o actoal ell taso Scale taso s a powel atheatcal tool o pocessg ages o etectg that ae abtaly scale. Hece t s se the class o lea steach vaat systes. Xaohog H 8 evelope Mell taso techqe o pobablty oelg o accate solto o pobles soe stal statstc. Factoal Mell taso gve by Aay as oe oe paaete agle to scale taso a hece t s also se patte ecogto pobles stal statstc. Moeove actoal Mell base coelatos ae se to obta te to pact a cotollg oets the avgato tas 7. The geealze actoal Mell taso we have toce ths pape s the eteso o actoal Mell taso gve by Aay a ca be se all above cases. The avatage o o geealze actoal Mell taso s t ca be se eve whe the sgals ctos ae sgla ctos. CONCLUSION We have gve the veso theoe o the geealze actoal cople Mel taso wth two leas. Gve soe applcatos vaos els o ths geealze actoal cople Mell taso. REFERENCES. T. Aleva a A. M. Babe Factals Egeeg : Factoal Foe Aalyss o Objects wth Scalg Syety CRC Pess Floa V. N. Mahalle A. S. Gahe a R. D. Taywae A Itocto to the Factoal Geealzato o Cople Mell Taso Cocate. 3. B. N. Bhosale a M. S. Chahay Factoal Foe Taso o Dstbtos o Copact Sppot Bll. Cal. Math. Soc

6 304 V. N. Mahalle et al.: Agla Cople Mell. 4. H. M. Ozatas Z. Zalvesy tay M. Alpe The actoal Foe Taso wth Applcatos Optcs a Sgal Pocessg Pb. Joh Wley a Sos Lt V. N. Mahalle A. S. Gahe a R. D. Taywae O Geealze Factoal Mell Taso Asa 9 Japa A. H. Zeaa Geealze Itegal Taso Ite Scece Pblshes New Yo Sazbo Zalevsy Meolvc : Usg Foe/Mell Base Coelatos a the Factoal Veso Navgatoal Tass Patte Recogto pp Xaohog H Ph. D. Thess Pobablty Moelg o Istal Statos sg Taso Techqe 995.