Analyticity and Operation Transform on Generalized Fractional Hartley Transform

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1 I Jourl of Mh Alyi, Vol, 008, o 0, Alyiciy d Oprio Trform o Grlizd Frciol rly Trform *P K So d A S Guddh * VPM Collg of Egirig d Tchology, Amrvi (MS), Idi Gov Vidrbh Iiu of cic d umii, Amrvi (MS), Idi Abrc I hi ppr w hv dicud h lyiciy horm d ivrio formul for h grlizd frciol rly rform d uig h w hv provd uiqu horm Alo w hv dicud frciol rly rform of lcd fucio d obid oprio rform formul for hi rform Kyword: Frciol Fourir rform, rly rform, Tig fucio pc, Grlizd fucio Iroducio: Now dy, frciol igrl rform ply impor rol i igl procig, img rcorucio, pr rcogiio, ccoic igl procig [8], [9] Fourir lyi i o of h mo frquly ud ool i igl procig d my ohr ciific dicipli Bid h Fourir rform for im frqucy rprio of igl, Wigr diribuio, h mbiguiy fucio, h hor im fourir rform d h pcrogrm r of ud g i pch procig rdr, quum phyic I mhmic lirur grlizio of h Fourir rform o h frciol Fourir rform i udr Nmi [0] iroducd h cocp of Fourir rform of frciol ordr, which dpd o coiuou prmr Th grlizio of ordiry fourir rform d i propri wr dicud i Criolro l [3] Zyd [] Drgom [4] c Frciol fourir rform i furhr grlizd o h igrl wih rpc o w mur dρ d w grlizd igrl rform w obid by Zyd [] Bhol d chudhry [] hd xdd frciol fourir rform o h diribuio of compc uppor Th frciol Fourir rform wih corrpod o h clicl Fourir rform d frcio l Fourir rform wih 0 corrpod o h idiy opror I [5] ohr igrl rform of Fourir cl, h i coi rform, i

2 978 P K So d A S Guddh rform d rly rform, r lo grlizd o h corrpodig frciol igrl rform d udid by diffr mhmici Th rly rform i rciv lriv d covi rl rplcm for h wll ow complx Fourir rform rly rform i gig grr imporc i vrl pplicio I [6] Brcwll ugg h h rly rform i f or fr h h Fourir rform d c b ud rplcm for h Fourir rform Uig h igvlu fucio, ud i frciol Fourir rform, diffr igrl rform i Fourir cl r grlizd o frciol rform by Pi [5] rly rform i lo grlizd o frciol rly rform by him hd how h for ll o giv igr m, m ( ) i h ig fucio of h rly rform d hd giv h formul for frciol rly rform, whr K K d, π [( i ) c(cc φ ) + ( + i ) c( ccφ ) ] I hi ppr fir w dfi grlizd frciol rly rform i cio d prov i lyiciy Th fucio f () i h rcovrd from by m of h ivrio formul i cio 3 Uig ivrio formul w hv provd h frciol rly rformbl grlizd fucio hvig h m rip of dfiiio d h m rform mu b idicl, which i md uiqu horm i cio 3 Frciol rly rform of om lcd fucio d oprio rform formul r obid i cio 4 d cio 5 rpcivly Lly cocluio i giv i cio 6 Alyiciy Of Frciol rly Trform : ' Th Grlizd Frciol rly Trform o E : R d S d 0 L S { R,, > 0 L, if icoφ i coφ i coφ K [( i ) c(ccφ ) + ( + i ) c( ccφ ) ], π whr π φ, h K E( R ) γ K up D K < if { < < r E ( R ) i h ig fucio pc ' Th grlizd frciol rly rform of E ( R ) i dfid,

3 Alyiciy d oprio rform 979, K, () ' whr E ( R ) i h dul pc of h ig fucio pc Alyiciy Thorm : ' Thorm : L ( ) f E R d i frciol rly rform i dfid () Th i lyic o Supp S { : R,, > 0 D R if h f h [ ](, D K Proof : L (, ) R, w fir prov h : i diffribl d [ { f ( )]( f ( ), K W prov h rul for, h grl rul follow by iducio For om 0, choo wo cocric circl C d C wih cr d rdii r d r rpcivly, uch h 0 < Δ < r 0 < r < r < L Δ b icrm, ifyig Coidr, ( + Δ ) Δ ( ) f ( ), K ) f ( ), ΨΔ ( ), ) Δ whr, ) [ K, ) K ] K ) ΨΔ For y fixd ( + Δ R, {( ccφ K + i coφ K ) D i coφ i coφ, φ d K C, φ, whr, C i coφ π K [i(ccφ ) + i co(ccφ )]

4 980 P K So d A S Guddh Sic for y fixd R fixd igr d rgig from 0 o, D K ) i lyic iid d o C, w hv by Cuchy igrl formul, Δ M D i (, ) ΨΔ ( ) π z Δ z ( )( ) C dz, whr,, z, ) ( + Bu for ll z C d rricd o compc ub of R, 0, M D K i boudd by co K Thrfor w hv, K DΨΔ ( ) Δ ( r r) r ub of ' E Thu Δ 0, D ΨΔ ( ) d o zro uiformly o h compc R hrfor i follow h () ΨΔ covrg i E( R ) o zro Sic w coclud h () lo d o zro hrfor ( i S Bu hi i ru for ll, c ( diffribl wih rpc o i lyic d D [ ](, D K 3 Ivr Ad Uiqu Thorm : 3 Ivr Of Frciol rly Trform: Grlizd frciol rly rform dfid i () c b wri icoφ i coφ i coφ ( i ) c(ccφ ) + ( + i ) c( ccφ ) d π ( i π ) i co φ f ( ) (cc φ + ( + i ) f ( ) (cc φ,

5 Alyiciy d oprio rform 98 Pu (ccφ ) v v viφ ccφ Tig rly ivr of boh id which i m rly rform, iv i i coφ φcoφ ( vi φ ) cv d ( i ) f ( ) + ( + i ) f ( ) i ( ) co φ i ( ) co φ If f () i v h f ( ) h, w g i coφ g ( v) c v dv, iv i φco φ φ whr g( v) v i ) i coφ g( v) c v dv, wh f () i v Puig g (v) d olvig, w g d K(, ) ccφ i coφ i coφ whr K c(ccφ ) icoφ Now if f () i odd h f ( ) g( v) c v dv ( i i i( φ + coφ ) ), i coφ g( v) c v dv Agi puig g (v) d olvig, w g + ( + i ) i coφ ( )

6 98 P K So d A S Guddh d K(, ) ccφ i coφ i( φ ) i coφ whr, K c(ccφ ) i coφ π 3 Uiqu Thorm : Thorm : If d { g( ) r frciol rly rform of f () d g () rpcivly for o d upp f S : S { : R, d upp g S : S { : R, ( { g( ) h f g i h of quliy of D ' (I ) if { ) Proof : By ivrio horm, f g lim N π N N K [ { g( ) ]d Thu f g i D ' (I ) 4 Frciol rly Trform Of Slcd Fucio : Frciol rly Trform of lcd fucio r buld follow

7 Alyiciy d oprio rform 983 TABLE Sr No Sigl { Frciol rly rform ( (φ i) i φ δ ( ) { δ ( ) i coφ π i + co φ [ co(ccφ ) i i(ccφ ) ] 3 δ ( ) { δ ( ) i coφ π 4 i { ( + i ( ) ( ) ) φ φ i i( cφ ) i 5 co { ( + co ( ) ( ) ) φ φ i co( cφ ) i 5 Oprio Trform Formul : I hi cio w prov om oprio rform formul for frciol rly rform, for which followig wo lmm c b ily provd 5 Lmm : Frciol rly rform giv i cio, c lo b xprd i coφ i coφ i coφ π 5 Lmm : { f ( ) icoφ i coφ i coφ π [ co(ccφ ) i i(ccφ ) ] [ co(ccφ ) + i i(ccφ ) ] d d

8 984 P K So d A S Guddh 53 Lmm : icoφ i coφ i i coφ φ [ i(ccφ ) + i co(ccφ ) ] d π i coφ iφ { f ( ) 54 Formul : If FrT h d ( i coφ [ { ] { f ( ) d Proof : Sic i coφ i coφ i coφ π co φ d i Cφ d Solvig w g ico [ co(ccφ ) i i(ccφ ) ] d ' [ co(cc φ ) i i(cc φ ) ] d [ { ] { f ( ) φ 55 Formul: If FrT { h + i φ d { f ( ) + i φ { d ' ( icoφ { { f ( Proof : { ) Q [ ] ) { 56 Formul: + c) (c+ c ) i coφ { co(ccφ c) + iφ i(ccφ c) f ( ) { Proof : w ow h, + c) ( + c) i coφ i coφ i coφ π (c+ c ) d + i φ { f ( ) + i φ d (c+ c ) i coφ i coφ { i(cc φ c) { co(ccφ( + c) ) i i(ccφ( + c) ) d (c+ c ) i coφ i coφ { co(ccφ c) i coφ { i(ccφ c) + iφ { i(ccφ c) f ( ) d 57 Formul : i coφ { f ( Proof : Coidr, d d d d d i coφ π ) { co(ccφ ) i i(ccφ ) d

9 Alyiciy d oprio rform 985 i coφ iφ iφ iφ i {coφ + coφ { f ( ) i coφ { f ( ) { f ( ) + i coφ 58 Formul : L b fixd rl umbrth mppig f ( ) i coiuou lir mppig o S o S d { f ( ) Whr i rl umbr i coφ i coφ [ co(ccφ ) i i(ccφ ) ] { Proof : By h dfiiio of frciol rly rform, { f ( ), i coφ i coφ i coφ [ co(ccφ ) i i(ccφ ) ] π Puig x d olvig w g { f ( ) f ( ) d i coφ i coφ [ co(ccφ ) i i(ccφ ) ] { 59 Formul: If FrT f ( ) ( h i i( ) co φ ( ) co φ Proof : By h dfiiio of frciol rly rform, i( ) coφ f ( ) ( i coφ i coφ i coφ [ co(ccφ ) i i(ccφ ) ] i( ) coφ Puig i π T d olvig w g f ( ) ( i ( ) co φ ( ) co φ f ( ) d

10 986 P K So d A S Guddh 6 Cocluio : Th grlizd frciol rly rform i dvlopd i hi ppr Th oprio rform formul provd i hi ppr c b ud, wh hi rform i ud o olv ordiry or pril diffril quio REFERENCES Ahmd I Zyd, A Covoluio d produc horm for h rciol Fourir rform, IEEE Sigl procig lr, Vol 5, No 4, April 998 B N Bhol d Chudhry M S, Frciol Fourir rform of Diribuio of compc uppor Bull cl Mh Soc, 94 (5), P , 00 3 Criolro,l, Mulipliciy of frciol Fourir rform d hir rliohip IEEE Tr o igl proc Vol48, No, J 000, P D Drgom Frciol Fourir rld fucio, Opic Commuicio, Vol 8, P 9-98, July Pi Soo-Chg, Ji-Jiu Dig, Frciol coi, i d rly rform, IEEE, Vol 50, No 7, July 00 6 R N Brcwll, Th rly Trform, Oxford U K Oxford Uiv Pr, R Scilr, Ergiv S d Ciiz N, Th u of h rly rform i gophyicl pplicio Gophyic, Vol 55, No, (Nov 990), P Ti Aliv d Bi Mri J, O Frciol Fourir rform mom, IEEE Sigl procig Lr, Vol 7, No, Nov Ti Aliv d Bi Mri J, Wigr diribuio d frciol Fourir rform for -dimiol ymmric bm, JOSA A, Vol 7, No, Dc 000, p Vicor Nmi, Th frciol ordr Fourir rform d i Applicio o quum mchic, J I Mh Appic, (998), 5, 4-65 Rcivd: Fbrury 4, 008

Advanced Engineering Mathematics, K.A. Stroud, Dexter J. Booth Engineering Mathematics, H.K. Dass Higher Engineering Mathematics, Dr. B.S.

Advanced Engineering Mathematics, K.A. Stroud, Dexter J. Booth Engineering Mathematics, H.K. Dass Higher Engineering Mathematics, Dr. B.S. Rfrc: (i) (ii) (iii) Advcd Egirig Mhmic, K.A. Sroud, Dxr J. Booh Egirig Mhmic, H.K. D Highr Egirig Mhmic, Dr. B.S. Grwl Th mhod of m Thi coi of h followig xm wih h giv coribuio o h ol. () Mid-rm xm : 3%