Operation Transform Formulae for the Generalized. Half Canonical Sine Transform

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Benjamin McGee
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1 Appl Mhmcl Scnc Vol 7 3 no 33-4 HIKARI L wwwm-hrcom Opron rnorm ormul or h nrl Hl Cnoncl Sn rnorm A S uh # n A V Joh * # ov Vrh Inu o Scnc n Humn Amrv M S In * Shnrll Khnlwl Coll Aol M S In luh@mlcom molmh@rmlcom Arc A nrlon o h rconl Con rnorm RC h cnoncl n rnorm CS h n u n vrl r nclun opcl nly n nl procn or prccl purpo hl cnoncl n rnorm mor uul Hnc n h ppr w hv prov om mporn rul ou Invron horm or hl cnoncl n Drnon propry Moulon propry Prvl Iny Scln propry or hl cnoncl n rnorm Kywor: Lnr cnoncl rnorm rconl ourr rnorm Inroucon h o h rconl powr o ourr opror ppr n mhmcl lrur rly n 93 I h n rcovr n qunum mchnc y Nm 5] H h vn ymc mho or h vlopmn o rconl nrl rnorm y mn o En vlu L c nc Nm n 98 vlop h nvlu mho or rconl ourr rnorm numr o ohr nrl rnorm hv n xn n rconl omn or xmpl Alm ] h u rconl ourr rnorm yw uh h uy rconl Mlln rnorm rconl Hlr rnorm h n vlop y Zy 9] Son uh 8] u numr o propry o rconl Hrly rnorm Joh uh ] u numr o propry o nrl cnoncl n rnorm c Bhol n Chouhry 3] h u mpr ruon; numr o pplcon o

Invron horm or hl cnoncl n Drnon propry Moulon propry Prvl Iny Scln propry or hl cnoncl n rnorm Kywor: Lnr cnoncl rnorm rconl ourr rnorm Inroucon h o h rconl powr o ourr opror ppr n mhmcl lrur rly n

2 34 A S uh n A V Joh rconl ourr rnorm n nl procn m procn lrn opc c u h rconl rnorm oun numr o pplcon n nl procn m procn qunum mchnc c urhr nrlon o rconl ourr rnorm nown lnr cnoncl rnorm w nrouc y Mohny 4] n 97 P Dn 67] h u n vlu pc Lnr cnoncl rnorm hr prmr lnr nrl rnorm whch h vrl pcl c rconl ourr rnorm rnl rnorm Chrp rnorm c Lnr cnoncl rnorm n ] / / LC or c or wh c Whr c n r rl prmr npnn on n P n Dn 6] h n cnoncl n rnorm CS n ] CS n uncon Spc ξ : An nnly rnl complx vlu uncon φ on R n lon o ξ R n or ch compc α S I whr } : > α α α n R S n or n R up < D I φ φ ξ γ Hl Cnoncl Sn rnorm: Dnon: h Hl Cnoncl Sn rnorm n R ξ cn n y } < K > whr K n Hnc h nrl hl cnoncl n rnorm o n R E cn n y } n

n ] / / LC or c or wh c Whr c n r rl prmr npnn on n P n Dn 6] h n cnoncl n rnorm CS n ] CS n uncon Spc ξ : An nnly rnl complx vlu uncon φ on R n lon o ξ R n or ch compc α S

3 Opron rnorm ormul 35 Snc h rn o nron or h hl cnoncl n rnorm ju ] n no or - un hl cnoncl n rnorm mor convnn hn un h cnoncl rnorm o l wh h vn uncon Invron horm or Hl Cnoncl Sn rnorm: I } hl cnoncl n rnorm o vn y } n ] hn } n Proo: h hl cnoncl n rnorm o vn y } n n n n C whr C n n n C whr By un nvr ormul n C n n n n

horm or Hl Cnoncl Sn rnorm: I } hl cnoncl n rnorm o vn y } n ] hn } n Proo: h

4 36 A S uh n A V Joh } n 3 Som opronl rul 3 Drnon propry o hl cnoncl n rnormon: I } no nrl hl cnoncl n rnorm o hn } } } Proo: W hv n } By nron y pr n n } n co ] n co n co } } } 3 Moulon propry o hl cnoncl n rnorm: I } no nrl hl cnoncl n rnorm o hn } ] ] co Proo: W hv co n } co co n } co

y pr n n } n co ] n co n co } } } 3 Moulon propry o hl cnoncl n rnorm:

5 Opron rnorm ormul 37 n n } co n n } co n n } co } ] ] co 33 I } no nrl hl cnoncl n rnorm o hn } ] ] n H CC Proo: W hv n n } n n n } n co co } n co co } n co co } n

6 38 A S uh n A V Joh } ] ] n 34 Scln propry o hl cnoncl n rnorm: I } no nrl hl cnoncl n rnorm hn ]} Proo: W hv n } Pun n } n n ]} 35 Drvv propry o hl cnoncl n rnorm: I } no nrl hl cnoncl n rnorm hn ] ]} } } Proo: W hv n } n }

7 Opron rnorm ormul 39 n co n co n ] co } ]} ] ]} } } 4 Prvl Iny or hl cnoncl n rnorm: I n r h nvron cnoncl n rnorm o n rpcvly hn n Proo: By non o } n Un h nvron ormul o n n complx conju w n n Chnn h orr o nron w n

8 4 A S uh n A V Joh Hnc prov Pun n quon 4 w l or hl cnoncl n rnorm SN } } co ] ] 3 n ] ] 4

9 Opron rnorm ormul 4 Concluon In h ppr r nroucon o h nrl hl cnoncl n rnorm r vn n Invron horm or hl cnoncl n Drnon propry Moulon propry Prvl Iny Scln propry Drvv propry or hl cnoncl n rnorm on whch wll uul n olvn rnl quon occurrn n nl procn n mny ohr rnch o nnrn Rrnc ] Alm LB 994: h rconl ourr rnorm n m- rquncy rprnon IEEE rn on Sn Proc Vol 4 No ] AV Joh n A S uh : nrl Cnoncl Sn rnorm Sc Rv Chm Commun: ] Bhol BN n Chouhry MS : rconl ourr rnorm o ruon o compc uppor Bull Cl Mh Soc Vol 94 No ] Mohny M97: Lnr cnoncl rnorm n hr unry rprnon Jour Mh PhyVol No 8 P ] Nm V 98: h rconl orr ourr rnorm n pplcon o qunum mchnc Jour In Mh App Vol ] P n Dn : Enuncon o Lnr Cnoncl rnorm Vol 5 No 7] P n Dn : rconl con n n Hrly rnorm IEEE rn On Sn Proc Vol 5 No ] Son uh 9: Convoluon n Rylh horm or nrl rconl Hrly rnorm EJPAM Vol No 6-7 9] Zy A IAu 998: Hlr rnorm Aoc wh h rconl ourr rnorm IEEE nl procn Lr Vol5 No8 Rcv: Ocor

Rv Chm Commun: 3 55-63 3] Bhol BN n Chouhry MS : rconl ourr rnorm o ruon o compc uppor Bull Cl Mh Soc Vol 94 No5 349-358 4] Mohny M97: Lnr cnoncl rnorm n hr unry rprnon Jour Mh PhyVol No 8 P 77-783

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