THETRIGONOMETRICTRANSFORMIncontrasttothehypercomplexdenitionin[20],we - PDF document

THETRIGONOMETRICTRANSFORMIncontrasttothehypercomplexdenitionin[20],wewanttodenerealvaluedgeneraltrigonometrictransforms,thatonlyconsistofscalarappearancesofsinesandcosines.Thereforeweusethefollowingnotation.Notation3.ForamultivectorA2Ip;q=fB2C`p;q;B22R�gC`p;qthatsquarestoanegativerealnumberandj2f0;1g,wedeneeAj:=(cos(jjAjj);ifj=0;sin(jjAjj);ifj=1(2)withthenormjjAjj=p �A22R.Lemma4.TheexponentialofamultivectorA2Ip;qC`p;qthatsquarestoanegativerealnumbersatiseseA=åj2f0;1g(A jjAjj)jeAj:(3)Denition5(Trigonometrictransform).LetA:Rm!C`p;qbeamultivectoreld,x;u2Rmvectors,F1;F2twoorderednitesetsofm,respectivelyn�m,mappingsRmRm!Ip;qC`p;q,andj2f0;1gm,k2f0;1g(n�m)multi-indices.TheTrigonometricTransform(TT)FFj1;Fk2isdenedbyFFj1;Fk2(A)(u):=ZRmmÕl=1e�fl(x;u)jlA(x)nÕl=m+1e�fl(x;u)kldmx(4)withe�f(x;u)jfromNotation3.Remark6.Thee�f(x;u)jl2Rareinthecenterofthegeometricalgebra,thereforethereisnousewithregardstocontenttodistinguishtheorderoftheirappearances.ItwillbehelpfulthoughtostresstheirrelationtotheGFT.Example7.ThestandardcosinetransformisaspecialcaseofthisdenitionwithF1=/0;F2=f2pixug;k2f0;1g1=0Fc(A)(u)=ZRA(x)cos(xu)dx=F/0;(2pixu)0(A)(u):(5)THETRUENATUREOFSEPARABLEGFTThedenitionofseparabilityhasalreadybeenintroducedin[5].Denition8.Wecallamappingf:RmRm!C`p;qx-separableorseparablewithrespecttoitsrstargument,ifitsufcesf=jjf(x;u)jji(u);(6)wherei:Rm!C`p;qisafunctionthatdoesnotdependonx.Analogouslywecallitseparableorseparablewithrespecttobotharguments,ifitsufcesf=jjf(x;u)jji;(7)withconstanti2C`p;q.Analogously,ageometricFouriertransformthatconsistsofseparablemappingsF1;F2iscalledseparable.Sepa-rabilityisacentralqualityformultiplication,shiftandconvolutionpropertiesofGFT.AlmosteverytransformfromapproachBandCisseparable.Ifthereexistanynonseparabletransformsthatareinvertible,isanissueofcurrentresearch.Therefore,theimportanceofthisclassofGFTsisobviousbecausetheapplicationsofatransformthatputsafunctionintoaspacefromwhichitmayneverreturnarerathersparse.Example9.FromalltheexamplesofspecialcasesofDenition1intheintroduction,onlysomecasesofthetwo-sidedtransform[15]andthecylindricaltransform[16]fordimensionshigherthantwoarenotseparable. 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