▁on ▁the ▁real ▁line ▁and , ▁just ▁as ▁for ▁periodic ▁functions , ▁these ▁functions ▁can ▁be ▁recovered ▁from ▁their ▁Fourier ▁transform s ; ▁and ▁ ▁Complex - valu ed ▁functions ▁on ▁a ▁finite ▁ab elian ▁group ▁have ▁discrete ▁Fourier ▁transform s , ▁which ▁are ▁functions ▁on ▁the ▁dual ▁group , ▁which ▁is ▁a ▁( non - can on ically ) ▁is omorphic ▁group . ▁Moreover , ▁any ▁function ▁on ▁a ▁finite ▁group ▁can ▁be ▁recovered ▁from ▁its ▁discrete ▁Fourier ▁transform . ▁ ▁The ▁theory , ▁introduced ▁by ▁Lev ▁Pon try agin ▁and ▁combined ▁with ▁the ▁H aar ▁measure ▁introduced ▁by ▁John ▁von ▁Ne umann , ▁André ▁We il ▁and ▁others ▁depends ▁on ▁the ▁theory ▁of ▁the ▁dual ▁group ▁of ▁a ▁locally ▁compact ▁ab elian ▁group . ▁ ▁It ▁is ▁analog ous ▁to ▁the ▁dual ▁vector ▁space ▁of ▁a ▁vector ▁space : ▁a ▁finite - dimensional ▁vector ▁space ▁V ▁and ▁its ▁dual ▁vector ▁space ▁V * ▁are ▁not ▁naturally ▁is omorphic , ▁but ▁the ▁end omorphism ▁algebra ▁( matrix ▁algebra ) ▁of ▁one ▁is ▁is omorphic ▁to ▁the ▁opposite ▁of ▁the ▁end omorphism ▁algebra ▁of ▁the ▁other : ▁ ▁via ▁the ▁trans pose . ▁Similarly , ▁a ▁group ▁G ▁and ▁its ▁dual ▁group ▁ ▁are ▁not ▁in ▁general ▁is omorphic , ▁but ▁their ▁end omorphism ▁rings ▁are ▁opposite ▁to ▁each ▁other : ▁. ▁More ▁categor ically , ▁this ▁is ▁not ▁just ▁an ▁isomorphism ▁of ▁end omorphism ▁al gebras , ▁but ▁a ▁cont rav ari ant ▁equivalence ▁of ▁categories ▁– ▁see ▁categor ical ▁consider ations