▁locally ▁compact ▁groups ▁G ▁the ▁space ▁ ▁does ▁not ▁contain ▁, ▁so ▁the ▁Fourier ▁transform ▁of ▁general ▁- functions ▁on ▁G ▁is ▁" not " ▁given ▁by ▁any ▁kind ▁of ▁integration ▁formula ▁( or ▁really ▁any ▁explicit ▁formula ). ▁To ▁define ▁the ▁ ▁Fourier ▁transform ▁one ▁has ▁to ▁resort ▁to ▁some ▁technical ▁trick ▁such ▁as ▁starting ▁on ▁a ▁dense ▁sub space ▁like ▁the ▁continuous ▁functions ▁with ▁compact ▁support ▁and ▁then ▁extending ▁the ▁is ometry ▁by ▁continu ity ▁to ▁the ▁whole ▁space . ▁This ▁unit ary ▁extension ▁of ▁the ▁Fourier ▁transform ▁is ▁what ▁we ▁mean ▁by ▁the ▁Fourier ▁transform ▁on ▁the ▁space ▁of ▁square ▁integra ble ▁functions . ▁ ▁The ▁dual ▁group ▁also ▁has ▁an ▁inverse ▁Fourier ▁transform ▁in ▁its ▁own ▁right ; ▁it ▁can ▁be ▁character ized ▁as ▁the ▁inverse ▁( or ▁ad joint , ▁since ▁it ▁is ▁unit ary ) ▁of ▁the ▁ ▁Fourier ▁transform . ▁This ▁is ▁the ▁content ▁of ▁the ▁ ▁Fourier ▁in version ▁formula ▁which ▁follows . ▁ ▁Theorem . ▁ ▁The ▁ad joint ▁of ▁the ▁Fourier ▁transform ▁restricted ▁to ▁continuous ▁functions ▁of ▁compact ▁support ▁is ▁the ▁inverse ▁Fourier ▁transform ▁ ▁where ▁ ▁is ▁the ▁dual ▁measure ▁to ▁. ▁ ▁In ▁the ▁case ▁ ▁the ▁dual ▁group ▁ ▁is ▁naturally ▁is omorphic ▁to ▁the ▁group ▁of ▁integers ▁ ▁and ▁the ▁Fourier ▁transform ▁special izes ▁to ▁the ▁computation ▁of ▁coefficients ▁of ▁Fourier ▁series ▁of ▁periodic ▁functions . ▁ ▁If ▁G ▁is ▁a ▁finite ▁group , ▁we ▁recover ▁the ▁discrete ▁Fourier ▁transform . ▁ ▁Note ▁that ▁this ▁case ▁is ▁very ▁easy