▁set ▁of ▁non - t rivial ▁( that ▁is , ▁not ▁ident ically ▁zero ) ▁multip lic ative ▁linear ▁function als ▁on ▁the ▁group ▁algebra ; ▁see ▁section ▁ 3 4 ▁of ▁. ▁This ▁means ▁the ▁Fourier ▁transform ▁is ▁a ▁special ▁case ▁of ▁the ▁G elf and ▁transform . ▁ ▁Plan cher el ▁and ▁ ▁Fourier ▁in version ▁the or ems ▁ ▁As ▁we ▁have ▁stated , ▁the ▁dual ▁group ▁of ▁a ▁locally ▁compact ▁ab elian ▁group ▁is ▁a ▁locally ▁compact ▁ab elian ▁group ▁in ▁its ▁own ▁right ▁and ▁thus ▁has ▁a ▁H aar ▁measure , ▁or ▁more ▁precisely ▁a ▁whole ▁family ▁of ▁scale - related ▁H aar ▁measures . ▁ ▁Theorem . ▁Cho ose ▁a ▁H aar ▁measure ▁ ▁on ▁ ▁and ▁let ▁ ▁be ▁the ▁dual ▁measure ▁on ▁ ▁as ▁defined ▁above . ▁If ▁ ▁is ▁continuous ▁with ▁compact ▁support ▁then ▁ ▁and ▁▁ ▁In ▁particular , ▁the ▁Fourier ▁transform ▁is ▁an ▁ ▁is ometry ▁from ▁the ▁complex - valu ed ▁continuous ▁functions ▁of ▁compact ▁support ▁on ▁G ▁to ▁the ▁- functions ▁on ▁ ▁( using ▁the ▁- norm ▁with ▁respect ▁to ▁ μ ▁for ▁functions ▁on ▁G ▁and ▁the ▁- norm ▁with ▁respect ▁to ▁ ν ▁for ▁functions ▁on ▁). ▁ ▁Since ▁the ▁complex - valu ed ▁continuous ▁functions ▁of ▁compact ▁support ▁on ▁G ▁are ▁- den se , ▁there ▁is ▁a ▁unique ▁extension ▁of ▁the ▁Fourier ▁transform ▁from ▁that ▁space ▁to ▁a ▁unit ary ▁operator ▁▁▁▁ ▁and ▁we ▁have ▁the ▁formula ▁▁▁▁ ▁Note ▁that ▁for ▁non - comp act