version ▁formula ▁for ▁L 1 - functions ▁▁ ▁The ▁dual ▁group ▁of ▁a ▁locally ▁compact ▁ab elian ▁group ▁is ▁used ▁as ▁the ▁underlying ▁space ▁for ▁an ▁abstract ▁version ▁of ▁the ▁Fourier ▁transform . ▁ ▁If ▁, ▁then ▁the ▁Fourier ▁transform ▁ ▁is ▁the ▁function ▁ ▁on ▁ ▁defined ▁by ▁ ▁where ▁the ▁integral ▁is ▁relative ▁to ▁H aar ▁measure ▁ ▁on ▁. ▁ ▁This ▁is ▁also ▁denoted ▁. ▁Note ▁the ▁Fourier ▁transform ▁depends ▁on ▁the ▁choice ▁of ▁H aar ▁measure . ▁It ▁is ▁not ▁too ▁difficult ▁to ▁show ▁that ▁the ▁Fourier ▁transform ▁of ▁an ▁ ▁function ▁on ▁ ▁is ▁a ▁bounded ▁continuous ▁function ▁on ▁ ▁which ▁van ishes ▁at ▁infinity . ▁ ▁Fourier ▁In version ▁Formula ▁for ▁- Function s . ▁For ▁each ▁H aar ▁measure ▁ ▁on ▁ ▁there ▁is ▁a ▁unique ▁H aar ▁measure ▁ ▁on ▁ ▁such ▁that ▁whenever ▁ ▁and ▁, ▁we ▁have ▁ ▁If ▁ ▁is ▁continuous ▁then ▁this ▁identity ▁holds ▁for ▁all ▁. ▁ ▁The ▁inverse ▁Fourier ▁transform ▁of ▁an ▁integra ble ▁function ▁on ▁ ▁is ▁given ▁by ▁ ▁where ▁the ▁integral ▁is ▁relative ▁to ▁the ▁H aar ▁measure ▁ ▁on ▁the ▁dual ▁group ▁. ▁The ▁measure ▁ ▁on ▁ ▁that ▁appears ▁in ▁the ▁Fourier ▁in version ▁formula ▁is ▁called ▁the ▁dual ▁measure ▁to ▁ ▁and ▁may ▁be ▁denoted ▁. ▁ ▁The ▁various ▁Fourier ▁transform s ▁can ▁be ▁class ified ▁in ▁terms ▁of ▁their ▁domain ▁and ▁transform ▁domain ▁( the ▁group ▁and ▁dual ▁group ) ▁as ▁follows ▁( note ▁that ▁ ▁is ▁Circle ▁group ): ▁ ▁As