▁an ▁example , ▁suppose ▁, ▁so ▁we ▁can ▁think ▁about ▁ ▁as ▁ ▁by ▁the ▁pair ing ▁ ▁If ▁ ▁is ▁the ▁Leb es gue ▁measure ▁on ▁E uclidean ▁space , ▁we ▁obtain ▁the ▁ordinary ▁Fourier ▁transform ▁on ▁ ▁and ▁the ▁dual ▁measure ▁needed ▁for ▁the ▁Fourier ▁in version ▁formula ▁is ▁. ▁If ▁we ▁want ▁to ▁get ▁a ▁Fourier ▁in version ▁formula ▁with ▁the ▁same ▁measure ▁on ▁both ▁sides ▁( that ▁is , ▁since ▁we ▁can ▁think ▁about ▁ ▁as ▁its ▁own ▁dual ▁space ▁we ▁can ▁ask ▁for ▁ ▁to ▁equal ▁) ▁then ▁we ▁need ▁to ▁use ▁ ▁However , ▁if ▁we ▁change ▁the ▁way ▁we ▁identify ▁ ▁with ▁its ▁dual ▁group , ▁by ▁using ▁the ▁pair ing ▁ ▁then ▁Leb es gue ▁measure ▁on ▁ ▁is ▁equal ▁to ▁its ▁own ▁dual ▁measure . ▁ ▁This ▁convention ▁minim izes ▁the ▁number ▁of ▁factors ▁of ▁ ▁that ▁show ▁up ▁in ▁various ▁places ▁when ▁computing ▁Fourier ▁transform s ▁or ▁inverse ▁Fourier ▁transform s ▁on ▁E uclidean ▁space . ▁( In ▁effect ▁it ▁limits ▁the ▁ ▁only ▁to ▁the ▁exponent ▁rather ▁than ▁as ▁some ▁mess y ▁factor ▁outside ▁the ▁integral ▁sign .) ▁ ▁Note ▁that ▁the ▁choice ▁of ▁how ▁to ▁identify ▁ ▁with ▁its ▁dual ▁group ▁affect s ▁the ▁meaning ▁of ▁the ▁term ▁" self - du al ▁function ", ▁which ▁is ▁a ▁function ▁on ▁ ▁equal ▁to ▁its ▁own ▁Fourier ▁transform : ▁using ▁the ▁classical ▁pair ing ▁ ▁the ▁function ▁ ▁is ▁self - du al , ▁but ▁using ▁the ▁( clean er ) ▁pair