▁In ▁mathematics , ▁specifically ▁in ▁harm onic ▁analysis ▁and ▁the ▁theory ▁of ▁topological ▁groups , ▁Pon try agin ▁du ality ▁explains ▁the ▁general ▁properties ▁of ▁the ▁Fourier ▁transform ▁on ▁locally ▁compact ▁ab elian ▁groups , ▁such ▁as ▁, ▁the ▁circle , ▁or ▁finite ▁cy clic ▁groups . ▁The ▁Pon try agin ▁du ality ▁theorem ▁itself ▁states ▁that ▁locally ▁compact ▁ab elian ▁groups ▁identify ▁naturally ▁with ▁their ▁bid ual . ▁ ▁The ▁subject ▁is ▁named ▁after ▁Lev ▁Sem en ov ich ▁Pon try agin ▁who ▁laid ▁down ▁the ▁found ations ▁for ▁the ▁theory ▁of ▁locally ▁compact ▁ab elian ▁groups ▁and ▁their ▁du ality ▁during ▁his ▁early ▁mathematical ▁works ▁in ▁ 1 9 3 4 . ▁Pon try agin ' s ▁treatment ▁re lied ▁on ▁the ▁group ▁being ▁second - count able ▁and ▁either ▁compact ▁or ▁discrete . ▁This ▁was ▁improved ▁to ▁cover ▁the ▁general ▁locally ▁compact ▁ab elian ▁groups ▁by ▁Eg bert ▁van ▁K amp en ▁in ▁ 1 9 3 5 ▁and ▁André ▁We il ▁in ▁ 1 9 4 0 . ▁ ▁Introduction ▁Pon try agin ▁du ality ▁places ▁in ▁a ▁un ified ▁context ▁a ▁number ▁of ▁observations ▁about ▁functions ▁on ▁the ▁real ▁line ▁or ▁on ▁finite ▁ab elian ▁groups : ▁▁ ▁Su it ably ▁regular ▁complex - valu ed ▁periodic ▁functions ▁on ▁the ▁real ▁line ▁have ▁Fourier ▁series ▁and ▁these ▁functions ▁can ▁be ▁recovered ▁from ▁their ▁Fourier ▁series ; ▁ ▁Su it ably ▁regular ▁complex - valu ed ▁functions ▁on ▁the ▁real ▁line ▁have ▁Fourier ▁transform s ▁that ▁are ▁also ▁functions