Harmonic Analysis on Vector Spaces over Finite Fields

• (i) G = ZN = Z/NZ = {0, 1, 2, ....., N − 1} with addition modulo N . For 0 ≤ n ≤ N − 1 let γn : G → S, γn(m) = exp(2πimn/N). Then {γ0, ....., γN−1} is a complete list of the characters so that ZN is isomorphic to ZN . An example of a primitive N ’th root of unity is ω := exp 2πi/N . • (ii) G = T = R/Z; for n ∈ Z let γn : G→ S, γn(x) = exp(2πinx). Then G∗ = {γn : n ∈ Z} so that G∗ is isomorphic to Z. • (iii) G = Z; for θ ∈ T let γθ : G → S, γθ(n) = exp(2πinθ). Then G∗ = {γθ : θ ∈ T} so that G∗ is isomorphic to T. • (iv) G = R; for ξ ∈ R let γξ : G → S, γξ(x) = exp(2πixξ). Then G∗ = {γξ : ξ ∈ R} so that G∗ is isomorphic to R. • (v) (G1 ×G2) is isomorphic to G1 ×G2.