ζ = (E^(((2  π I)/8)))

a = (∑_(k = 0)^7)(k + 1) (x^k)

alist = CoefficientList[a, x]

φ = 
      Chop[Fourier[alist,  FourierParameters -> {1, 1}]]

Table[a /. {x -> (ζ^k)}, {k, 0, 7}] // N

InverseFourier[φ, 
      FourierParameters -> {1, 1}]

b = (∑_(k = 0)^7) (x^k)

blist = CoefficientList[b, x]

alist = PadRight[alist, 16]

blist = PadRight[blist, 16]

φ = 
      Chop[Fourier[alist,  FourierParameters -> {1, 1}]]

ψ = 
      Chop[Fourier[blist,  FourierParameters -> {1, 1}]]

φ*ψ

inv = 
      Chop[InverseFourier[φ*ψ,  
          FourierParameters -> {1, 1}]]

CoefficientList[a*b, x]