Classical Matrix-Vector multiplication Complexity of standard matrices Why are standard unitary transforms such as the Fourier and the Hadamard transforms believed to have a multiplicative complexity (number of multiplications) of $O(n^{1+\delta_{m}})$ and an additive complexity of $O(n^{1+\delta_{a}})$ (number of additions) where both $\delta_{m}$ and $\delta_{a}$ cannot be $0$ simultaneously?

(Please refer: Section 2.2 Page 15 of "Complexity Lower Bounds Using Linear Algebra", by Satyanarayana V. Lokam, for candidate matrices conjectured to have an arithmetic complexity (multiplicative + additive complexity)  of $\Omega(n^{1+\delta})$).
Essentially, your are asking why people believe that DFT or DHT do not have linear size arithmetic circuits. First, researcher tried to find such circuits for decades but did not succeed. Second, there are superlinear lower bounds (Morgenstern's theorem, see the book by Bürgisser, Clausen, Shokrollahi) in restricted models. From Morgenstern's theorem, in follows that in linear arithmetic circuits of linear size over the complex numbers  for the DFT, the absolute values of the constants used in the circuit go to infinity as $n$ goes to infinity.