Convolution and Square in Abelian Groups III

We know that the functional equation \(f\!\star \! f (2\,t) =\lambda f(t)^2\) on the cyclic group of odd order d has a non-zero solution f when \(\lambda =\sqrt{a}\!+\!i\sqrt{b}\) where a, b are positive integers with \(a\!+\!b=d\) and \(a\equiv \frac{(d+1)^2}{4}\; \textrm{mod}\;4\) . We show here that in this case the function f can be chosen to be equal to the conjugate of its Fourier transform.