My question is this: if $h(k)$ preserves the complex conjugation property (in other words, $h(k) = h(-k)$, $k$ can be just $-n$, $-n+1$, ..., $0$, $1$, ..., $n-1$),then $ikh(k)$ also preserves complex conjugation property which is easy to derive. But actually $ikh(k)$ does not preserve the complex conjugation property. Because we know $h(-n)$ must be real value for $h(k)$ is conjugation. Then $ikh(-n)$ must be image value which destroy the complex conjugation property. How to explain this?