Show that both side of the Plancherel formula are linear in $f$.

Question

Plancherel Formula: Let $f$ and $h$ be functions on G. Then, $$\sum_{s\in G}f(s^{-1})h(s)=\frac{1}{|G|}\sum_{i}d_{i}Tr(\hat{f}(\rho_{i})\hat{h}(\rho_{i}))$$.

I understand $f$ is linear if $f_1+f_2(x)=f_1(x)+f_2(x)$ and $(cf)(x)=cf(x)$, but I'm not sure how to physically show that each side in fact linear in $f$.

Answer 1

To elaborate on J.W. Tanner's comments above, "linear in $f$" means the following:

• For functions $f,g$ on $G$, $$\sum_{s \in G} (f + g)(s^{-1}) h(s) = \frac{1}{|G|} \sum_i d_i \operatorname{Tr}((\widehat{f+g})(\rho_i) \hat{h}(\rho_i)).$$
• For a scalar $c$, $$\sum_{s \in G} (cf)(s^{-1}) h(s) = \frac{1}{|G|} \sum_i d_i \operatorname{Tr}((\widehat{cf})(\rho_i) \hat{h}(\rho_i)).$$