Generalized small-dimension lemma and d’Alembert type functional equation on compact groups

Let $${\mathbb {C}}$$ be the set of complex numbers and $$\sigma $$ be a continuous automorphism and $$\tau $$ be a continuous anti-automorphism such that $$\sigma ^{2}=\tau ^{2}=id.$$ The purpose of this paper is to generalize the small-dimension lemma [20, Small Dimension Lemma] and by help of it we find on any compact group G the non-zero continuous solutions $$f:G\rightarrow {\mathbb {C}}$$ of the functional equation $$\begin{aligned} f(x\sigma (y))+f(\tau (y)x)=2f(x)f(y), \ \ \ x,y \in G, \end{aligned}$$ in terms of continuous characters of G.