# Question about regular representation of compact group.

I first define the setting for my question. Let $$G$$ be a compact group with probability Haar measure $$\mu_G$$. Denote by $$\lambda$$ the left regular representation on $$L^2(G)$$ defined for $$f \in L^2(G)$$ and $$g \in G$$ by $$(\lambda(g) f)(x) = f(g^{-1}x)$$ for $$x \in G$$. For $$f \in L^2(G)$$ we denote by $$f^{*}$$ the function defined by $$f^{*}(g) = \overline{f(g^{-1})}$$ for $$g \in G$$. Moreover, for $$f_1,f_2 \in L^2(G)$$ we define the convolution $$f_1*f_2$$ as $$(f_1 * f_2)(g) = \int f_1(h)f_2(h^{-1}g) \, d\mu_G(h)$$

For my question, let $$V \subset L^2(G)$$ be a finite dimensional subrepresentation of $$\lambda$$ of the form $$V = \langle \lambda(G)v \rangle$$ for some $$v \in L^2(G)$$. How to prove the following claim?

Claim: There exists a unique function $$f_v \in C(G)$$ with the properties that $$f_v = f_v * f_v = f_v^{*} \quad\text{ and }\quad C(G) * f_v = V.$$ Moreover $$\dim(V) = ||f_v||_2^2.$$