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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 $$\dim(V) = ||f_v||_2^2.$$

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