I am studying about fourier series on a finite abelian group $G$. In the text book are defined the translation-invariant subspaces of $L(G)$ (the algebra of all functions of $G$ in $\mathbb{C}$ with the convolution product) as the subspaces $S$ of $L(G)$ such that for every $f \in S$ and for all $g \in G$ the function $f_g(x) =f(x-g)$ belongs to $S$.
There is a Theorem that indicates that "Every one-dimensional translation-invariant subspace of $L(G)$ is spanned by a character of $G$."
But, what happens to the translation-invariant subspaces of a larger dimension than one?. Can you give me a example, please?
Thanks.
There is what I noticed :
One-dimensional translation-invariant subspace means $$\forall (x,g) \in G^2,\qquad f(x-g) = C(g) f(x)$$ and $g \mapsto C(g)$ is clearly a character spanning the subspace.
Two-dimensional translation-invariant subspace means $$\forall (x,g) \in G^2,\ \ (a,b) \in \mathbb{C}, \quad a f_1(x-g) +b f_2(x-g) = c(g,a,b) f_1(x)+ d(g,a,b) f_2(x)$$ The linear operator $T_g: \mathbb{C}^2 \to \mathbb{C}^2$ sending $(a,b)$ to $c(g,a,b),d(g,a,b)$ fulfill $T_{gg'} = T_g T_{g'}$ and $g \mapsto T_g$ is a $2$-dimensional representation of $G$, but I'm not sure how it spans the subspace