Any function $f:\mathbb{F}_q \rightarrow \mathbb{C}$ has a unique representation $f(x)=f_{\delta}\delta(x)+\sum_{\chi} f_{\chi}\chi(x)$

While going through an article I have come across the following fact:
Any function $f:\mathbb{F}_q \rightarrow \mathbb{C}$ has a unique representation $$f(x)=f_{\delta}\delta(x)+\sum_{\chi} f_{\chi}\chi(x)$$ where the sum ranges over all multiplicative characters of $\mathbb{F}_q$ and $\delta$ is the function taking 1 if $x=0$ and 0 otherwise.
My question is, what is the proof of this statement? Please suggest some proof or some material I can go through, because I have no idea what kind of theory is required. Once this statement holds, one can evaluate the constants $f_{\delta}$ and $f_{\chi}$.

The multiplicative group of units of $\mathbb{F}_q$ is a cyclic group. As with any finite abelian group, the characters form a basis for functions from the units of $F_q$ to $\mathbb{C}$. By adding $\delta$ you therefore extend this to a basis of functions from all of $F_q$ to $\mathbb{C}$.

The above result concerning finite abelian groups, is a special case of a general result about finite groups: The rows of the character table of a finite group $G$, give a basis for functions $G\to\mathbb{C}$, which are invariant under conjugation: e.g. $f(g)=f(hgh^{-1})$.

Any text on group representation theory will cover this near the start.