Suppose we have a fixed positive integer $n$ and three functions $f:\mathbb N \longrightarrow \mathbb N$ and $g:\mathbb N\times \mathbb N\longrightarrow \mathbb N$ and $a:\mathbb N\rightarrow \mathbb N$ with the relation: $$\text{for each}\;k\in \mathbb N,\;f(k)=a(1)\,g(1,k)+a(2)\,g(2,k)+\cdots+a(n)\,g(n,k)$$ can we apply Mobius inversion theorem to determine the coefficients $a(i)$ in terms of the functions $f$ and $g$?
I know that over $(\mathbb N,\leq)$ the Mobius function has a simple expression, that is, $\mu(i,i)=1$ and $\mu(i-1,i)=-1$ and all the other $\mu(j,i)=0$ for $j\not = i$ and for $j\not = i-1$.
but i could not see how to use this invesion in my situation. Thanks for your help!
In general, you can not determine each $a(i)$ if $f$ and $g$ are known. For example, let $n = 2$ and let $g(i,j)= 1$ for all $ i = 1, 2$ and all $j = 1, 2, \ldots$. Then for all $k$ we have $f(k) = a(1) + a(2)$ (in particular $f$ must be constant). If we knew, for example, that $f(k) = 3$ for all $k$ then we could not determine whether $[a(1), a(2)] = [1,2]$ or $[a(1), a(2)] = [2,1]$.