I was wondering if there is a way to make more sense out of a fourier-like series like the following:

Suppose we want to decompose a function f(x) as

f(x)= sum_{k=1}^ \infty a_k e^ {2pi i /k x}.

Normally the power is like 2pi i k x, but I was wondering about a sum of lower and lower frequencies as opposed to higher frequencies.

I think a better way to approach this concept would be to suppose the function can be decomposed as

f(x) = sum_{k=0}^ N a_k e^ {2pi i x/2^ k}.

Then you can recover these a_n by integrating:

For any M >= N, for all 0 <= n <= N,

a_n = 1/2^M \int_0^ {2^ M} f(x)e^ {-2pi i x/2^ n}dx.

Idk, I'd like to see something in this vein but more rigorous.