Edit: I added smoothness, hoping to simplify the problem with this additional assumption.

Let me motivate this question first: In signal analysis it is often of interest to understand when a certain function has a fast decaying representation with respect to some basis. I encountered an example where I expected such a fast decaying representation but was unable to show it myself:

The situation is that one is given a sequence of functions whose supports are only overlapping with their nearest neighbors. Intuitively, I expected that if I would write any of those function in terms of the Gram-Schmidt orthonormalized sequence of all functions, then the expansion coefficients should decay fast.

Let me now make this more precise:

Take a family of intervals $I_{2i}:=(i,i+1)$ and $I_{2i+1}=(\frac{1}{2}+i,\frac{1}{2}+i+1)$ with $i \ge 0$ Those intervals are almost disjoint in the sense that $I_i \cap I_j=\emptyset$ if and only if $\left\lvert i-j \right\rvert >1,$ so we only have a nearest neighbor overlap.

We consider a family of smooth functions $h_i:I_i \rightarrow \mathbb{R}$ that are $L^2$ normalized (but not orthogonal) where $h_{i+1}$ is just a translate of $h_i$ by $\frac{1}{2}$ to the right.

We assume that $\langle h_0,h_1 \rangle \neq 0$ and $\langle h_1,h_2 \rangle \neq 0$ to avoid trivialities. Due to the condition on the domain, all other $h_i$ however (besides $h_0$, $h_1,$ and $h_2$) have zero overlap with $h_1$.

We now consider the Gram-Schmidt orthonormalized sequence $(g_i)$ of the $h_i$, i.e. $g_0:=h_0$ and $g_1:=\frac{h_1-\langle h_1,g_0\rangle g_0}{\left\lVert h_1-\langle h_1,g_0\rangle g_0 \right\rVert}$ and so on. Note that also $g_2,g_3,...$ and so on may now have non-vanishing overlap with $h_1$, although this one should intuitively still be small.

Clearly, $h_1=\sum_{n=0}^{\infty} a_n g_n$ for some $a_n$.

I would like to know: How rapidly do the coefficients $a_n$ decay? In particular, is it true that (the following notation means up to a constant) $\left\lvert a_n \right\rvert \lesssim L^n$ for some $L\in (0,1)$?