It is well-known that the Haar wavelet system forms an orthonormal basis for $L^2[0,1]$. I am interested in forming a similar orthonormal basis for $L^2[0,\frac58]$. We write $[0, \frac58] = [0,\frac12] \cup [\frac12, \frac58]$ and let $e_1 = \frac{1_{[0,1/2]}}{\|1_{[0,1/2]}\|}$ and $e_2 = \frac{1_{[1/2,5/8]}}{\|1_{[1/2,5/8]}\|}$. Since $\frac58$ is a dyadic rational the support of some of the Haar wavelets are contained in $[0, \frac58]$. I am interested to know whether $e_1$ and $e_2$ along with the Haar wavelets whose support is contained in $[0,\frac58]$ forms an orthonormal basis.
With some computations we can easily get that this sequence of wavelets is at the minimum orthonormal. To show that its closed span is dense I am trying to argue that if there is $f \in L^2[0,\frac58]$ such that $f \perp \{e_n\}_{n=1}^\infty$, then $f = 0$ a.e. Any help is welcomed!