I define the mother Haar wavelet to be: \begin{align} \phi(t) = \begin{cases} 1 &\mbox{if } 0 \leq t < 1/2 \\ -1 & \mbox{if } 1/2 \leq t \leq 1 \\ 0 &\mbox{otherwise}. \end{cases} \end{align} Now, we have that the sequence of rescaled wavelets is:
$$ \phi_n(t) = 2^{j/2} \phi(2^jt - k), \quad n = 2^j + k. $$
Basically, we have that any integer $n$ can be uniquely written as $2^j + k$ for some $j \geq 0$ and $0 \leq k < 2^j$.
From here, I would like to show that $\phi_n$ is a complete and orthonormal sequence within $L_2 [0,1]$.
The orthonormal part is trivial for me, however, to show that it is also complete, I would like to use Parseval's Theorem.
Essentially, it claims that for any orthonormal sequence in $L_2 [0,1]$, if the only element $x \in L_2 [0,1]$ that satisfies $E(x\cdot\phi_n) =0$ for all $n$ is $x=0$, then the finite linear combinations of the functions $\{\phi_n\}$ form a dense subset of $L_2 [0,1]$. Then by definition, if the finite linear combinations of the functions $\{\phi_n\}$ form a dense subset of $L_2 [0,1]$, we have that it is a complete orthonormal sequence.
I start this by first choosing $x \in L_2 [0,1]$. Then, I have that $E(\phi_n(t)\cdot x) = E(2^{j/2} \phi(2^jt - k)\cdot x)$ (where $E$ is the expectation). Here, I want to show that this is $0$ only if $x=0$. However, I am not sure how to handle the expectation, and am not sure with what I am integrating to. Does anyone have a hint or know if my approach in using Parseval's Theorem is the right way to go? thanks!
Add-on: The version of Parseval I am using is from Stochastic Calculus and Financial Applications by Steele:
