Will inverse DWT of approx. coefficients results in an approximate signal in original space?

Question

I have a signal $y$ in real-time space $V_0$. Hence assume $y = y^0$, the approximation coefficients of the signal at level $V^0$ (as per Mallat's pyramidal algorithm). I applied DWT and obtained approximation coeffcients $y_a^1 \in V^1$ and detail coefficients $y_d^1 \in W^1$. Now, I apply inverse DWT to $y_a^1$ by appending zero to all detail coefficients (i.e. $y_d^1 = 0$) and obtain $\hat{y}^0$. My question is whether $\hat{y}^0$ lies in $V^0$. As we neglected all the detail coefficients, does it result in different space? I am not a mathematician, but would like to know the answer to complete a proof in controllability.

Answer 1

The inverse DWT produces a linear combination of basis functions of $V_0$, thus of course results in an element of $V_0$, whatever the average and detail coefficients are.

Presumably your DWT is orthogonal, then the setting to zero of the detail coefficients is a projection, thus the norm of the reconstructed signal is smaller than the norm of the original signal.

You should care for terminology, $V_j,W_j$ are subspaces of $L^2(\Bbb R)$, as you also initially say. So if $y^0$ is a function of continuous time, so are $y^1_a,y^1_d$. The spaces of coefficient sequences are always the full $\ell^2(\Bbb Z)$, thus there are no questions of inclusion in subspaces.