Multiresolution analysis

Remarks

Some authors only ask in clause 4 that the translates of $\varphi$ form a Riesz basis of $V_0$; an orthonormalization step then yields a function as in clause 4. Clauses 1 to 3 imply that $f \in V_0$ if and only if $f(2^j \cdot) \in V_j$. Let $W_j$ be the orthogonal complement of $V_j$ in $V_{j+1}$. Then $L^2(\R)$ is the orthogonal direct sum of all $W_j$. There is an orthonormal wavelet $\psi$ whose integer translates form an orthonormal basis of $W_0$.

Example: let $V_j$ be the set of $f \in L^2(\R)$ that are constant on each interval $[k 2^{-j}, (k+1) 2^{-j})$, $k \in \Z$. Nesting and scaling are clear, and step functions on dyadic intervals are dense in $L^2(\R)$. A function in every $V_j$ is constant on $[0, 2^{-j})$ and on $[-2^{-j}, 0)$ for all $j \le 0$, hence constant on $[0, \infty)$ and on $(-\infty, 0)$, hence $0$ since it is square-integrable. With $\varphi = 1_{[0, 1)}$, clause 4 holds; this is the Haar multiresolution analysis.

Non-example: $V_j = L^2(\R)$ for all $j$ satisfies nesting, scaling and density, but $\bigcap_j V_j = L^2(\R) \ne \set{0}$, so separation fails.