We have a $L^2(\mathbb{R})$ multiresolution $(V_j)_j$ through the family of scaling functions $\phi_j$ and the orthogonal wavelets $\psi_j$ such that $$\phi(x)=\sum_{n\in\mathbb{N}}a_n\phi(2x-n)$$ and $$\psi(x)=\sum_{n\in\mathbb{N}}b_n\psi(2x-n)$$ with $b_n=(-1)^na_{1-n}$
We know that $\psi_{j,k}(x)=2^{j/2}\psi(2^jx-k)$ and $\phi_{j,k}=2^{j/2}\phi(2^jx-k)$
I must show that $$\psi_{j,k}=\sum_{n\in\mathbb{N}}b_{n-2k}\phi_{j-1,k}$$ and $$\phi_{j,k}=\sum_{n\in\mathbb{N}}a_{n-2k}\phi_{j-1,k}$$
All I was able to do through simple substitution is find that
$$\psi_{j,k}=2^{-1/2}\sum_{n\in\mathbb{N}}b_{n-2k}\psi_{j+1,k}$$ and $$\phi_{j,k}=2^{-1/2}\sum_{n\in\mathbb{N}}a_{n-2k}\phi_{j+1,k}$$
Anyone know how to proceed ? Am I missing something ? Thanks.