Wavelet series ($m={\sum }_k{a}_{0,k}{\varphi }_{0,k}+{\sum }_{j\ge 0,k}{b}_{j,k}{\psi }_{j,k}$)

For an orthonormal wavelet and its scaling function, set ${\varphi }_{j,k}(x)=2^{j/2}\varphi (2^{j}x-k)$ and similarly for $\psi$. The coarse scaling functions together with nonnegative-resolution wavelets form an orthonormal basis of $L^2(\mathbb{R})$. The coefficients are the corresponding L2 inner products. The series converges in the L2 norm, and the Parseval identity for a Hilbertian basis equates squared norm with the sum of squared coefficients. Pointwise convergence requires additional hypotheses.