= Wavelet series
{title2=$m=\sum_ka_{0,k}\phi_{0,k}+\sum_{j\geq0,k}b_{j,k}\psi_{j,k}$}
For an <orthonormal wavelet> and its <scaling function>, set $\phi_{j,k}(x)=2^{j/2}\phi(2^jx-k)$ 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.
Back to article page
