Coiflet

Example

Coiflets are a family of compactly supported orthonormal wavelets constructed by Ingrid Daubechies and named after Ronald Coifman. For an even integer $L \ge 2$, the coiflet of order $L$ is a real orthonormal wavelet $\psi$ with scaling function $\varphi$ such that:

1. $\psi$ has $L$ vanishing moments: $\int_{\R} x^l \psi(x) \, dx = 0$ for $l = 0, 1, \dots, L-1$.
2. $\int_{\R} \varphi(x) \, dx = 1$ and $\int_{\R} x^l \varphi(x) \, dx = 0$ for $l = 1, \dots, L-1$.

The low-pass filter of the coiflet of order $L$ has $3L$ coefficients, so $\varphi$ and $\psi$ are supported on intervals of length $3L - 1$. For $L = 2$ the filter has $6$ coefficients.