Coiflet

Coiflets are discrete wavelets designed by Ingrid Daubechies, at the request of Ronald Coifman, to have scaling functions with vanishing moments. The wavelet is near symmetric, their wavelet functions have $N/3$ vanishing moments and scaling functions $N/3-1$, and has been used in many applications using Calderón–Zygmund operators.G. Beylkin, R. Coifman, and V. Rokhlin (1991),''Fast wavelet transforms and numerical algorithms'', Comm. Pure Appl. Math., 44, pp. 141–183Ingrid Daubechies, ''Ten Lectures on Wavelets'', Society for Industrial and Applied Mathematics, 1992,

Theory

Some theorems about Coiflets:

Theorem 1

For a wavelet system $\$, the following three
equations are equivalent:

:
\begin
\mathcal(0,l] = 0 & \textl =0,1,\ldots,L-1 \\
\sum_n (-1)^n n^l h 0 & \textl =0,1,\ldots,L-1 \\
H^(\pi)=0 & \textl=0,1,\ldots,L-1
\end
and similar equivalence holds between $\psi$ and $\tilde$

Theorem 2

For a wavelet system $\$, the following six equations
are equivalent:

:
\begin
\mathcal(t_0,l] = \delta & \text l=0,1,\ldots,L-1 \\
\mathcal(0,l] = t_0^l & \text l=0,1,\ldots,L-1 \\
\hat^(l)(0)=(-jt_0)^t& \textl=0,1,\ldots,L-1 \\
\sum_n (n-t_0)^l h \delta & \text l=0,1,\ldots,L-1 \\
\sum_n n^l h t_0^l & \text l=0,1,\ldots,L-1 \\
H^(0)=(-jt_0)^t & \text l=0,1,\ldots,L-1 \\
\end

and similar equivalence holds between $\tilde$ and $\tilde$

Theorem 3

For a biorthogonal wavelet system $\$, if either $\tilde$ or $\psi$
possesses a degree L of vanishing moments, then the following two equations are equivalent:

:
\begin
\mathcal(t_0,l] = \delta & \text l=0,1,\ldots, \bar-1 \\
\mathcal(t_0,l] = \delta & \textl=0,1,\ldots, \bar-1 \\
\end

for any $\bar$ such that $\bar \ll L$

Coiflet coefficients

Both the scaling function (low-pass filter) and the wavelet function (high-pass filter) must be normalised by a factor $1/\sqrt$. Below are the coefficients for the Wavelet#Scaling function, scaling functions for C6–30. The wavelet coefficients are derived by reversing the order of the scaling function coefficients and then reversing the sign of every second one (i.e. C6 wavelet = ).

Mathematically, this looks like $B_k = (-1)^k C_$, where ''k'' is the coefficient index, ''B'' is a wavelet coefficient, and ''C'' a scaling function coefficient. ''N'' is the wavelet index, i.e. 6 for C6.

Matlab function

F = coifwavf(W) returns the scaling filter associated with the Coiflet wavelet specified by the string W where W = 'coifN'. Possible values for ''N'' are 1, 2, 3, 4, or 5.{{cite web, title=coifwavf, url=http://www.mathworks.com/help/wavelet/ref/coifwavf.html, website=www.mathworks.com/, accessdate=22 January 2015

References

Orthogonal wavelets
Wavelets