Coiflet

Para un sistema de ondículas${\displaystyle \{\phi ,{\tilde {\phi }},\psi ,{\tilde {\psi }},h,{\tilde {h}},g,{\tilde {g}}\}}$Las siguientes tres ecuaciones son equivalentes:

${\displaystyle {\begin{array}{lcl}{\mathcal {M_{\tilde {\psi }}}}(0,l]=0&{\text{para }}l=0,1,\ldots ,L-1\\\sum _{n}(-1)^{n}n^{l}h[n]=0&{\text{para }}l=0,1,\ldots ,L-1\\H^{(l)}(\pi )=0&{\text{para }}l=0,1,\ldots ,L-1\end{array}}}$

y una equivalencia similar se cumple entre${\displaystyle \psi }$y${\displaystyle {\tilde {h}}}$

Para un sistema de ondículas${\displaystyle \{\phi ,{\tilde {\phi }},\psi ,{\tilde {\psi }},h,{\tilde {h}},g,{\tilde {g}}\}}$Las siguientes seis ecuaciones son equivalentes:

${\displaystyle {\begin{array}{lcl}{\mathcal {M_{\tilde {\phi }}}}(t_{0},l]=\delta [l]&{\text{para }}l=0,1,\ldots ,L-1\\{\mathcal {M_{\tilde {\phi }}}}(0,l]=t_{0}^{l}&{\text{para }}l=0,1,\ldots ,L-1\\{\hat {\phi }}^{(}l)(0)=(-jt_{0})^{t}&{\text{para }}l=0,1,\ldots ,L-1\\\sum _{n}(n-t_{0})^{l}h[n]=\delta [l]&{\text{para }}l=0,1,\ldots ,L-1\\\sum _{n}n^{l}h[n]=t_{0}^{l}&{\text{para }}l=0,1,\ldots ,L-1\\H^{(l)}(0)=(-jt_{0})^{t}&{\text{para }}l=0,1,\ldots ,L-1\\\end{array}}}$

y una equivalencia similar se cumple entre${\displaystyle {\tilde {\psi }}}$y${\displaystyle {\tilde {h}}}$

Para un sistema de ondículas biorthogonales${\displaystyle \{\phi ,\psi ,{\tilde {\phi }},{\tilde {\psi }}\}}$, si alguno${\displaystyle {\tilde {\psi }}}$o${\displaystyle \psi }$ Si posee un grado L de momentos nulos, entonces las siguientes dos ecuaciones son equivalentes:

${\displaystyle {\begin{array}{lcl}{\mathcal {M_{\tilde {\psi }}}}(t_{0},l]=\delta [l]&{\text{para }}l=0,1,\ldots ,{\bar {L}}-1\\{\mathcal {M_{\psi }}}(t_{0},l]=\delta [l]&{\text{para }}l=0,1,\ldots ,{\bar {L}}-1\\\end{array}}}$

para cualquier${\displaystyle {\bar {L}}}$de tal manera que${\displaystyle {\bar {L}}\ll L}$