# Meyer wavelet

The Meyer wavelet is an orthogonal wavelet proposed by Yves Meyer. It is infinitely differentiable with infinite support and defined in frequency domain in terms of function ${\displaystyle \nu }$ as:

${\displaystyle \Psi (\omega ):={\begin{cases}{\frac {1}{\sqrt {2\pi }}}\sin \left({\frac {\pi }{2}}\nu \left({\frac {3|\omega |}{2\pi }}-1\right)\right)e^{j\omega /2}&{\text{if }}2\pi /3<|\omega |<4\pi /3,\\{\frac {1}{\sqrt {2\pi }}}\cos \left({\frac {\pi }{2}}\nu \left({\frac {3|\omega |}{4\pi }}-1\right)\right)e^{j\omega /2}&{\text{if }}4\pi /3<|\omega |<8\pi /3,\\0&{\text{otherwise}},\end{cases}}}$

where:

${\displaystyle \nu (x):={\begin{cases}0&{\text{if }}x<0,\\x&{\text{if }}01.\end{cases}}}$

There are many different ways for defining this auxiliary function, which yields variants of the Meyer wavelet. For instance, another standard implementation adopts

${\displaystyle \nu (x):={\begin{cases}{x^{4}}(35-84x+70{x^{2}}-20{x^{3}})&{\text{if }}0
Spectrum of the Meyer wavelet.

The Meyer scale function is given by:

${\displaystyle \Phi (\omega ):={\begin{cases}{\frac {1}{\sqrt {2\pi }}}&{\text{if }}|\omega |<2\pi /3,\\{\frac {1}{\sqrt {2\pi }}}\cos \left({\frac {\pi }{2}}\nu \left({\frac {3|\omega |}{2\pi }}-1\right)\right)&{\text{if }}2\pi /3<|\omega |<4\pi /3,\\0&{\text{otherwise}}.\end{cases}}}$
Meyer scale function.

In the time-domain, the waveform of the Meyer mother-wavelet has the shape as shown in the following figure:

Meyer wavelet.

In 2015, Victor Vermehren Valenzuela and H. M. de Oliveira gave the explicitly expressions of Meyer wavelet and scale functions:

${\displaystyle \phi (t)=\left\{{\begin{array}{ll}{\frac {2}{3}}+{\frac {4}{3\pi }}&t=0,\\{\frac {\sin({\frac {2\pi }{3}}t)+{\frac {4}{3}}t\cos({\frac {4\pi }{3}}t)}{\pi t-{\frac {16\pi }{9}}t^{3}}}&otherwise,\end{array}}\right.}$

and

${\displaystyle \psi (t)=\psi _{1}(t)+\psi _{2}(t)}$

where

${\displaystyle \psi _{1}(t)={\frac {{\frac {4}{3\pi }}(t-{\frac {1}{2}})\cos[{\frac {2\pi }{3}}(t-{\frac {1}{2}})]-{\frac {1}{\pi }}\sin[{\frac {4\pi }{3}}(t-{\frac {1}{2}})]}{(t-{\frac {1}{2}})-{\frac {16}{9}}(t-{\frac {1}{2}})^{3}}},}$

and

${\displaystyle \psi _{2}(t)={\frac {{\frac {8}{3\pi }}(t-{\frac {1}{2}})\cos[{\frac {8\pi }{3}}(t-{\frac {1}{2}})]+{\frac {1}{\pi }}\sin[{\frac {4\pi }{3}}(t-{\frac {1}{2}})]}{(t-{\frac {1}{2}})-{\frac {64}{9}}(t-{\frac {1}{2}})^{3}}}.}$

## References

• Meyer (Y.), Ondelettes et Opérateurs, Hermann, 1990.
• Daubechies, (I.), Ten lectures on wavelets, CBMS-NSF conference series in applied mathematics, SIAM Ed., pp. 117–119, 137, 152, 1992.
• Victor Vermehren Valenzuela and H. M. de Oliveira, Close expressions for Meyer Wavelet and Scale Function, 2015, p.4.