The Meyer wavelet is an orthogonal

wavelet A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...

proposed by

Yves Meyer Yves F. Meyer (; born 19 July 1939) is a French mathematician. He is among the progenitors of wavelet theory, having proposed the Meyer wavelet. Meyer was awarded the Abel Prize in 2017. Biography Born in Paris to a Jewish family, Yves Meyer ...

. As a type of a

continuous wavelet {{Unreferenced, date=December 2009 In numerical analysis, continuous wavelets are functions used by the continuous wavelet transform. These functions are defined as analytical expressions, as functions either of time or of frequency. Most of the co ...

, it has been applied in a number of cases, such as in

adaptive filter An adaptive filter is a system with a linear filter that has a transfer function controlled by variable parameters and a means to adjust those parameters according to an optimization algorithm. Because of the complexity of the optimization algorit ...

s, fractal

random fields In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as \mathbb^n). That is, it is a function f(x) that takes on a random value at each point x \in \mathbb^n(or some other ...

, and multi-fault classification. The Meyer wavelet is infinitely differentiable with infinite support and defined in frequency domain in terms of function

\nu

as :

\Psi(\omega) := \begin
 \frac  \sin\left(\frac  \nu \left(\frac -1\right)\right) e^ & \text 2 \pi /3<, \omega, < 4 \pi /3, \\
 \frac  \cos\left(\frac  \nu \left(\frac-1\right)\right) e^ & \text 4 \pi /3<,  \omega, < 8 \pi /3, \\
 0 & \text,
\end

where :

\nu (x) := \begin
 0 & \text x < 0, \\
 x & \text 0< x < 1, \\
 1 & \text x > 1.
\end

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

\nu (x) := \begin
 x^4 (35 - 84x + 70x^2 - 20x^3) & \text 0 < x < 1, \\
 0 & \text.
\end

The Meyer scale function is given by :

\Phi(\omega) := \begin
 \frac & \text ,  \omega,  < 2 \pi/3, \\
 \frac \cos\left(\frac \nu \left(\frac - 1\right) \right)  & \text 2\pi/3 < , \omega,  < 4\pi/3, \\
 0 & \text.
\end

In the

time domain Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the c ...

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

Close expressions

Valenzuela and de Oliveira give the explicit expressions of Meyer wavelet and scale functions: :

\phi(t) = \begin
 \frac + \frac & t = 0, \\
 \frac & \text,
\end

and :

\psi(t) = \psi_1(t) + \psi_2(t),

where :

\psi_1(t) = \frac,

\psi_2(t) = \frac.

References

External links

{{commons, Wavelet, Wavelet
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