Meyer Wavelet

The Meyer wavelet is a particular family of wavelet functions that form an orthogonal set and possess compact support in the frequency domain. Unlike many other wavelets whose definitions are given piece‑wise in time, the Meyer construction starts with a smooth transition function defined in frequency space and then translates it back to the time domain. The result is a wavelet that is infinitely differentiable as a function of time while still being strictly limited to a finite band of frequencies.

Because it balances smoothness with precise frequency localisation, the Meyer wavelet is especially useful for multi‑resolution analysis, where one wishes to examine a signal at several levels of detail without introducing spurious artefacts. Its orthogonal nature means that any square‑integrable function can be represented exactly as a sum of scaled and shifted copies of the wavelet, making it a convenient basis for numerical approximation, signal compression, and denoising tasks.

You will encounter the Meyer wavelet in contexts ranging from image processing and audio analysis to the numerical solution of partial differential equations. It also appears when researchers need a mathematically tractable wavelet that still respects the demands of physical modelling, such as representing electron density functions or other smooth fields in quantum‑chemical calculations.