Hermite polynomials

The Hermite functions are obtained from the Hermite polynomials by this multiplied by the density of the Gaussian distribution.

They are a very good example of the definition (creation ) of an orthonormal basis, similar to the Sinus-/Kosinusfunktionen. While the latter are able, by means of spectral analysis ( Fourier analysis) to decompose a periodic signal into a frequency spectrum, the Hermite functions allow the description of singular events.

They have an important meaning in physics to construct the orthonormal solution functions of the quantum harmonic oscillator. Motivated by the creation and annihilation operators of quantum mechanics one obtains the following recursive representation of the Hermite functions

While the operator is defined by

Singular events are generally characterized by intensity, mean and standard deviation. However, these characteristics can be identical for different, very different events so that they are not sufficient for characterization. Therefore, we determined the so-called "higher statistical moments " as comparison figures. However, these are very sensitive to noise and drift of the zero line and therefore only of limited use. Expanding distribution into Hermite functions, the coefficients are very stable, as the functions live only in the central region and thus further dampen external data suitable.

The development of an event according to a function representing the Hermite functions has a certain similarity to the wavelet transformation.