Hermite–Gauss wavelets: synthesis of discrete forms and investigation of properties

Considered new results of studies of eigenvectors and vector functions of discrete and continuous Fourier transforms. It is known that such eigenvectors are products of the Gauss function on Hermite polynomials, a name is proposed for the functions obtained on the basis of this product: Hermite-Gauss wavelets. In the paper studies on the base of mathematical analysis methods of continuous functions and numerical methods, the properties and methods of synthesis of eigenvectors and vector functions of discrete and continuous Fourier transforms are investigated. Expressions for calculating the scale parameter and the normalizing factor for discrete forms of Hermite-Gauss wavelets are obtained. The studies performed to prompt that the scale parameter of the discrete form of Hermite-Gauss wavelets depends on the number of samples, and the norm depends on the number of samples and the number of the wavelet. The form of the Fourier transform matrices is obtained which has good conditionality when calculating eigenvectors in the form of Hermite-Gauss wavelets. Hermite-Gauss wavelets form a basis, and therefore can be used in tasks of signal decomposition and synthesis. For choosing a mother wavelet for decomposition and synthesis, firstly one should be guided by the features and properties of the shapes formed by it. For some signals, Morlaix or Daubechy wavelets can give compact decomposition, for others, Hare wavelets, and there are also signals for which Hermite-Gauss wavelets are most effective for spectral decomposition.