Investigation of the properties of the Fourier transform of Hermite-Gauss wavelets and application of the results obtained in radio electronics problems

Hermite-Gauss wavelets are obtained as a result of the product of Hermite polynomials by the Gauss function. Hermite-Gauss wavelets form a basis in the space of real numbers and therefore allow decomposition of functions satisfying the Dirichlet condition. The set of orthonormal forms of Hermite-Gauss wavelets are eigenfunctions of the Fourier transform, such a feature determines the use of wavelets in a wide range of theoretical and practical engineering problems. An analysis of the literature shows that insufficient attention has been paid to the Fourier transform of various forms of Hermite-Gauss wavelets, their mutual expression through the Fourier transform and connections with other functions having the form of a product of polynomials by a Gauss function. Therefore, according to the authors, the research is relevant. The subject of research is to increase the efficiency of the Fourier transform of some types of polynomials based on the Hermite-Gauss wavelet decomposition. In the course of the research, five interrelated theorems have been formulated and proved, forming the basis of the work. Two types of decomposition are used to formulate and prove theorems. The first decomposition method is based on a generalized Fourier transform based on the Hermite-Gauss wavelet basis. The second method of decomposition of polynomials is based on sequential division with remainder. The main result of the work is the proposed mathematical apparatus in the form of formulated and proven theorems applicable to the Fourier transform of arbitrary polynomials of one variable multiplied by the Gauss function, and the transformation is performed analytically and without using the Fourier integral. According to the authors, the obtained result contributes to the development of the theory and practice of signal processing in the field of radio electronics, optoelectronics, electrical engineering and the theory of automatic control. In the final part of the paper, some examples of the use of the developed theory are presented and a method for synthesizing the responses of systems with a fractional-rational transfer function and a method for synthesizing radio signals based on Hermite-Gauss wavelets is proposed.