## 1 Introduction  
The application of the Fourier-Galois transform covers a wide range of scientific and technical disciplines [1, 2, 3], where it is used for signal analysis, solving differential equations, and optimization in various systems. This transform combines the algebraic methods of Galois theory with the harmonic analysis of the Fourier transform, which makes it especially useful in problems involving the use of symmetrical features and group structure. One of the important areas of application of the Fourier-Galois transform is coding theory and cryptography [4, 5]. It is used to analyze and create resistant ciphers based on Galois theory [6, 7]. For example, working with finite fields and cyclic groups, which are used in modern encryption systems such as RSA and ECC, can be simplified using this transform [8, 9]. The Fourier-Galois transform has found application in digital signal processing [10, 11], where it helps to solve problems of filtering, compression and signal analysis. An example is in the efficient recovery of signals from noisy data, where the symmetric properties of the transform in question help improve the filtering of high-frequency noise [12, 13]. In mathematical physics, the Fourier-Galois transform is used to solve linear differential equations, especially those involving operators with certain symmetric properties [14, 15]. An example is the solution of equations in electrodynamics or quantum mechanics, where the symmetry of the system allows for simpler calculations [16]. In number theory, the Fourier-Galois transform is used to study the properties of numbers and to solve problems related to the distribution of prime numbers [17]. For example, it is used in the analysis of algebraic structures of number fields [18] and in solving problems related to the distribution of residues by modules [19]. In the field of computational mathematics, the Fourier-Galois transform is used to optimize FFT algorithms in systems with finite fields [20]. This is relevant in problems related to big data processing and modeling of complex systems, such as meteorological models or control systems. In quantum information science, the Fourier-Galois transform is used to develop quantum algorithms [21]. For example, it is used in factorization and period search problems [22], which is important for quantum cryptography and quantum search algorithms. In telecommunications, the Fourier-Galois transform helps improve modulation and signal coding systems. An example is its application in OFDM (Orthogonal Frequency Division Multiplexing) [23; 24; 25], where it is used to improve noise immunity and increase data transmission speed. Thus, the Fourier-Galois transform finds application in a wide range of disciplines, from number theory and cryptography to quantum information science and digital signal processing, offering powerful tools for solving problems related to symmetry and algebraic structures. A number of other areas of application of the Fourier-Galois transform, as well as the Fourier transform, are associated with systems described by the convolution operation. Indeed, it is permissible to pose the question that the description of a system, reduced to the operation of digital convolution, can then be reduced to a description in terms of the transformation of the spectrum formed based on the corresponding orthogonal set of basis functions. There is an obvious analogy here. Linear systems of arbitrary nature, possessing the property of invariance with respect to the time shift operation, are described by the operation of classical (analog) convolution, which is used, for example, in Fourier optics [26]. One of the key applications of the convolution theorem is signal and image filtering [27]. In filtering, such as smoothing or removing noise from an image, convolution allows filters to be applied efficiently.