## 1 Introduction
In addition to the Walsh basis, there are other orthogonal bases that allow one to move to the spectra of digital functions. One of them is the Haar basis [44], which consists of piecewise constant functions that divide segments into two parts. The Haar basis is used for data compression and the analysis of signals with abrupt changes [45; 46]. Another important example is the discrete cosine transformation, which is often used in image processing such as JPEG compression [47]. It is effective for computing convolution in problems involving low-frequency components of an image. The next step in this direction is the transition to bases constructed using the Residue Number System (RNS). It is appropriate to note that RNSs are currently increasingly used, including for fast arithmetic [48; 49]. The RNS has a number of significant advantages that make it attractive for use in computing systems, especially in parallel processors and cryptography. The first advantage of RNS is that it breaks long integers into several smaller numbers that can be processed independently of each other. This property allows arithmetic operations to be performed in parallel, which significantly speeds up computations, especially in systems where high processing speed is important, such as digital signal processing [50; 51] or cryptographic systems [52; 53]. The second advantage of RNS is that it reduces the complexity of arithmetic operations. Unlike a traditional number system, where addition, subtraction, and multiplication require carries, in a remainder system, modulo operations are performed for each separately, eliminating the need for carries. This results in simpler and faster operations, especially for large numbers. RNS also makes it easy to implement complex operations such as modulo inverses, making it particularly useful for cryptographic applications such as RSA encryption [54]. Finally, RNS is highly error-tolerant and secure. Because each number is represented as a set of remainders over different moduli, single errors in one of the remainders do not always cause serious failures in the system. This property can be used to design systems with increased reliability [55], as well as to build schemes that are resistant to side-channel attacks in cryptography. Furthermore, the digital convolution theorem applies to functions that take values in Galois fields. It is worth emphasizing that a function that takes values in any finite algebraic system (Galois field, finite algebraic ring, etc.) can serve as a model of any digital signal that changes in a finite range of amplitudes. It is only important that the number of signal levels does not exceed the number of elements in the given system. Traditionally, signal models are functions that take values on a set of real or complex numbers, but this is nothing more than a matter of agreement and convenience [40]. When using a function that takes values in some finite algebraic ring containing zero divisors, the convolution operation is decomposed into a set of convolutions that can be called partial, and which are calculated in terms of the corresponding Galois fields [56]. This work is the next step in relation to the work [56]. Namely, the goal of this work is to find such sets of basis functions that would allow partial convolutions (in calculations in algebraic rings) to be reduced to operations on the Fourier-Galois spectra of their individual components. Thus, this work allows us to find an analog of transfer functions for any systems that are described by partial convolutions.