## 4 Results
### Example  
The convolution of the model function shown in Figure 1, curve 1 with the model convolution kernel, Figure 2, is used. The function, curve 1, takes discrete integer values in the range from 0 to 6 on discrete values of the current integer variable \(k\). These values are shown by blue dots (similarly for the curve in Figure 2). The type of model function was chosen based on the convenience of demonstrating the results obtained. Curve 2 in Figure 1 is the result of calculating the convolution using the rules of ordinary arithmetic. It also takes values on discrete values of the current integer variable \(k\) (red dots), but the range of its variation increases. Let us show that the result of the convolution operation, shown in Figure 1, curve 2, can also be obtained by using the digital convolution theorem for a set of partial convolutions. The table in supplemental materials File S1 presents functions corresponding to 24-cycle bases in the fields \(GF\big{(}5^{2}\big{)}\) and \(GF\big{(}7^{2}\big{)}\), respectively. More precisely, functions with a 24-term period were selected from the basis functions corresponding to the field \(GF\big{(}7^{2}\big{)}\). Figure 1: Model function \(f\), curve 1, and the result of the convolution \(f*K\) of this function with the model convolution kernel K, Figure 2 (curve 2). Figure 2: Model convolution kernel \(K\). These bases were used to calculate the real and imaginary parts of the Fourier-Galois spectra of the functions  
\[f_{1}\equiv f(mod\ 5);\ f_{2}\equiv f(mod\ 7);\ K_{1}\equiv K(mod\ 5);\ K_{2} \equiv K(mod\ 7) \tag{42}\]  
The calculations were carried out in a Python program; the program code is presented in supplemental materials File S2. In the diagrams in Figures 3 and 4, the real and imaginary components of the Fourier-Galois spectra of the functions \(f_{1}\) and \(f_{2}\) are shown separately as an example. Figure 4: Imaginary parts of the Fourier–Galois spectra \(ImF[f_{1}]\) and \(ImF[f_{2}]\) of the functions \(f_{1}\) and \(f_{2}\), defined in the fields \(GF(5)\) and \(GF(7)\), respectively; blue color—field \(GF(5)\); red color—field \(GF(7)\). Figure 3: Real parts of the Fourier–Galois spectra \(ReF[f_{1}]\) and \(ReF[f_{2}]\) of the functions \(f_{1}\) and \(f_{2}\), defined in the fields \(GF(5)\) and \(GF(7)\), respectively; blue color—field \(GF(5)\); red color—field \(GF(7)\). The calculations were carried out using the formulas  
\[F_{1}(f_{1})=q_{1}{\sum_{j=1}^{24}}\theta^{\bar{k}j}f_{1}(j) \tag{43}\]  
where \(\theta\) is a primitive element of the field \(GF\big{(}5^{2}\big{)}\), and  
\[F_{2}(f_{2})=q_{2}{\sum_{j=1}^{24}}\xi^{\bar{k}j}f_{1}(j) \tag{44}\]  
where \(\bar{\xi}\) is the square of the primitive element of the field \(GF\big{(}7^{2}\big{)}\), the bar over the symbol \(\bar{k}\) means that the number (degree) corresponding to the conjugate sequence is used in the calculations. Similar formulas were used to calculate the components of the spectrum of the kernels \(K_{1}\) and \(K_{2}\). At the next step, the products \(F_{1}(K_{1})F_{1}(f_{1})\) and \(F_{2}(K_{2})F_{2}(f_{2})\) were calculated. We emphasize that these products were calculated in the sense of the fields \(GF\big{(}5^{2}\big{)}\) and \(GF\big{(}7^{2}\big{)}\), respectively; i.e., in one case, Formula (23) was used, and in the other, (24). The corresponding diagrams are shown in Figures 5 and 6. The diagrams also show the real and imaginary components separately. Then, the reverse transition to partial convolutions \(K_{1}*f_{1}\) and \(K_{2}*f_{2}\) was carried out according to the following formulas:  
\[K_{1}*f_{1}=q_{1}^{-1}{\sum_{j=1}^{24}}\theta^{\bar{k}j}z_{1}(j) \tag{45}\]  
\[K_{2}*f_{2}=q_{2}^{-1}{\sum_{j=1}^{24}}\xi^{\bar{k}j}z_{2}(j) \tag{46}\]  
where \(z_{1}(j)\) is the \(j\)-th component of the spectrum \(F_{1}(K_{1})F_{1}(f_{1})\), and \(z_{2}(j)\) is the \(j\)-th component of the spectrum \(F_{2}(K_{2})F_{2}(f_{2})\). The results of partial convolution calculations are shown in Figures 7 and 8.