## 3 Statement of the Problem  
A digital convolution theorem of the form [40] is applicable to each of the partial convolutions (12).  
It consists of the following:  
Sequences of the following form a complete basis [40].  
\[\overset{\rightarrow}{w}_{k}=\Big{(}\theta^{k},\theta^{2k},\theta^{3k}, \ldots,\theta^{(p-2)k},1\Big{)}, \tag{13}\]  
where \(\theta\) is a primitive element whose powers up to and including the \((n-1)\)th are given by all nonzero elements of the used Galois field \(GF(p)\), \(k=1,\ldots,p-1\). The last element in this sequence is always equal to 1, since any nonzero element of the Galois field satisfies the equation  
\[x^{p-1}-1=0 \tag{14}\]  
where \(p\) is the number of elements of the field, including the zeroth one.  
Sequences (13) can also be considered as piecewise constant functions taking values in the field \(GF(p)\). They are direct \"digital\" analogs of harmonic functions; in particular, they form a complete basis and are orthogonal in the following sense [40]  
\[\sum_{j=1}^{j=p-1}w_{k_{1}}^{(j)}w_{k_{2}}^{(j)}=\Big{(}\begin{matrix}\"n\",k_{1 }+k_{2}\equiv 0(mod\ p-1)\\ 0,k_{1}+k_{2}\
ot\equiv 0(mod\ p-1)\end{matrix} \tag{15}\]  
Formula (15) contains the notation \"\(n\)\" in quotation marks. This notation implies that if the condition \(k_{1}+k_{2}\equiv 0(mod\ p-1)\) is met, all terms under the sum sign on the left side of this formula turn into 1. However, the sum of n units, generally speaking, in different Galois fields (even if the number of such terms remains the same) can take on different values.  
The condition on the right side of Formula (15) is an analog of the following relation for harmonic functions  
\[\int_{0}^{2\pi}\exp\left(-i\frac{2\pi mt}{T_{0}}\right)\exp\left(i\frac{2\pi nt }{T_{0}}\right)\sim\delta_{m,n} \tag{16}\]  
That is, the fulfillment of the condition \(k_{1}+k_{2}\equiv 0(mod\ p-1)\) means that the sequences \(w_{k_{2}}^{(j)}\) and \(w_{k_{1}}^{(j)}\) are conjugate to each other.  
For a sequence conjugate to a sequence \(\overset{\rightarrow}{w}_{j}\), the notation \(\overset{\rightarrow}{w}_{j}^{T}\) will be used below.  
Relation (16) allows us to immediately move on to the spectral representation of the signal in the form  
\[\overset{\rightarrow}{u}=\sum_{j=1}^{j=p-1}z_{j}\overset{\rightarrow}{w}_{j} \tag{17}\]  
where the spectral components (Fourier-Galois spectrum) can be calculated as  
\[z_{j}=q(\overset{\rightarrow}{u},\overset{\rightarrow}{w}_{j}^{T}) \tag{18}\]  
where \(q\) is a factor determined from the condition \(q(\overset{\rightarrow}{w}_{j},\overset{\rightarrow}{w}_{j}^{T})=1\), which is different for different Galois fields.  
It is essential that the spectrum defined by Formula (18) describes the piecewise constant function (or sequence) \(\overset{\rightarrow}{u}\) with exhaustive completeness, provided that this function is periodic, and its period contains exactly \(p-1\) cycles.  
The digital convolution theorem is expressed by the formula [40]  
\[\left(\overset{\rightarrow}{U}_{out,m},\overset{\rightarrow}{w}_{j}^{T} \right)\sim\left(\overset{\rightarrow}{K}_{m},\overset{\rightarrow}{w}_{j} ^{T}\right)\left(\overset{\rightarrow}{U}_{m,inv},\overset{\rightarrow}{w} _{j}^{T}\right) \tag{19}\]  
It can be seen that it is a complete analog of the classical convolution theorem.  
However, there is a significant nuance. Sequences of (13), as emphasized above, contain exactly \(p-1\) elements. Accordingly, for different numbers \(p_{i}\), which are factors of the number \(P\), the number of cycles on which the orthogonal basis is determined will also be different. This does not allow for applying the convolution theorem in the form (19) to Formula (10) directly.  
The problem solved in this paper can be formulated as follows.  
Find such orthogonal bases that will allow applying the digital convolution theorem to Formula (10), thereby determining the transfer function for digital convolution, represented through a set of partial convolutions.  
This problem is solved using the method of algebraic extensions.