Monoids of natural numbers in the numerical-theoretical method in the approximate analysis

For every monoid $M$ of natural numbers defined a new class of periodic functions $M_s^\alpha$, which is a subclass of a known class of periodic functions Korobov $E_s^\alpha$. With respect to the norm $\|f(\vec{x})\|_{E_s^\alpha}$, the class $M_s^\alpha$ is an inseparable Banach subspace of class $E_s^\alpha$.
It is established that the class $M_s^\alpha$ is closed with respect to the action of the Fredholm integral operator and the Fredholm integral equation of the second kind is solvable on this class.
In this paper we obtain estimates of the image norm of the integral operator, which contain the kernel norm and the $s$-th degree of the Zeta function of the monoid $M$. Estimates are obtained for the parameter $\lambda$, in which the integral operator $A_{\lambda,f}$ is a compression. The theorem on the representation of the unique solution of Fredholm integral equation of the second kind in the form of Neumann series is proved.
The paper deals with the problems of solving the partial differential equation with the differential operator $Q\left(\frac{\partial }{\partial x_1},\ldots,\frac{\partial }{\partial x_s}\right)$ in the space $M^\alpha_{s}$, which depends on the arithmetic properties of the spectrum of this operator.
A paradoxical fact is found that for a monoid $M_{q,1}$ of numbers comparable to 1 modulo $q$, a quadrature formula with a parallelepiped grid for an admissible set of coefficients modulo $q$ is exact on the class $M_{q,1,s}^\alpha$. Moreover, this statement remains true for the class $M_{q,a,s}^\alpha$ with $1<a<q$ when $q$ is a Prime number. Since the functions of class $M_{q,a,s}^\alpha$ with $1<a<q$ do not have a zero Fourier coefficient $C(\vec{0})$, then for a simple $q$ the sum of the function values at the nodes of the corresponding parallelepipedal grid will be zero.