The algorithm for constructing a differential operator of 2nd order and finding a fundamental solution

Abstract

The main area of application of various spaces of generalized functions lies in the theory of differential equations and in the theory of quadrature and cubature formulas. Therefore, it becomes necessary to study spaces of generalized functions, one way or another connected with various areas in Rn. The theory of differential equations in the space of generalized functions differs from the theory of these equations in the space of ordinary functions. Deriving these equations and finding their solutions are important in applications. The discrete analogue Dm,n [β] of the polyharmonic operator Δm=(∂2∂x21+∂2∂x22+⋯+∂2∂xn2)m plays an important role in constructing optimal quadrature and cubature formulas in the spaces W2(m)(Rn) and L2(m)(Rn). In the first in the space L2(m)(Rn) by constructing and studying the properties of the inversion of the convolution operator with the function Gm,n [β], where Gm,n (x) - is the fundamental solution of the polyharmonic operator, i.e. properties of such a function of discrete argument Dm,n [β], which satisfies the equality Dm,n[β]*Gm,n[β] = δ[β], where δ [β] is equal to one at β = 0, equal to zero at β ≠ = 0, S.L. Sobolev [1]. The theory of quadrature and cubature formulas was developed in periodic spaces by S.L. Sobolev W˜2(m)(Rn) and L˜2(m)(Rn), and for non-periodic space S.L. Sobolev W2m results are comparatively small. The main goal is that until 2 now in our 2 studies to find the discrete analogue of Dm [β], the fundamental solution Vm(x) was used, but the explicit form of the differential operator, which was Dm [β], was not known. It can be especially noted that the problem of constructing a differential operator and improving the fundamental solution of which, a discrete analogue of this operator Dm [β] is used to find the optimal coefficients of quadrature, cubature and interpolation formulas for a non-periodic space of S.L. Sobolev W2m is actual.