RECOVERY FUNCTION FOR SOBOLEV SPACES WITH DOMINATING MIXED DERIVATIVE

Linear approximation is the approximation of a function from a certain class by elements of a fixed finitedimensional subspace of that same class. For instance, for one-dimensional periodic functions, such elements are trigonometric polynomials. In the multidimensional case of interest, for functions periodic in each variable, this subspace is the set of trigonometric polynomials with a spectrum from a step hyperbolic cross. However, the question of selecting coefficients for these polynomials arises. This paper presents an apparatus for recovering functions from Sobolev spaces with a dominating mixed derivative based on given points, and establishes error estimates for the recovery. The method is based on constructing a recovery function in the form of a polynomial with a spectrum from a step hyperbolic cross, where the coefficients are calculated using the given points. The approximation error is of the order of the orthowidth, which is an optimal result for such polynomials. The proposed method is exact for polynomials with a spectrum from a step hyperbolic cross. Additionally, a functional that recovers the Fourier coefficients for functions from the indicated spaces is derived. Due to the explicit expression of the recovery function, the obtained formula can be used to solve applied problems.