Leskevych T. Spline approximation of multivariate functions.

The object is the multivariate functions defined on multidimentional parallelepiped with parallel to coordinate axes edges. The aim is finding the exact value of error of approximation some classes of multivariate functions by linear, multilinear and harmonic splines, also the construction of asymptotically optimal adaptive methods of approximation particular functions by different types of splines. Methods are the general methods of solving of the approximation theory extremal problems, general facts or the functional analysis, the theory of functions and the mathematical physics, also adaptive approximation of particular functions methods. In uniform norm the exact value of error of approximation by linear and multilinear splines classes of multivariate functions with given convex upwards majorant by specified in a certain way modulus of continuity is determining. In certain cases the exact value of the integral norm error of approximation by harmonic splines class of multivariate functions, given by bordering the integral norm of Laplace operator action is determining. The sharp asymptotic of the optimal integral norm error of adaptive approximation by harmonic splines twice continuously differentiable functions by harmonic splines is providing. The sharp asymptotic of the optimal integral norm for adaptive approximation smooth functions by splines, which specifed by given linear continuous projection operator is providing. Scope - the approximation theory, the functional analysis, the mathematical physics, the learning process.